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Application of the Flory--Rehner equation and the Griffith fracture criterion to paint stripping.

[C] FSCT and OCCA 2009

Abstract Chemical paint stripping is a part of paint technology that is very necessary but for which quantitative understanding has been comparatively unexplored. Traditional paint strippers were based on very effective solvents, such as dichloromethane, that are now recognized as being dangerous for people and the environment. Optimal replacement by environmentally sound, and effective, paint strippers requires a better understanding of the action of the traditional materials. This communication links the stripping of cured paint films by aggressive solvents to the effect that swelling has on the cohesive and adhesive properties of a crosslinked polymer network. Swelling, described by the well-known Flory--Rehner equation, can be used to estimate the reduction in strength of the coating film. Thus, its resistance to fracture, described by the Griffith equation, is decreased so that the swelling stresses cause severe weakening and sometimes spontaneous removal. The equations lead to a description that fits well with common experience and so may be useful in selecting materials for future, less toxic, and more easily disposed, stripping formulations.

Keywords Stripping, Swelling, Solvent, Internal stress, Flory--Huggins, Griffith, Adhesion, Fracture strength

Introduction

In order to function efficiently, a solvent-based paint stripper must diffuse into the polymer, swell it considerably, render it very friable and reduce its adhesion so that it may easily be removed. A traditional paint stripper (1), (2) utilizes small molar volume solvents with moderate evaporation rates, such as dichloromethane and affects only the polymeric coating and not its substrate. A mixture of solvents, such as methanol or tetralin (commonly with dichloromethane), may be used to control evaporation or broaden the range of coatings that a stripper could attack. Other ingredients, such as wax, may be added to a formulation to reduce the evaporation rate. Cellulosic thickeners may prevent rapid run-off and surfactants are included if the waste is to be washed off with water. The function of dichloromethane and such solvents is to swell the polymer network and thus disrupt it, and possibly allow the penetration of more aggressive ingredients of the formulated stripper, e.g. acids, alkalis, amines, phenol, etc., to attack the chemical bonds in the polymer network. Some ingredients of the conventional paint strippers have more than one function, e.g., phenol may be a good solvent, but it may also attack the oxide film on a metallic substrate, thus diminishing adhesion. (3) Paint removers may also use saponification to degrade the polymer. This commonly occurs in binders that contain ester groups in the polymer network and is achieved by the use of alkalis, commonly sodium hydroxide, which is otherwise well known to polymer chemists in various incarnations of "base baths" for cleaning laboratory equipment. In other polymers, ether linkages would usually better attacked by acidic ingredients.

Although there are a number of patents for paint stripping compositions and tools for their use, there is very little analysis in the open scientific literature about how paint strippers function. Dichloromethane/phenol paint strippers are very effective, but they pose environmental (use and disposal) and health problems. Their use requires personal protective equipment, containment, and disposal of the mixture containing the paint must be as hazardous waste. Alternative materials are, and should be, sought. If environmentally benign paint strippers are to become as effective as the traditional materials, greater understanding is necessary.

Previously, the action of solvents in paint strippers was characterized using the interaction between cross-linked polymer and solvent using solubility parameters (including the more modern Hansen version that uses the dispersion, polar and hydrogen bonding components) (3), (4) and the Dimroth parameter, which expresses the polarity of a solvent and is part of the [pi] (*) scale.(5) There is also some research on coating removal by art conservators in order to better repair old paintings by removing overpainted or varnished layers without harming the fundamental artwork underneath. However, the need in art conservation is to provide rather gentle, gradual removal that is easily controlled, but the work does provide some systematic insight into the chemical compatibility requirements of stripper and paint layer. (6) Conservators often use the Teas plot to characterize solvents. (7) Wollbrinck (8) gives a useful survey of commercially useful strippers in terms of ingredients and their functions in paint strippers.

This note provides a complement to the prior, chemical compatibility oriented, work, and a rationale for understanding and designing solvent-based paint strippers. It was motivated by the realization of the applicability of some venerable but very useful descriptions of polymer materials' properties.

Throughout this note the term "conventional" paint strippers will be used to denote the solvent-based type of formulated chemical paint stripper, e.g., dichloromethane/phenol.

Action of the solvent on the crosslinked polymer coating

Swelling of the polymer film

First, the chosen solvent molecules must be small enough to permeate quickly between the polymer chains and they must be very compatible with the polymer chains so that they solvate them and cause considerable swelling and thus loss of cohesion. The Flory--Huggins, F--H, theory (9) provides a more suitable theory, for describing the interaction of polymers and solvents, than regular solution theory, which is the basis for solubility parameters. F--H incorporates the entropy of interaction as well as explicitly the degree of polymerization and the size of the solvent molecules. Thus, it contains more and fundamental knowledge about the necessary properties of the interacting species in paint stripping. An extension to F--H is well known as a method of estimating crosslink density through swelling measurements, using optical microscopy or simple gravimetry, via the Flory--Rehner equation (10), (11):

1n(1-[phi]) + [phi] + [chi][phi].sup.2] + [upsilon]V([[phi].sup.[1/3]] - [phi]/2) = 0 (1)

where [phi] = volume fraction of the polymer in the swollen system, so the volume swelling ratio, S = 1/[phi]; v = crosslink density (mole/volume); V = molar volume of the solvent, and [chi] = Flory--Huggins parameter representing the solvent--polymer interaction energy. Typical use of this equation is to measure the crosslink density by measuring the swelling induced by a solvent of known [chi]. More detailed calculations of swelling in crosslinked polymers have been developed, (12), (13) but the Flory--Rehner equation is simple, familiar, and gives an estimate suitable for the calculations here.

If necessary, a F--H parameter can be calculated from the more widely known solubility parameters, [[delta].sub.i], for the solute (1) and solvent (2), e.g.:

[chi] = 0.6 [V/RT][[([[delta].sub.d1] - [[delta].sub.d2]).sup.2] + 0.25[([[delta].sub.p1] - [[delta].sub.p2]).sup.2] + 0.25[([[delta].sub.h1] - [[delta].sub.h2]).sup.2]] (2)

Several expressions linking the F--H parameter and solubility parameters exist. Equation (2) is somewhat empirical, but separates the solubility parameter into the well-known dispersion, polar and hydrogen bonding components. (14) If one wants to go further, there are methods to calculate the cohesive energy density, that underlies the solubility parameter, for a variety of crosslinking polymers. (15)

Equation (1) can be solved numerically by standard bisection techniques of finding its root and was used to calculate the swelling of a polymer network of relative density 1.2 when imbibed with a solvent of molar volume 64 [cm.sup.3]/mol (same as dichloromethane, Table 1), see Fig. 1a, and another of molar volume 136.3 [cm.sup.3]/mol (same as tetralin), Fig. 1b. In each case, the effect of the compatibility between the solvent and polymer was investigated by varying the F--H parameter between positive values of 0 (perfect compatibility, solubility parameters equal) and 1 (very unlikely to be a solvent for the uncrosslinked polymer). For completeness, the range of molecular weight between crosslinks used in calculations here goes to higher values than are typical of most crosslinked coatings.
Table 1: Molar volumes of solvents

Solvent                Density,        Molar mass,
                    g [cm.sup.-3]   g [mol.sup.-1]

Dichloromethane          1.33              84.9

Tetralin                 0.97             132.2

Water                      1               18

Acetone                  0.79             58.08

Solvent               Molar volume,
                 [cm.sup.3] [mol.sup.-1]

Dichloromethane            64.0

Tetralin                  136.3

Water                      18

Acetone                    73.5


Here, the effect of solvent molar volume is demonstrated by solving equation (1) with molar volume as the variable, Figs. 1c and 1d. Figure 1c was calculated for reasonably good solvents with [chi] = 0.1, and Fig. 1d for theta solvents ([chi] = 0.5), which are probably the poorest solvents that might be used. Water has a very low molar volume, but is not very compatible with conventional, heavy, and medium duty polymer networks. Acetone is a very useful solvent which is almost universally used for cleaning surfaces of paint and polymer smears, but its low density increases its molar volume above that of dichloromethane.

Figures 1a and 1b show how the swelling varies greatly with solvent--polymer compatibility, [chi], and how important is the degree of crosslinking. Also, as expected, solvents with smaller molecules are much more effective in swelling polymer networks, see Figs. 1c and 1d. Dichloromethane is effective because its chemical compatibility makes it very useful and its comparatively high density reduces its molar volume significantly. We can see that much greater swelling will be possible if there is an ingredient in the paint stripper formulation that attacks the network and reduces the crosslink density, i.e., increases [M.sub.c], the molecular weight between crosslinks.

[FIGURE 1 OMITTED]

Fig. 1: Swelling as a function of the Flory--Huggins parameter (and molecular weight between crosslinks), caused by a slovent having a molar volume of (a) 64 [cm.sup.3]/mol, (b) 136.3 [cm.sup.3]/mol. (c) Sweling as a function of the solvent molar volume (and molecular weight between crosslinks), with solvents having Flory--Huggins parameter of 0.1 and (d) swelling as a function of solvent molar volume (and molecular weight between crosslinks) for theta solvents, having Flory--Huggins parameter of 0.5

Strength of swollen polymer network

Having calculated how swelling is affected by material parameters, we can now examine how the coating film is weakened by the swelling and thus may be stripped.

Here, the Griffith fracture criterion (16) will be used as a simple and robust means of quantifying the strength of the paint film. The Griffith approach is the basis for linear elastic fracture mechanics and was extended by Irwin (17) and Orowan (18) to include plastic effects and further via the J-integral. (19) These extensions to the Griffith equation will not be explored in the very simple approach here.

The Griffith equation calculates the stress, [sigma], necessary to cause fracture, from the tensile modulus of the material, E, the surface energy density, [gamma], necessary to create the two surfaces either side of the crack and the size of the existing (penny-shaped) crack, a. The stress at failure may have contributions, not only from externally applied stress, but also from any internal stress caused by swelling. These material parameters will be affected by the swelling of the polymer.

[SIGMA] = [square root of [[E2[lambda]]/[[pi]a]]] = [[SIGMA].sub.external] + [[SIGMA].sub.[gamma]internal] (3)

One of the results of the common, statistical rubber elasticity theory (20) is that the modulus of a crosslinked material is proportional to the concentration of crosslinks, [n.sub.c] (mole/volume).

E = [3n.sub.c]RT (4)

where R = molar gas constant and T = absolute temperature.

There are more modern, complete theories describing polymer networks, (12), (21) but the simple approach is very common in use and meets the needs of this discussion. One common method of measurement of crosslink density is done by measuring the modulus well above the glass transition temperature, [T.sub.g], so that the effect of physical chain entanglements can be neglected. The imbibtion of considerable solvent, from a paint stripper, will cause the crosslinked polymer to be effectively well above its [T.sub.g] even at normal ambient temperatures. Thus, equation (4) will be used to calculate modulus which will diminish according to the swelling, S, because the crosslink density will be diminished. Molecular weight between crosslinks, [M.sub.c], is readily calculated from the crosslink density and the density, [rho], by using:

[M.sub.c] = [[rho]/[n.sub.c]] (5)

If the subscript '0' is used to denote the property before swelling, the crosslink density will be reduced to a value of [n.sub.c0]IS (or [n.sub.c0][delta]) and thus modulus would be reduced to a value of [n.sub.c0]/S (or [E.sub.0][delta]).

In a similar, simple way one might expect the surface energy necessary to create new crack surface, [gamma]0, to be reduced in a swollen polymer by a factor that reflects the reduction in the number of polymer chains (or groups) that lie in, or cross, a given area. Thus, [gamma]0 would be reduced by the swelling to [gamma][equation]/S2B]. Any flaw that threatened fracture before swelling will, presumably, be increased from a linear size [a.sub.0] to a swollen size [a.sub.0][S.sup.1/3] Thus, the effect of swelling is to reduce the stress necessary to fracture the film, and thereby facilitate stripping. If all these factors due to swelling are included in equation (2), the effect of the swelling can clearly be seen since the fracture stress has been reduced by the swelling ratio. Surface energy, [gamma], has been used until this point to provide some physical insight, but hereafter in this note, strain energy release rate, G, will be used to replace 2[gamma] as is usually done in linear elastic fracture mechanics.

[SIGMA] = [1/S][square root of [[[E.sub.0]2[[gamma].sub.0]]/[[pi][a.sub.0]]]] = [[SIGMA].sub.external] + [[SIGMA].sub.[gamma]internal] (6)

Any aggressive ingredient of the formulated stripper that reduces the crosslink density by chemically degrading the polymer will not only permit greater swelling (see previous section), but it will also further reduce the modulus and the fracture surface energy and, again, enlarge the flaw size thus further reducing the fracture stress by another factor. This additional factor depends on the fraction of chemically degraded polymer chains and might be calculated, and included, if the change in molecular weight between crosslinks were known.

Internal strains in the bulk of the coating (before swelling) arc usually tensile due to densification continuing after the gel point due to solvent loss or continued crosslinking, and because the area of the coating is largely constrained by adhesion to the substrate. These intrinsic internal strains, that are typically a few percent in magnitude, (22) will be opposed and swamped by the change in volume due to swelling by the stripping solvent. If we neglect these intrinsic internal strains, the internal strain due to swelling can be readily estimated. Since swelling implies very large deformations, the neo-Hookean (23) calculation should be used for strain via the swelling ratio. Thus the prevailing internal stress, in the swollen polymer, in a state of plane stress (although this may well not apply in a highly swollen polymer mass) will be given by: where the modulus is reduced by the swelling, as before, and v is the Poisson's ratio. Substituting for the value of [E.sub.0] from above gives the internal stress in terms of the molecular weight between crosslinks:

[[SIGMA].sub.internal] = [[E.sub.0]/[3S]] [[[S.sup.[2/3]] - [S.sup.[ - 1/3]]]/[1 - v]] (7)

[[SIGMA].sub.internal] = [[[[rho].sub.0]RT]/[M.sub.C]][1/S] [[([S.sup.[2/3]]

- [S.sup.[[ - 1]/3]])]/[1 - v]] (8)

Now the effect of the internal, swelling stress on the toughness of the polymer film can be estimated by using equation (6) with equation (8) using the same method of calculating the modulus from the crosslink density. In Figs. 2a-2c Poisson's ratio was assumed to be 0.5, as in an incompressible solid, and a suitable value of [G.sub.0], the strain energy release rate in the unswollen polymer, was taken as 0.3 kJ/[m.sup.2] (typical of epoxy and other polymers) from the literature. (23), (24)

A swelling ratio of 1 describes an unswollen paint film. Regardless of the effect of the flaw size, one can see, in Fig. 2, how the strength of the polymer film is greatly diminished, and thus its ease of removal greatly facilitated, by swelling and by having a lower cross-linking density (higher Mc).

The choice of flaw size for calculating the breaking stress has a large role in the results. The order of magnitude of the ultimate strength, in tension, of typical unswollen paint films may be 1 MPa or more. In Fig. 2a, the flaw size was chosen to be 1 mm which might be typical of the cracks and wrinkles that appear in a paint film subjected to an effective stripping formulation. Calculations of wrinkle wavelengths can be construed from models proposed elsewhere (25), (26) and are not included in the scope of this note. These calculations match expectations, but they are approximate. With a flaw size of 1 mm, the strength of the film is very low compared to the strength of an unaffected paint film and, above a certain values of the swelling ratio, the swelling stresses are a large fraction of the strength of the films and may cause failure themselves, without external help.

[FIGURE 2 OMITTED]

Fig. 2: (a) Fracture stress of swollen polymer networks, in the presence of a flaw sized 1 mm, and internal swelling stress. In this case, the swelling stress may be greater than the strength of the polymer. (b) Fracture stress of swollen polymer networks, In the presence of a flaw sized 100 [micro]m, and internal swelling stress. (c) Fracture stress of swollen polymer networks, in the presence of a flaw sized 10 [micro]m, and internal swelling stress. In this case, the fracture stress remains higher than the internal stress at any value of swelling

In the case of a film where the flaw size is 10 [micro]m, perhaps determined by a scratch or the size of a filler particle, the overall strength is much higher, Fig. 2c, and approaching 1 MPa (in the unswollen state) which matches reasonably well the expectations for competent coating films. Here, the swelling stress (not affected by the flaw size) is a much less important factor and more external stress would always be necessary to remove the coating film. Figure 2b shows the intermediate effect in films with a flaw size of 100 [micro]m where only the films with low crosslink density are overcome by the swelling stress. Wrinkling, due to the swelling that takes place, would introduce much larger flaws and thus weaken a paint film significantly. In fact, wrinkling may be a manifestation of an effective stripping formulation.

Adhesion

Ideally, paint strippers should also substantially reduce adhesion between polymer and substrate. The solvent or solvent combination should have an affinity for the polymer and spread across the interface at the substrate (or pigment or filler particle surfaces) and penetrate fissures in the film. (2) The rate at which fissures will be penetrated might be approximately evaluated by looking at the Lucas--Washburn equation for capillary rise, but this will not be included here. An effective stripper swells the polymer network to the extent that most of the molecules encountered in the walls of a fissure are solvent molecules, so the normal conditions of capillary rise are not met. As mentioned earlier, some ingredients in stripper formulations could be selected for materials that dissolve surface oxide layers on inorganic substrates and thus specifically attack the adhesion interface.

The Griffith failure criterion can be applied to adhesive bonds (27), (28) as well as to cohesion. In adhesion, the strain energy release rate is replaced by the interfacial work of adhesion, [W.sub.ad]. Similarly, one might expect the interfacial work of adhesion to be reduced by the reduction in polymer moieties close to the substrate. One could attempt to calculate how changes at the adhesion interface would affect the work of adhesion (27), (29), (30) via changes in concentration of polar groups etc., perhaps ultimately leaving only Van der Waals forces to sustain adhesion, but this is not attempted here because it would depend on the chemistry of each substrate, as well as the chemistry of each adherend. If the solvent has a greater affinity for the substrate than the polymer, then presumably the adhesion of the polymer would eventually be eliminated. If the polymer is not displaced completely from the substrate by the solvent, and if the work of adhesion depends not only on those polymer moieties immediately adjacent to the substrate, but also depends on the next nearest neighbors, etc., the work of adhesion would be reduced by a factor proportional, at the most, to the swelling. However, for discussion, an intermediate behavior would be where the adhesion is reduced proportionately to the effect of the area occupancy of polymer chains, as proposed in the discussion on cohesive strength, above.

[FIGURE 3 OMITTED]

Fig. 3: Swelling constrained by the presence of a substrate in crosslinked polymer networks, by a solvent having a molar volume of 64 [cm.sup.3]/mol

In a glue line, the Young's modulus would be replaced by the effective Bulk modulus of the composite joint. If we assume that the substrate is unaffected by the stripping chemicals, the relevant modulus remains that of the polymer which is reduced by swelling in much the same way as discussed above for the effect on cohesive strength. The flaw size now relates to possible delamination at the coating--substrate interface (or glue thickness in a glued joint), but again will presumably be increased by the degree of swelling at the interface, if wetting conditions permit. Thus, for discussion here, the effect on adhesion would be analogous to the reduction in cohesive strength by swelling.

Thus the effect of swelling would be as before. However, in a confined polymer, swelling is limited somewhat. (12),(31) If the presence and constraint of the substrate limit the possibility of swelling, the Flory--Rehner equation is modified (25):

In(1 - [phi]) + [phi] + [[chi] [phi].sup.2] + vV(1/[phi] - [phi]/2) = 0 (9)

The effect of the constraints on swelling can be seen in Fig. 3 where the calculation of the swelling ratio for dichloromethane clearly shows much lower values, which are approximately 1/3 of those in Fig. 1. Nevertheless, the stress required to overcome adhesion would still be considerably reduced, in a similar way to cohesion.

Spontaneous loss of adhesion occurs when a coating film is thick enough to store enough mechanical energy, by virtue of internal strain in the plane of the coating, to overcome the work of adhesion, (28), (32) provided that strain energy is released in no other way. In an otherwise flawless film, the thickness, d0, above which spontaneous adhesive failure occurs for an unswollen, ideal, linear elastic material, is given by:

[d.sub.0] = [[W.sub.ado]/U] (10)

where U = strain energy density in the film, which for an unswollen film with internal strain,[epsilon.sub.0], from the curing process is:

[d.sub.0] = [[[W.sub.ado](1 - v)]/[[E.sub.0][e.sub.0.sup.2]]] (11)

This equation is obtained from a simple energy balance, related to the derivation of the Griffith equation. In a coating system with good adhesion, one finds values of do to be several hundreds of micrometers (32) which is much higher than typical coating thicknesses and thus internal strain is not a problem for competent films.

In an isotropically swollen polymer network, the strain energy density is given by (12), (33):

U = [1/2]E[[phi].sup.[1/3]]([[phi].sup. - 2/3] - 1) = [1/2][[E.sub.0]/S][1/[S.sup.1/3]]([S.sup.2/3] - 1) (12)

If swelling reduces modulus and work of adhesion, equation (10) becomes:

d = [[W.sub.ado]/[S.sup.2/3]] [S/[E.sub.0]][[S.sup.[1/3]]/([S.sup.2/3] - 1)] = [[W.sub.ado]/[E.sub.0]] [[S.sup.[2/3]]/([S.sup.2/3] - 1)] (13)

If a swelling ratio, S, typical of that found in Fig. 3 is chosen as 3, then the value of [S.sup.(2/3)] is 2.08 so the effect of the swelling ratio, S, in the numerator and denominator of equation (13) is to give a factor of approximately 2.

Values of internal strain in an unswollen film are [10.sup.-2] to [10.sup.-3] after typical film formation. (32) Thus, [[esp].sub.0.sup.2] in equation (11) is of order [10.sup.-3]--[10.sup.-5], so it is immediately clear that the value of d in equation (13) will be a factor of [10.sup.3] to [10.sup.5] smaller, in a well-swollen film. Thus, the thickness (of the swollen film) at which spontaneous loss of adhesion occurs would be a small fraction of a micrometer, and would be much less than the prevailing coating thickness. It is very unlikely that a swollen, friable polymer network could transmit all this strain energy to overcome adhesion at the substrate, but it does indicate that an effective stripper could readily provoke delamination, thus aiding adhesive failure.

Concluding remarks

A basis for studying solvent-based paint strippers has been assembled by making some simple assumptions about what happens to the material parameters that determine fracture toughness, i.e. modulus, strain energy release rate, and flaw size, when a polymer network is swollen. The extent of swelling was readily determined by use of the Flory--Rehner equation, over a range of crosslink density, solvent compatibility and solvent molar volume that is likely to span most crosslinked polymers, and the solvents in stripping formulations. The equations used in the illustrative examples were, in each case, the simplest available, but more complete models are referenced that could be incorporated if more exacting studies were undertaken.

The calculated effects of swelling on the cohesive strength and adhesive strength of the swollen polymer networks agree well with common experience of which solvents are most useful for paint stripping and show the effect of polymer--solvent compatibility, solvent molar volume and the utility of aggressive chemicals in a stripping formulation for reducing crosslink density. The equations here could form the basis for further, quantitative research on environmentally benign replacements for the traditional, very effective (and much less benign) paint stripping formulations.

S. G. Croll (*)

North Dakota State University, Fargo, ND, USA

e-mail: Stuart.Croll@ndsu.edu

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Date:Jan 1, 2010
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