APPLICATION OF ABSOLUTE RATETHEORY OF ADHESION
July, 1956
At pH 2, where there is little charge on the particles, the addition of an electrolyte has very little effect. In the aforementioned sols containing 3.77 and 6.63% silica, addition of 1.5% of sodium sulfate did not change the viscosity. It is necessary to measure the viscosity promptly when the pH of the sol has been reduced to 2, even though the rate of gelling of colloidal silica is a minimum at about this point.9 Over a period of several days a t 25”, aggregation of the particles occurs, and eventually the solution will form either a precipitate or gel. However, during the first hour or so, a t ordinary temperature, colloidal solutions a t the indicated concentrations increase very little in viscosity. It should be pointed out that the volume fraction of the dispersed phase, C, may be increased not only by the amount of water associated with the surface of the individual silica particles, but also by the degree of aggregation of the particles. If a number of particles were joined together to form a porous aggregate, the water within the pores would be essentially immobilized from a hydrodynamic standpoint, so that the “dispersed phase” would include more water than if the particles were not aggregated. Thus aggregation decreases the value of 8. This effect of aggregation accounts for the low value of S (Le., 53%) for the sols mentioned above. However, since the sols referred to in Table I are not aggregated, the relationship between viscosity and surface area can be accounted for by the hydration of the surface, as shown below. Calculation of the Composition of the “Dispersed Phase.” -It is known that the hydrated surface of amorphous silica is covered with silanol groups (-SiOH).’O The composition of discrete spherical particles, in terms of SiOz and HzO present as hydroxyl groups, can be calculated from the specific surface area. Thus A = 2720/d where A = specific surface area of the particles in square mfters per gram and d = average particle diameter in millimicrons. The particle composition9 has been calculated as 22?r x = - d8 and y = (2.80d)2 6 where the composition is represented as ( S ~ O Z ) ~ ( H ~ O ) ~ .
2“
(9) R. K. Iler, “The Colloid Chemiatry of Silica and Silioates,”
The Cornel1 Presa, Ithaca, N. Y., 1965, p. 45.
957
Let us now assume that in addition to the chemically bound layer of silanol groups, there are n layers of water molecules adsorbed or fixed in some way to the surface as far a~ their hydrodynamic behavior is concerned. The association of hydrogen-bonded water molecules with the silanol layer, in view of the large volume of the oxygen atom relative to hydrogen, suggests that in the first layer there may be one water molecule associated with each underlying silanol group.11 Since the hydrogen atoms of two silanol groups are equivalent to one molecule of water reacted with one unit of anhydrous SiOz, the composition of the “dispersed phase” may be represented as
(Si02)=(HzO),(HzO ) n ( ~ u ) Thus for every molecule of water present in the silanol layer there would be two water molecules in each physically bound water layer. From the above formula, using the formula weights for SiOz and H20, the dispersed phase has the composition, by weight
Substituting for x, y and d from the above relationships 1 8480(100
n = -2 [
- S)
SA
-
Degree of Hydration.-Applying this formula to the values of S and the specific surface area, A , in Table I, the values of n for each sol were calculated. For example, for sol E having a specific surface area of 612 m.z/g. and a viscosity corresponding to a value of S of 83%, n was calculated to he 0.9. For n to have been 0 or 2, the required value of S would have been about 94 or 74%, respectively. Such values are well outside the range of experimental error in deterniiniiig viscosity. Thus, for discrete silica particles ranging in size from 3.3 mp diameter in sol A to 7.5 mp in sol I, the value of n remains constant a t a value of about unity. It therefore appears that there is about a monomolecular layer of water molecules immobilized a t the hydroxylated surface of the silica particles. (11) Ref. 9, p. 240.
(10) Ref. 9, pp. 99, 103, 234.
APPLICATION OF THE ABSOLUTE RATE THEORY TO ADHESIOIV* BY MARSHALL R. HATFIELD AND GEORGE B. RATHMANN Contribution No. 96, Central Research Department, Minnesota Mining & Manufacturing Company, St. Paul 1, Minnesota Received January 19, 1966
Attempts to treat adhesion of a deformable adhesive to a rigid surface in terms of a simple fuse model for the bond are shown to be inadequate for explaining the time dependence of adhesion failure. Modification of this model to include the effects of a time dependent modulus of elasticity for the adhesive does not appear to be a satisfactory solution. The application of the absolute rate theory is based upon the assumption that bonding and debonding are rate processes and do not occur instantaneously which automatically results in a time dependent failure. The general implications of the theory are that: (1) loads below a critical force will never produce failure (2) the work of adhesion will have a profound effect on the time to breakage for a given load, although it represents a smafl fraction of the energy expended, and (3) the free energy of activation for viscoelastic flow of the adhesive also has a profound effect on the time to breakage for a given load. Precise and simplified forms of the rate equation are given for adhesion failure of a bond under tensional load. For comparison of similar systems, the simplified form may be adequate. I n addition, proper use of the simplified form may permit some information concerning work of adhesion to be derived from experimental adhesion data.
Introduction Adhesion, as discussed in this paper will be restricted to the bonding together of two solids, one soft and deformable (adhesive) and the other rigid and non-deformable (adherend). The primary purpose is to demonstrate that the rate theory offers a plausible mechanism for the direct approach t o the
* Presented at the 128th National American Chemical Sooiety Meeting, Cincinnati, Ohio, April, 1955.
failure of such an adhesive bond under tensional stress. The fact that small variations in the applied stress have a pronounced effect on the time required for bond failure is well established experimentally. The introduction of the rate theory provides a natural explanation for this time-dependency. Other explanations, which consider the adhesive bond as a fuse, introduce time-dependency through the viscoelastic properties of the adhesive
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MARSHALL R. HATFIELD AND GEORGE B. RATHMANN
which are known to be time dependent. Such explanations will be grouped as the fuse theory of adhesion. 9 The Fuse Theory.--Consider a rigid solid attached to an elastomeric solid by adhesive bonds a t the interface. If the applied force exceeds the product of the number of bonds and the force required to break each bond, instantaneous failure results. The analogy to a fuse in electrical circuits is obvious. This picture clearly needs alteration to introduce time-dependency since failure would occur immediately a t forces above a critical value and never a t forces ,below this value. A superficially attractive modification incorporates the modulus behavior of the elastomeric adhesive. If a constant load i s applied to the system such that the stress (load per unit cross-section) a t the interface is not sufficient to break the bonds, failure does not occur immediately. Creep of the adhesive and the resulting smaller cross-section will result in higher stresses within the elastomer. Sufficiently high stresses are developed, it may be argued, to break the adhesive bonds. Actually, however, only the time-dependency of cohesive failure could be accounted for in this way, since there is no change in the cross-section of the interface. A further modification of the theory postulates changes in geometry in the stretching adhesive to account for a stress build-up a t the interface. A precise formulation of the stress build-up resulting from geometry changes would be, a t best, complicated. Distributions of bond energies and more elaborate mechanisms for developing high interfacial stress as a result of increasing stress within the adhesive offer no substantial improvement. One modification of the fuse theory introduces the concept of tensional fluctuation. Even for forces insufficient to break the bonds instantly, statistical fluctuations in tension will result in random rupturing of the bonds. As the bonds fail, the stress a t the interface increases and complete failure eventually occurs. Such an approach is mathematically as complicated as the application of the absolute rate theory, and does not offer any advantages. The rate theory by focusing attention upon the interface where failure is known to occur, provides a means of examining bond formation as well as bond failure.’ The Rate Theory.-Instead of considering bonds as fuses it will be assumed that the formation and failure of bonds involves the passage over a free energy barrier. Molecules at the interface must be thermally activated before passing into the unbonded state. The time to failure is, then, the time required for all the bonded molecules to acquire sufficient energy by thermal fluctuations to pass over the barrier. Following Eyring2 a free energy profile can be drawn as in Fig. 1. Here, the difference in free energy of the bonded and unbonded states, W , is the free energy of adhesion per bond. It is assumed that in the unbonded (1) W. M. Bright, Paper presented at Case Institute Symposium on Adhesion, Cleveland, 1952. “Adhesion and Adhesives, Fundamentals and Practice.” Editors Clark, Rutzler, and Savage, John Wiley Sons, New York, N. Y . , 1954,P. 130. ( 2 ) S. Glasstone, I
