ADHESION OF POLYMERS AT THE INTERFACE WITH SOLIDS
2.5 THE STRENGTH OF ADHESION JOINTS
The strength of adhesion joints determines the main properties of PCM. When evaluating the adhesion strength one should take into account many factors, in-cluding the processes of crack development, distribution of stresses in the sys-tem, the existence of inner stresses, etc. These questions are thoroughly discussed elsewhere.4In our analysis of the adhesion strength we give more em-phasis to the relationship between thermodynamic evaluation of adhesion and adhesion joint strength, which have no direct correlation. One has to distinguish between two definitions: thermodynamic work of adhesion as an equilibrium value which does not depend on the conditions of test, application of adhesive to a surface, roughness of the surface, etc. This value, as as has been shown, de-pends only on the thermodynamic characteristics of adhesive and adherent. The adhesion strength, similar to the strength of any solid, is a kinetic value depend-ing on the conditions of failure, defects in the material structure, weak boundary layers, etc.
It is known that the theoretical strength of a solid does not correspond to the real strength. The first is determined by the molecular forces, whereas the second depends on the structure of the material. The deformation of a solid is a non-equilibrium process dependent on energy dissipation. The lack of correla-tion between thermodynamic work of adhesion and strength of adhesion joints is a direct consequence of the non-equilibrium failure. It may be predicted that the correspondence between these two values may only be reached if the strength was determined in the thermodynamically equilibrium conditions, i.e., at deformation with infinitely low rate.
The thermodynamic work of adhesion is an invariant value determined only by the nature of interacting surfaces, whereas the strength of a joint de-pends on many factors. From this point of view, only thermodynamic evaluation may have physical meaning. However, thermodynamics does not take into ac-count nonelastic deformations, defects at the phase boundary, the inner stresses in the adhesion joints which arise during the joint formation, concentration of
stresses connected with the difference in modulus of the adhesive and adherent, and many other factors.
According to Bikerman,8,47the failure of adhesion contact never has a pure adhesion character. Depending on the conditions, various types of failure may be observed:
• adhesive in which a full separation of adhesive from substrate occurs
• cohesive failure where the destruction of adhesive contact proceeds either in adhesive or in the solid phase
• mixed failure where partial destruction of contact takes place in both phases.
Only adhesive failure should characterize adhesion, even though it is a rather rare case.
It is evident that the theory of failure of adhesion joints should be based on the general principles of solid destruction. However, the transfer of the classical concept of Griffith’s theory to two-phase systems is very complex.52The difficul-ties are related to determination of two main parameters in the equation for crit-ical stress of fracture:
σf = K EGc l
12
[2.65]
where E is elastic modulus, Gcis the work of deformation of a crack, and l is its length. The work of crack deformation is stored as a free surface energy of a solid,γs, and energy,ψ, needed for other processes which accompany the crack propagation:
Gc= 2γ2+ψ [2.66]
For a two-phase system, it is necessary to introduce two new parameters,δ1 andδ2, which characterize the thickness of two border layers in adhesive and ad-herent where the process of the energy dissipation proceeds. Good52proposed the form of functional dependencies for some cases of fracture at various values and signs of∆G and∆E, which allow one to draw some conclusions regarding ad-hesion in two-phase systems. If∆G and∆E have the same sign, and if the forces
acting along the interface are sufficiently high, the most probable place of crack origin is in one of the contacting phases. If∆G and∆E have the opposite signs and∆logG >∆logE, and interface forces are sufficiently high, the fracture pro-ceeds near the surface in the phase with lower G. At∆logG< ∆logE the fracture takes place near the surface in the phase with either lower G or E. This case in-cludes the results which can be explained from the point of view of weak bound-ary layers. In the case when the interface forces are weak, as a rule, true adhesion fracture proceeds. The fracture is determined by the nature of forces acting at the interface. In strong interfacial interactions, the fracture begins in the phase with lower G, whereas in weak interactions, it begins at the phase bor-der. Gutowski53 proposed a very interesting model to describe the adhesion strength which takes into account the surface properties of materials involved.
The force of interaction F12between two materials can be estimated based on the energy function, U(r), relating the energy of interaction to the separation dis-tance, r:
F = dU (r )
12 dr12 12
12
[2.67]
The energy of interaction between two materials can be estimated from the Eq 2.2. The total thermodynamic work of adhesion and thus the interaction en-ergy comprises three main contributions: dispersive component, acid-base in-teraction term, and dipole-dipole inin-teraction term. In order to apply the thermodynamic relationship to estimate the interaction forces, the surface en-ergy of materials 1 and 2 should be described in terms of their bulk properties, including the equilibrium separation on distances r1and r2, pertinent to these materials. This becomes possible due to the fact that the surface energy of any material is given by:
γ ω
= h (2)π 32 r
o
2 o2 [2.68]
where h is the Planck constant,ωois the specific adsorption frequency, and r is the equilibrium separation distance. Considering the values of surface energies of materials 1 and 2 for the energy of interaction, we obtain:
U = h (2)
This equation describes the energy of interaction as a function of bulk properties of the matrix and filler and the relevant separation distance specific to these materials.
The resultant force of interaction (per unitary area), or adhesion between the matrix and reinforcement is given by the following expression:
F h (2)
Eq 2.71, expressing the interaction force as a function of surface energy of the filler and the ratio of surface energies of filler and matrix, is applicable to the systems adhering through physical interactions typically failing at the filler-matrix interface.
The value Gcin Eq. 2.65 may be presented as Gc= Wc+ Wp, where Wcis the energy of cohesion and Wpis the energy of viscoelastic losses. Taking this into ac-count, Gutowski proposed the following expression for the force of adhesion (stress) between the filler and a matrix:
F 3
( )
2
( ) (a,T, )
12 1
1 2 2
1 2 3 v
∝ −
Φ
γ γ γ γ γ ε
/ / & [2.72]
where &a is the rate of crack propagation, T is temperature,εis strain, andΦv is the viscoelastic loss function. Gutowski has shown that the total fracture energy Gc is always proportional to the intrinsic fracture energy G = Wc A12, which in turn depends upon the surface properties (surface energy) of the matrix and filler. It was also shown experimentally that adhesion in terms of strength of ad-hesive bonds increases up to some maximum value with the increase of WA12and then begins to decline along with any further increase in WA12 (Figure 2.2).
It is well known that when evaluating the peel adhesion strength, only a part of mechanical work is consumed for real fracture of the adhesion bonds, whereas energy balance is spent on various by-processes. The part of energy spent on the deformation of polymer film during peeling is of special importance.
Figure 2.2. The theoretical relationship between the relative strength of adhesion, F12/γ1, and thermodynamic work of adhesion, WA12
/γ1. [Adapted by permission from W. Gutowski in Controlled Interphases in Composite Materials, Ed. H. Ishida, Elsevier, New York, 1990, p.
This example indicates that it is necessary to know the individual processes leading to adhesion joint failure. These factors should be taken into account when determining the so-called quasi-equilibrium work of adhesion.54 The method is based on the analysis of the dependence of the peeling force on time at a given rate of peeling (Figure 2.3). Continuous increase of force (part AB) is ob-served, then force remains almost constant (part BC). The constant force corre-sponds to the peel stress at a given rate of detachment P. The work of adhesion of coating is determined from the average stress value at detachment relative to the unitary width of the film, and it depends on the film thickness and deforma-tion rate. However, if at point C the peeling is stopped, the stresses created in the film lead to a further detachment of the film from the substrate surface as a result of the relaxation of stresses created in the film.The peeling process pro-ceeds till the stresses in the film are not balanced by the adhesion forces (part CD). At this point, the peeling process stops
(point D). The value of the residual stress at the point D serves as a quasi-equilibrium value of adhesion strength, since this value does not de-pend on the rate of peeling nor film width. The mechanical work of adhesion bond failure and corresponding heat effects can be recorded us-ing a method of deformational calorimetry,55 which allows one to measure the heat and me-chanical energetic effects during deformation of an adhesion joint at various rates of deforma-tion. The method allows one to distinguish be-tween the energetic effects at the interface dependent on a failure of interfacial bonds and the deformation of an adhesive. The disadvan-tage of the two methods mentioned above is that they cannot be applied to the composites containing particulate filler. For such systems, other methods should be used.
Photoelectron X-ray spectroscopy is one of applicable methods. The method comprises
ele-Figure 2.3. Dependence of the peeling force, p, on time, τ, (see the text). [Adapted by permission from A. N.
Kuksin, L. M. Sergeeva, and Y. S. Lipatov, J. Adhes., 6, 275 (1974)]
mental analysis of thin surface layers. From the data on the intensity of spectral lines of polymer, Ip, and filler, If, it is possible to evaluate the fracture surface oc-cupied by filler particles. In cohesive failure of adhesion joints, Ip= 1 and If= 0; in adhesive failure, Ip= 0 and If= 1. For quantitative evaluation, there is a need to use a model capable of accounting for the shape of filler particles and distribu-tion of stresses at the filler particle surface. In the simplest case, when filler par-ticles are spherical in adhesion failure, the following calculations can be performed. Suppose that the tensile strength of a matrix exceeds the adhesion strength,σp >σafrom the interfaces into the bulk of a polymer matrix. The sur-face of the fracture is represented in Figure 2.4.
If the filler particles are placed in the cubic lattice having period, d, the fol-lowing expression is obtained:
γ π σ σ
= II = R [1 ( ) ] 2d
f fo
o2
a p 2
2
− /
[2.73]
From the equation, it follows that
[ ]
σa=σp 1 (2− γ π/ )(d / R )o2 12 [2.74]
If the volume fraction of a filler, isφ= 4 3(R d)π/ o / 3, we obtain:
Figure 2.4. Scheme of the fractured surface of composites containing particulate filler (for expla-nation see the text).
[ ]
σa=σp 1 (2− γ π π // )(4 3 )φ 2/3 1/ 2 [2.75]
The quantitative data obtained from these relationships can be considered only an approximation because the simple model assumes pure adhesive fail-ure. Also, there is no proof that all bonds in the cross-section are destroyed si-multaneously. The method allows evaluation of the relationship between adhesion and cohesion failure, which is its main advantage.
To establish the character of failure (adhesion or cohesion), microscopic methods and visual analysis of the fracture surface are used.53Also, methods based on the electron scattering (β-rays) are used because scattering depends on the surface composition.58If the sample is composed of, for example, metal and adhesive film, current arising whenβ-rays are scattered is connected to free metal surface. The fraction of adhesion failure, A, can be estimated from:
A (%) = (Cβ- Ca)/(Cm- Ca)100 [2.76]
Ca and Cm are counts ofβ-rays, respectively for the surface of adhesive and metal, and Cβis an averaged count for a great number of regions of fracture.
There exists a linear relationship between the breaking stress of the adhesive joint and the fraction of bonds destroyed at the interface. Using this technique, the profile of the stress distribution was evaluated,58which allows us to estab-lish a correlation between the breaking stress and the fraction of the surfaces with cohesion and adhesion failureθcandθa:
σf ≈K[θ σa ac +θ σc cp] [2.77]
where σac andσcpare the average stresses for adhesion and cohesion failure and K is proportionality coefficient. This equation may be presented in a more conve-nient form:
σf ≈θ σa( a −σp) +σp [2.78]
whereσaandσpare values proportional toσac andσcp. From the linear depend-ence ofσfandθa, the characteristics of adhesion and cohesion strength for mixed character of the failure of adhesion joints can be found by extrapolation. The use
of this approach allows one to estimate the characteristics of various layers of an adhesion joint (weak layer formed by impurities, polymer layer, oxide layer, etc.).
The above-mentioned methods are based on the analysis of the fracture surface, which may not be always convenient and possible. Also, analysis of frac-ture surface assumes a simultaneous break of bonds over the whole area of the adhesion contact, which is not the case with dispersed fillers. The strength of the adhesion contacts on the filler particle surface determines the mechanical stress in the interphase region at the binder-filler interface, at which the filler parti-cles begin separating from the binder resulting in formation discontinuities.59 To estimate this stress, various methods were proposed, including determina-tion of changes in the sample volume by ultrasound waves to detect discontinu-ities in the deformed composition.
The analysis of the change in the modulus of elasticity of the material as a result of a sample preloading is still another technique which can be used.60The modulus is determined twice: in the initial sample, and after a fraction of the ex-isting adhesion bonds in the material is broken. Such an approach ensures a lin-ear viscoelasic behavior of the binder and provides the possibility to analyze the interrelation between the modulus of elasticity and the volume concentration of separated filler particles, based on two known facts: increasing the filler concen-tration proportionally increases Young’s modulus of a composite if there is suffi-ciently strong adhesion bond between the binder and a filler (see Chapter 6);
conversely, Young’s modulus is decreased if the adhesion bond is absent or is not sufficiently strong. In the latter case, the binder is separated from the filler sur-face determination during elasticity modulus determination even at low stress.61,62After the adhesion contact in filled polymer is partially destroyed, the Young’s modulus of the composite decreases. The effect is well-known as Mullin’s effect, used for estimation of adhesion strength. The decrease in the Young’s modulus of a filled polymer after a preset mechanical action can be used as evidence that separation of binder from a fraction of filler particles has oc-curred. If change of the Young’s modulus for varying concentration of separated filler particles,φx, is determined, then the fraction of separated particles can also be determined from the Young’s modulus change. For this purpose, Zgaevsky62proposed the relation, derived theoretically:
E E = 2
3(1 )
f o
− φf [2.79]
where Eois the Young modulus of the binder, Efis the modulus of filled polymer, andφfis the concentration of the filler which not bonded to the filler. It should be noted that this relationship poorly describes the experimental results in the re-gion of low filler concentration. The experimental results can be much better ap-proximated by the empirical equation:60
E Ef = e
o
-3φf [2.80]
It can be expected that when some filler particles are separated from binder in the course of a mechanical action, the remaining filled polymer with unbroken adhesion bonds serves as a “binder”. Eq 2.80 can be applied to predict the properties of such a system. When a polymer filled with adhesively bonded particles is treated as a binder characterized by the modulus Ef, then, in the presence of such “binder”, the same relation should exist between the composite modulus Exand filler concentration,φx:
E Ex = e
f -3
φx [2.81]
The value of Ef can be determined experimentally in an initial sample;
hence equation 2.81 makes it possible, in principle, to calculate the concentra-tion of the separated-off filler.
The volume concentration of debonded particles or the fraction of the debonded filler,φo=φx/φfcan be determined by measuring the Young’s modulus of a filled elastomer before and after preloading that breaks adhesion contacts.
When studying the filler separation in the model samples, it was noted that not all particles debond simultaneously, and that the number of debonded particles depends not only on the magnitude of the disturbing stress, but also on the time
of duration of the disturbing load, s. It was found that an increase of s results in a decrease of the Ex/Efratio, i.e., in a growth ofφx.
Consequently,φxis a function of disturbing stress,σo dist. An analysis of the form of the function can be useful in calculating the strength of adhesion,σa. Possible dependences ofφxandσo distare shown in Figure 2.5. Curve 1 character-izes the case of filler separation when a certainσis reached, followed by cata-strophic failure. A characteristic point of the curve is that corresponding to the σo distvalue at which a sharp increase in the separation rate occurs. Proceeding from the assumption that the entire sample’s resistance force is concentrated solely at the cross-sectional area of all the filler particles in the sample cross-sec-tional plane, one can calculateσa from the formula:
σa=σo dist /2φf2/3 [2.82]
whereσo distis the stress corresponding to the beginning of the catastrophic sepa-ration of the filler. The dependence shown by curve 2 is observed as well. One of the causes of such a behavior may be a nonuniformity of the stress-strain state of binder interlayer; as a result, the filler debonding occurs at locations where the
Figure 2.5. Typical form of the dependence of debonded filler concentration on average stress in sample. See text for explanation of both curves. [Adapted by permission from Y. Lipatov, V.
Babich, and T. Todosijchuk, J. Adhes., 35, 187 (1991)]
stress has reached the level of the adhesion or cohesion strength. Withσo dist in-creasing, more such sites are created and an increasing number of filler parti-cles debond. To evaluate the adhesion strength of such a case, consider the simplest model of an elementary cell of a filled polymer, illustrated in Figure 2.6.
The cell is a polymeric cube with an edge length, a. Filler particles of a spherical shape have a diameter, D. The shortest distance between particle surface is d. If such a cell is deformed, the absolute deformation is∆a; then the strain in poly-mer at point K isε =∆a/a. The absolute deformation of the polymeric interlayer at point P is also ∆a because filler is a high-modulus material and, therefore, it practically does not deform. The strain at point P is alsoε = ∆a/a, hence:
εp /εf = a / d; a / a = a / d∆ ∆ [2.83]
Since a > d, then, from Eq 2.83, the deformation, a, in sites where particles come closest to each other, and, consequently, the stress in the same sites in the binder interlayer, are appreciably greater than in material bulk. A purely geo-metric analysis shows that the volume concentration of the filler isφf= πD3/6a3, and:
Figure 2.6. Elementary cell of filled polymer. [Adapted, by permission, from Y. Lipatov, V. Babich, and T.Todosijchuk, J. Adhes., 35, 187 (1991).]
a = D
This allows the a/d ratio to be expressed in terms of the filler concentration,φf:
This allows the a/d ratio to be expressed in terms of the filler concentration,φf: