Langmuir 1996, 12, 4497-4500
4497
Soft Adhesives† P. G. de Gennes Colle` ge de France, 11 place Marcelin Berthelot, 75231 Paris Cedex 05, France Received October 17, 1995. In Final Form: February 1, 1996X We attempt a simple theoretical picture for the strong adhesion properties of weakly cross linked rubbers. These ideas are then extended to zero crosslinking -ie the problem of tack for a linear polymer.
I. General Aims Most fracture processes inside a glue are dominated by dissipation near the fracture tip; this occurs for instance with polymers which craze, where most of the separative energy (per unit area) G is due to a precursor craze.1,2 However, there is another possibility, which is important for very soft materials, in practice, for weakly cured rubbers; inside the network, we have many chains which are free and many chains which are tied at one end only. In cases like this, the low frequency modulus µ0 (related to the network) is small. But the high frequency modulus µ∞ (which contains the effects of the entangled free chains and of the dangling ends) is high. Typically, we can achieve:
λ ) µ∞/µ0 ∼ 100
(1)
It was shown by Gent and Petrich3 that poorly cross-linked systems of this type have a very anomalous curve G(V) (adhesive energy/velocity) as shown on Figure 1. The anomaly disappears upon further “curing” (when the network is more cross-linked and µ0 raises to become comparable to µ∞). Here, we are concerned only with what Gent and Petrich call the first transition: from a soft rubber (modulus µ0 due to a few cross links) to a hard rubber (modulus µ∞ dominated by entanglements). Both states are assumed to be fluid. The second transition (from hard rubber to a glass) will not be discussed here. We also restrict our attention to bulk fracture and do not consider any detailed interfacial features, such as transitions between cohesive and adhesive modes. In section II, we construct a simple picture of viscoelastic fracture in a weakly cross-linked rubber.4 The major idealization which we impose is to assume that mechanical relaxation inside the network is described by a single relaxation time τ. Of course, this is very crude; the dangling ends have a wide distribution of length, etc. But the one time approximation allows an understanding of what happens in space, far behind the fracture tip. In section III, we proceed to the more general case where there are many relaxation times. Various relations between the fracture energy G(V) and the complex modulus µ(ω) have been proposed, often assuming that ω and V are related via a fixed length l (ω ) V/l). We arrive here at a formula which is quite different, where G(V) is an integral over many frequencies (eq 21). This equation † Presented at the Workshop on Physical and Chemical Mechanisms in Tribology, held at Bar Harbor, ME, August 27 to September 1, 1995. X Abstract published in Advance ACS Abstracts, Sept. 15, 1996.
(1) Brown, H. Macromolecules 1991, 24, 2752. (2) de Gennes, P. G. Europhysics Lett. 1991, 15, 191. (3) Gent, A.; Petrich, R.; Proc. R. Soc. London 1969, A310, 433. (4) de Gennes, P. G. C. R. Hebd. Seances Acad. Sci. 1988, 307, 1949.
S0743-7463(95)00886-9 CCC: $12.00
Figure 1. Peel force P versus peel rate R for an un-crosslinked butadiene-styrene rubber adhering to a PET polyester film after ref 3. The symbols C and I denote cohesive failure and interfacial failure, respectively.
will be derived here but will not be discussed in detail; the subject is so vast that a separate publication will be required. Section IV discusses the fracture energy G(V) for uncross-linked melts, where µ0 ) 0. This, as we shall see, can lead to very large fracture energies and is clearly an important component of tack. Of course, our discussion is still very far from real materials, such as pressure sensitive adhesives; in these systems, the second transition of Gent and Petrich (to a glassy state) plays a major role. But, once again, and independently of the detailed process involved in the transition, we believe that we can give a simple insight. In all this text, our approach is mainly based on scaling laws; we gain in simplicity, but we ignore all numerical coefficients. II. Bulk Fracture: The “Viscoelastic Trumpet” Let us start with some remarks on the viscoelastic features. The simplest formula for the complex modulus µ(ω) as a function of frequency (ω) is the following:
iωτ µ(ω) ) µ0 + (µ∞ - µ0) 1 + iωτ
(2)
Usually, with one relaxation time τ, we think that we have to cope with two regimes ωτ > 1 and ωτ < 1. Here, in fact, when λ ) µ∞/µ0 . 1, we have three regimes: (i) At very low ωµ ) µ0, we expect a soft solid. (ii) When 1 > ωτ > λ-1, we can approximate
µ(ω) ∼ (µ∞ - µ0)iωτ ) iωη
(3)
The modulus is purely imaginary; we are dealing with a © 1996 American Chemical Society
4498 Langmuir, Vol. 12, No. 19, 1996
de Gennes
of the density F and the local velocity v reduce to
F
Dv ) -∇σ = 0 Dt
(7)
where we set Dv/Dt ) 0, because we are dealing with velocities much smaller than the sound velocity; inertial terms are negligible. Thus the equations are simply ∇σ ) 0 and are the same for any viscoelastic medium. The stress components must also satisfy certain compatibility conditions (because they derive from a displacement field). But these geometric conditions, again, are the same for any viscoelastic medium. (iii) In an elastic medium of modulus µ, with a stress field described by eq 5, the displacement field u is simply related to the stress by the scaling law
Figure 2. Viscoelastic “trumpet”, fracture profile when a single relaxation time is present.
liquid of viscosity
η ) (µ∞ - µ0) τ ∼ µ∞τ
(4)
(iii) At high frequencies (ωτ > 1) we recover a strong solid (µ ) µ∞). What are the consequences of this on a moving fracture? When the fracture velocity V is not too small, so that Vτ is larger than the size of the fracture tip, we can think in terms of a continuum, and we find three regions in our rubber (Figure 2). Near the fracture tip, we are discussing small spatial scales, or short times, and we have a strong solid. At intermediate distances, Vτ < r < λVτ, we have a liquid, giving a large dissipation. At higher distances r > λVτ, we have a soft solid. You might think that the effects in the liquid zone are not very important, because the stresses here are relatively smallswe are far from the fracture tip. On the other hand, because r is large, we have a huge volume of fluidlike behavior and the overall dissipation is increased. It will turn out that this volume factor dominates. Although we are talking about a complex viscoelastic medium, the scaling law for the stress as a function of distance is still simple, and equivalent to what we have in a simple elastic medium5
σ(r) ∼ K0/r1/2
(5)
σ ) µ∇u
(8)
u ) (K0/µ)r1/2
(9)
and this imposes
If we now take the scaling structure of the product (σu) along the fracture (where u now measures the opening of the fracture), we find that it is independent of distance
σu ) K02/µ ) G0
(10)
where G0 is the corresponding adhesion energy. For our more complex problem, we shall again find the separation energy G from the product (σu), calculated at larger r (in the soft medium); measuring at large distances we incorporate all dissipation effects. Let us now return to Figure 2 and find out the overall shape of the fracture profile u(x). In the strong solid region, we have a classical parabolic shape
u ) K0/µ∞x1/2
(11)
and σu ) G0. These equations hold provided that x is larger than the (microscopic) size l of the zone, where nonlinear processes are dominant. (We expect l j 100 Å for our materials). On the other hand, the strong solid region is restricted to x < Vτ. Thus, our discussion holds, provided that l , Vτ. In the liquid zone, σu is not constant. The scaling law relating u to σ is based on a viscous stress
d2u d du )ηV 2 σ)η dx dt dx
( )
(12)
where the factor K0 is associated with the adhesion energy G0 due to local processes near the tip; again ignoring all coefficients, we have the standard relation from fracture mechanics
and with σ ∼ x-1/2 (eq 5) this gives u ∼ x3/2. Thus the product σu increases linearly with x
G0 = K02/µ∞
When we reach the soft solid region (x ) λVτ), we find
(6)
Three remarks concerning the stress distribution: (i) Equation 5 is simply a scaling law, ignoring all specific details for different components of the stress; for instance, on the fracture tip, the normal stress component σzz must vanish identicallysbut the other components still follow the scaling form (5). (ii) Why does this simple form remain valid in a viscoelastic medium? The equations of motion, in terms (5) See, for instance: Lake, G.; Thomas, A. In Engineering with rubber; Gent, A., Ed.; Hanser: Munich, 1992.
σu ) x/VτG0
σu ) λG0
(13)
(14)
and this gives us the overall adhesion energy G
G/G0 ) λ ) µ∞/µ0
(15)
A more rigorous derivation of eq 15 has been given by Hui.5 The result deserves a number of comments. Note first that the viscoelastic corrections give a multiplicative effect; they do not add up a constant term to G0. Second, we see that the ratio G/G0 can be very large if the material
Soft Adhesives
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complex modulus γ ) σ/µ(ω), where ω is the local sweeping frequency
ω ) V/r
(19)
Thus, γ˘ ) iωσ/µ(ω). We see that our scaling estimates all depend only on r; thus we may replace the integral
∫ dx dy f ∫ 2π r dr f const ∫ r dr Collecting all these results, we arrive at
V G(V) ∝ Figure 3. Truncation of the viscoelastic trumpet in a thin slice of adhesive material.
is very poorly cross-linked (µ0 f 0). This also explains why the enhancement in G disappears upon further curing (µ0 f µ∞). All this discussion held for a thick glue in the cohesive mode (bulk fracture). But very often the glue is in the form of a thin slab (thickness W) as shown in Figure 3. The opening process is then cut off at distances x ∼ W. Thus when λVτ > W, the dissipation zone is restricted in size and we have
G ) G0W/Vτ
2
) K02V
G(V) = µ∞ G0
∫0ω
µ′(ω) ) µ′′(ω) = ω
R
(17)
where R is a certain critical exponent. In this case, a simple extension of the above argument (with a distribution of relaxation times which is fixed by R) does show that G ∼ VR. This could appear as a rather exceptional situation; however, since this region near the connectivity threshold is the region of long τ and large G, many practical materials may be purposely chosen to lie in this region. (2) General Relation between G(V) and µ(ω). Our starting point here is the dissipation TS˙ (per unit length of the fracture line).
TS˙ ≡ V G(V) )
∫
(20)
µ′′(ω)
max
2
2 1/2
[µ′(ω) + µ′′(ω) ]
dω ω
(21)
For bulk fracture, the cut off frequency ωmax is
ωmax ) V/l
(22)
where l is the size of the nonlinear region near the fracture tip. (Typically l ∼ 10 to 100 Å). Two remarks concerning these formulas are as follows: (a) When there is a single relaxation time τ, the domain of important frequencies in eq 21 is
III. Other Types of Relaxation (1) Sol/Gel Transitions. What happens when we have not one single relaxation time, τ, but a broad distribution of relaxation times, giving a complex form to the dissipative modulus µ(ω)? A natural attempt, proposed long ago, assumes that G(V) is proportional to the modulus µ′′(ω) measured at a frequency ω ) V/l, where l is some characteristic length of the rubber network. However, this is not correct in general, as pointed out in particular by Barber et al. on simple examples.6 There is one case where it holds, namely, when the network is just at its gelation point, at the transition between isolated clusters and one infinite cluster plus finite molecules. At this point, the elastic modulus µ(ω) is given by a power law
1 Im( ∫ dω ω µ(ω))
where Im(f) denotes the imaginary part of f. Remembering that the bare fracture energy is G0 ) K02/µ∞, we end up with
(16)
In this regime, G(V) is a decreasing function; the pulling force drops if the velocity increases. This often generates mechanical instabilities in the fracture process. We believe that this instability is the source of the peak in the experimental G(V) curve observed by Gent and Petrich.
2
σ σ iω = ∫ r dr iω ∫ r dr µ(ω) µ(ω)
1 1
