Influence of pigmentation on the mechanical properties of paint films

Progress in Organic Coatings, 2 (1973/74) 237-267 0 Elsevier Sequoia Sk, Lausanne - Printed in Belgium

INFLUENCE PROPERTIES

OF PIGMENTATION OF PAINT FILMS

ON THE MECHANTCAL

A. TOUSSAINT Laboratoire

de la ProJksion,

I. V.P.,

I342-Limelette

(Belgium)

CONTENTS

Introduction, 237 1 Influence of the PVC on the stress-strain curves, 238 2 Influence of the PVC on glass transition temperature T,, 244 3 Quantitative interpretation of the reinforcement, 247 3.1 The modulus, 247 3.2 Changes in the ultimate properties, 251 3.2.1 iufiuence of the experimental conditions upon the ultimate properties, 253 3.2.2 Ultimate strengths of tensile strengths, 258 3.2.3 Ultimate strains, 262 4 The Mullins effect, 263 5 Multiaxial tensions, 264 Conclusions, 266 References, 266 INTRODUCTION

Generally, amorphous and homogeneous elastomers are of little use because of their low tensile strength. This is the reason why, in practice, one incorporates into them tilers, finely divided mineral substances, to increase substantially their mechanical properties. The last ten years have brought a great deal of experimental results on the reinforcing effect, also called stiffening, of the mechanical properties of polymers due to the presence of small solid particles. Some theories have been advanced and useful generalisations produced, but they are limited to elastomers. On the other hand, in the paint field, very little experimental work has been done with a view to interpreting or explaining moditications of the mechanical properties. Most of the existing work is qualitative. The first reference dealing with the pigment volume effect is due to Calbeckl. Other work followed in which properties such as gloss, resistance to degradation agents, permeability etc. were examined. In addition, it would be interesting to be able to predict the mechanical behaviour of a paint m, modulus and ultimate properties, on the basis of the specific properties governing the physical and/or chemical interactions between pigments and binders. Let us recall here that all the curves describing a “property” in function of the pigment volume concentration (PVG are characterised by a particular value of the .

238 concentration beyond which all the properties undergo thorough modifications (Fig. 1). This value is called the critical pigment volume concentration (CPVC). According to Asbeck and van Loo it should correspond to a state of the paint in which the amount of binder would be that exactly necessary to ml the existing voids between pigment particles, this being in its state of maximum settling allowed by the dispersion degree of the system and after evaporation of solvents and diluentszg8.

‘\

--\_

2

‘.

\

\

\7 \

\

\

\ \

\

\

\ /I _

----_-__

_

/

*c

=x

=

.XE--__

____--_--#

PVC Fig. I_ l-gloss,

Z-corrosion,

3-blistering,

4-permeability.

In the first section we will qualitatively summarise the modifications undergone by the mechanical properties of binders and polymers in general when finely divided particles are incorporated. In the second section we will discuss the influence of pigmentation on the glass transition temperature, Finally, we will review the main theoretical studies attempting to explain and to quantify the observed behaviour. I. INFLUENCE

OF THE PVC ON THE STRESS-STRAIN CURVES

In 1925, Wiegand” classified pigments and extenders into two categories: - those which contribute significantly to increasing the ultimate strength, the abrasion resistance and the modulus: they are reinforcing; - those which only increase density and hardness without modifying the stresses: they are fillers. This classification is, as we shall see later, quite rough. Indeed, it appears that certain reinforcings for rubber do not play that role for synthetic resins, and conversely that some mers for rubber become definitely reinforcing for other resins. These diEerences in behaviour should depend on the reactivity of the materials towards binders in which they are dispersed. A quantitative evaluation of the forces involved is difficult to make due to uncertainty in the chemical or physical nature of the interactions. In a wide sense, these interactions include the interfacial forces between solids

239 and polymers,

the orientation

effect in the vicinity of the interface,

the formation

of

a contact zone, etc. In 1953, Elm” showed that the extensibility of a paint film decreased whilst its ultimate strength increased with the PVC. Bosch3s1 1 noted a similar behaviour with alkyd resins, and further showed that the observed modifications were dependent on the nature of the pigments, e.g. zinc oxide was more efficient than china clay. Experience showed also that when replacing part of the zinc oxide by china clay, the ultimate stress diminished without modification of the extensibility. Bosch also found that the milling conditions and the time elapsed between paint manufacture and its application influence the mechanical properties. Becker and Howell’ ’ studied the behaviour of latex paints on varying the PVC from 30 to 65%; they found that the ultimate stresses of free films go through a maximum value at the CPVC while the strains diminish continuously_ Bernardi’ 3 came to similar conclusions with other emulsion systems. Similar results have been obtained by Mika14 with epoxy resins, by Jaffe’ 5 with polyvinyl acetate, by Zorll’6 with an alkyd/melamine, by Oosterhof” with vinyl acetate/versatic acid ester copolymers, and by Schallerig with an acrylic resin. Recently, Suryanarama’g determined the CPVC of four red iron oxides in linseed oil and the ultimate properties of oils and alkyds pi_mented with zinc oxide and titanium dioxide. He showed that the tilms had optimum properties of permeability, ultimate mechanical properties in the vicinity of the CPVC and that this latter vaiue found by mechanical tests is identical with those found by other methods. It has been shown that zinc oxide and magnesium oxide react with acid groups of polyesters”. Such effects have also been obtained with titanium dioxide. It must be concluded that the reactivity of a pigment or of a filler, which depends on the biader used, has a marked effect on the mechanical properties. However, in the absence of chemical interactions it is the finest particles that show the greatest reinforcing power. In the case of a non-reactive or slightly reactive binder pigmented with chalks, titanium dioxides, barium sulfates (natural and precipitated), and china clay of variable granulometry, Toussaint et aLzl have shown that the shape of the curves for the extension and stress ratios E/E, and a/~~ versus PVC is determined mainly by the morphology of the particles. Curves are classified into 3 categories (Figs. 3, 3): (a) carbonates (cylinders), (b) titanium dioxides and barytes (spheres), and (c) clays (lamelIar). In each class the curves are somewhat itiuenced by the granulometry. This latter has a greater influence on the O/G, curves than on the E/E,curves. E and a0 are the ultimate strains of the pigmented and unpigmented films, and u and a0 are the corresponding tensile strengths_ This influence is also more pronounced for the small deformation rates than for the large ones. Summarising, experience shows that, generally, the tensile strength of a detached pigmented 6lm increases with PVC as compared with that of an unpigmented film, reaching a maximum value at the CPVC. For values higher than the CPVC the tensile strength diminishes quickly to reach a zero value. On the other hand, the ultimate

240

j-

Some

I

1

70

20

Fig. 2. e chalk OMYA

30

I

1

40

50 PVC %

BLP 3, v microtalc

legend

as fig.2

1

I

I

10

20

30

I

1

40

50

PVC %

IT extra, A titanium

dioxide KRONOS

EWO 425. Binder,acrylic; R.H. 55%; r”, 25°C; strainrate, 1 cm n&-l.

R, q baryte

Fig. 3. Same legend as Fig. 2.

strain decreases continuously with the PVC (Fig. 4). The reinforcing effect is more or less pronounced, depending on the binder/solids system involved, but in every case the changes in property are approximately proportional to the PVC. Generally one observes an increase in the Young’s (E) and torsion (G) modulus, and the damping is roughly proportional to the PVC. Zorll i 5v1 6 showed that the Young’s and rigidity modulus of an alkyd melamine pigmented with TiO, increase with the PVC, especially around 40-50% ; the same happens for the glass temperature, Tg. IHe also found that the modulus shows a discontinuity in its change for a value somewhat higher than 50%; this value is considered to be the CPVC (Fig. 5). Similar tidings were made by Le Besnarais and Piston” with a pigmented vinyl acetate/acrylic copolymer, by Galperin23 (Fig. 6) with titanium dioxide pigmented epoxy resins, by Oosterhoff” with latex and by Bender24 with a number of alkyds differing.by oil llngth or inco@oration of other resins. Beyond the CPVC,

241

/

I I I

I

1

10

20

I

1

30

40

50

I

60 PVC %

Fig. 4.

100

ION

k t

n” ?

z

0.1 -

‘W

0.011

1

-30

I

0

1

I

30

60

I

90

t=c

Fig. 5. Variation of the modulus (torsion) with the PVC (after Zorll’5*‘6). 3 PVC lo%, 4 PVC 20%, 5 PVC 30%, 6 P\rC 40%, 7 PVC 50%.

1 PVC O%, 2 PVC 5%,

1 2 3 4 5 6 7 8 9

I

-20

60% 57.5

C.P.V.

8

C.P.V.

47.5 C.P.V. 45 C.P.V. 42.5 CP.V. Copolymke

seul

-10

10

0

20

30

40

50

tot

60

70

Fig. 6. Variation of the modulus (torsion) with the PVC (after Le Besnaraisz2). Binder, vinyl acetateacrylic copolymer; pigment, Ti02.

direct contact is supposed to exist between the particles: being no longer separated by binder layers they would slip less easily over one another, thus resulting in an increase in rigidity of the films. Baker and Morisz5, studying the reinforcement of silicone elastomers by various fine particles, showed that the modulus increase must be attributed to a physical interaction which modifies the elasticity of the network. These authors also suggest that the increase in tensile strength would be due to increased viscosity of the material, this increased viscosity being greater, for a given PVC, the finer the particle size. On the contrary, the existence of chemical links between binders and particles would result in a decreased viscosity. Ackay26 pointed out the effect of surface treatment of pigments on the mechanical properties (Fig. 7). Untreated titanium dioxide behaves normally as foreseen by the theory of reinforcement. On the other hand, if the surface is treated with mineral substances, Al,Os for example, one observes a lowering of the Ts and an increase in the top of the curves tan 6 (loss angle) V~ESZLS temperature. The surface treatment of the pigment confers higher flexibility and impact resistance while maintaining the hardness of film that is observed in films pigmented with untreated titanium dioxide. One must also observe that the modulus increase is, for a given PVC, generally higher for polymers in their rubbery state than in their vitreous state. In fact, the stress-strain curves for elastomeric materials loaded with inert fillers have several distinct regions similar to those for unfilled elastomeric materials.

243

- 1.1 -1.0

- 0.9

-0.8

- 0.7

1010x1

1

0 Kronos RN59 A Kronos RA44 0 Kronos RA45 L0 i3aver titan

-06

10 E

- 0.5 m

,,L

notional

RSM-3 - 0.4

- 0.3

- 0.2

- 0.1

.

20

.

I

40

9

60

8

I

80

Temperature

Fig.

7. (After

8

.

100

.

*

120

I

#

140

a

-0

(‘C)

Ackay26.)

The curves have an initial region in which the stress increases with strain and the modulus depends on the amount of filler and the properties of the binder. This is followed by a second region in which the stress may be roughly independent of the strain or may increase at a rate markedly less than in the first region. This is typical of the curves of rubbers at low temperature which cold-draw or neck. In the third region the stress increases more rapidly with strain than in the second region and continues to increase until the sample breaks. Thus, both tiled and unfilled materials have sigmoidal stress-strain curves with or without distinct yield points. However, as stated by Smith -s’ , med systems and especially those in which the binder wets, but does not react chemically with the filler exhibit this sigmoidal-type curve for different reasons than do rubbers and plastics. In filled systems the binder adheres to the filler initially, which reinforces the system. In the transition and second regions of the stress-strain curves, adhesive bonds between binder and particles break

244 and vacuoles form around the solid particles; this process is called dewetting. Thus since the curves show precisely the elongation at which dewetting or adhesive failure begins, one can conveniently study methods for modifying the adhesive strength. Summarising, the mechauical characteristics of binders are modified by incorporation of finely divided solid particles, giving an increase of the modulus and tensile strength and a decrease of the ultimate strain. These changes are proportional to the PVC. However, the magnitude of the change depends on several factors: size, morphology and chemical nature of the particles, surface treatment, polymer physical state, humidity etc. 2. INFLUENCE

OF THE

PVC

ON

GLASS

TRANSITION

TEMPERATURE

To

The glass transition temperature is one of the most important parameters in polymer characterisation, and it is known that many properties are modified around that temperature. For example, the modulus, generally high and fairly constant at temperatures below T’, diminishes quite rapidly once the temperature is above T8. The ultimate strain, which slowly increases at temperatures below T,, increases markedly with temperaeJres above TE. The thermal expansion coefficient shows a discontinuity in its value at T,. It is therefore important, for evident technological reasons, to know if the Z; is modified by incorporation of pigment and, if so, how. Little work has been done on this subject. Newman and Cox2’ observed a Tg increase in SiO,-charged polydimethylsiloxane. This behaviour is interpreted by the reaction of silica with the residual reactive sites in the siloxane chains. Galperin 23 also found a T’, increase in polyurethanes filled with glass beads proportional to their concentration. The same happens with polymethacrylate and polystyrene tYled with glass beads”. Droste and Di Benedetto, studying the behaviour of a thermoplastic epoxy polymer filled with glass beads and kaolin, showed not only that the T. increases with the PVC but also that the higher the sp.xific area, the more pronounced tbe increase. These results are interpreted in terms of interactions between solids and polymers, which reduce the mobility and the flexibility of polymer chains in the vicinity of the interphases. Some authors, on the contrary, &Id a decrease in T’.. Butta and Baccareda27 showed that the T’ remained quite unchanged in a polyisobutylene filled with carbon black but that it diminished if the elastomer was vulcanised. They attribute this phenomenon to a cross-linking difference (in its degree or in network regularity). Kumins and Roteman2* found two transition temperatures in a polyvinyl chloride/ vinyl acetate copolymer; the higher is attributed to the vinyl skeleton whereas the lower is attributed to the acetate groups. These authors found that the higher glass transition temperature first diminishes with the PVC and increases afterwards; both glass transition temperatures tend to coincide for a particular PVC value. The explanation of the first behaviour should be that the acetate groups are adsorbed on the TiOZ polar sites, diminishing so far the number of hydrogen bonds between polymer chains; the result of this is an increase of the degree of freedom and a T’ decrease for low

245 PVC values. Above a certain PVC value, however, the number of adsorbed acerate groups becomes so high that the degree of freedom of the acetate group and of the main chain are strongly diminished, with the result that both 7’pvalues approach each other and tend toward a single limiting value. Finally, Kraus and Gruver31 did not observe any Tg changes in the case of a styrene-butadiene copolymer filled with carbon black. Toussaint et al.” found no evidence of any significant changes of Tg in acrylic and styrene-butadiene resins containing various pigments ~‘crsus their PVC. As we see, these results are often contradictory since some observe changes of T, cer.su~ PVC, others a decrease, while some do not find any significant changes. What can be concluded from this? It seems evident that, according to the degree of wetting of the pigment by the binder, there will be formed a contact layer or a contact phase more or less continuous at the pigment-binder interface. This phase will be preferably tightly bound to the solid particle, so that the phenomena (physical or chemical) involved between pigments and binders and during wetting are energetically important. It is also obvious that the polymer structure in that phase may be modified according to the nature and the magnitude of the interactions: preferential orientation effect of polymer segments at the particle surface, chemisorption (hydrogen bonding for example, or primary bonds formation between solid particles and acidic groups attached to the polymer backbone, etc.). Solid particles may also impede the polymer chain mobility. The different types of interaction (hydrodynamic, physical adsorption, chemisorption) de&e the possible coupling scale and, while different energetically, they all three possess the property of diminishing the chain or chain segment mobility and flexibility and consequently increasing 7”. These three e_ffects may of course act

simultaneously. It must be stated that T’ increases have been observed with the crystallite contents in partially crystalline polymers. It is therefore not unreasonable to compare a filled polymer in which there is an excellent pigment/binder interaction with partially crystalline polymers. In those materials T, increases with the degree of crystailinity, and shifts in relaxation times appear also. It must be said that the T’ (of a polymer as well as of a composite) is really the temperature at which the amorphous nature of the polymer phase shows changes in its dynamic properties, the inorganic material remaining unmodified or undergoing any changes. If the Tg is accurately measured, its value could be a measure (or a means of measuring) the modiEcations of the properties of the polymer phase as the result of the incorporation of a second phase. It has been shown that the adsorbed polymer layer thickness is about 30-80 A. Consequently the mobility restriction is also practically limited to that thickness, or in any case its effect on the neighbouring chains or segments drops rapidly when the distance from the surface increases. Kwei6’ reports that the particle’s sphere of influence could extend up to 1500 A from the particle surface. In that zone, which in fact deals only with a few percent of the polymer (for values beIow the CPVC), the T’ can show an increase of 10°C 32 . The Tg of the bulk polymer must remain, in principle, unchanged. _

246 The existence of two distinct polymer phases has been shown in an NMR band width study for a polyester filled with sodium perchlorate. It is interesting to note the work of Kraus and Gruve? ’ on a styrene-butaciiene copolymer filled with carbon blacks having different reidorcing powers. They found that the polymer expansion coefficient in the rubbery state is not affected by the solid particles, whereas it strongly diminishes in the vitreous state. Since the thermal expansion coefficient.of the polymer is much higher, in both states, than that of the mineral solids, it follows that stresses will be set up around the solid particles on cooling for thermoplastic or thermosetting resins; these stresses are incapable of quick relaxation, in the vitreous state. The same phenomena will happen for films being built at ambient temperature from solvent or dispersion paints when the T. of the binder is higher than ambient temperature. The polymer will be submitted to a multiaxial stress (which in fact diminishes Lam the particle surface to the polymer bulk), and since the Poisson ratio of polymers in their vitreous state is less than l/2, all happens as if the polymer expanded. In other words, there is an apparent free volume increase and the Tg of the polymer ought to decrease, at least in the vicinity of the interfaces. On the contrary, if the transition temperature lies below the ambient temperature, the stress will be relaxed and no change in its value will be observed. So the polymer fraction in contact with the solid particles and in their immediate vicinity must show a 7” increase owing to chain segments mobility decrease; the Tg of the polymer bulk must remain unaffected or show a slight increase only (due also to restricted mobility) if its value is below the ambient temperature, or show a decrease (due to the apparent free volume increase) in the opposite case. In practice one should observe an enlargement of the vitreous transition zone in both cases, the more pronounced the higher the PVC. In the former case the enlargement will occur on the high temperature side with a higher mean Tg value as a result, while in the latter case it occurs more or less symmetrically with the result that the mean TY value remains practically untiected or diminishes slightly. If VA is the polymer fraction in the interaction zone of the solid particles, V, the total volume of the composite, 4 the volume fraction of solid, S the specific area (cmZ/cm3) and 6 the interaction zone (cm” polymer/cm3 particles) then:

All the polymer will be in the interaction zone when VA = (1 - 4) Vr and the T’ of the composite (T’,) will attain a limiting value when #J = l/(1 +S6). Droste3’ showed that for thermoplastic epoxy resins, change in TB can be represented by: T,,-TT,,

= AT’,11

-exp

GJWI

in which ATgm is the highest deviation observed, pure polymer, and B is an empirical constant.

T’.Ois the glass temperature

of the

247 3. QUANTITATiVEi

INTERPRETATION

OF THE

REINFORCEMENT

3.1 The modihs Using Einstein’s equation relative to the viscosity of liquid suspensions, G~th~~*‘~ showed that Young’s modulus E (elastic modulus) of filled elastomers is related to the modulus E, of the uufilled poiymer by the equation: 14.1 c$‘)

E = E,(l+Z.S#+

(I)

4 being the solid particles volumic fraction. This relation is only valid for spherical shaped particles and for values of 4 lying between 0 and 10%. Other equations may be found in the literature, e.g. Eilers33:

volume of filler)/(true volume of filler). where V = (sedimentation The relation found by NieIsen34: (3)

E = E&l --c#J”~)

is valid, when there is good adhesion between polymer and Nler. In the case of high values of C#(aggregate formation) or for particles having other shapes than spherical, the following relationships are applicable: Guth3’:

E = E,(1+0_67f+-+

1.62f242)

wherefis the shape factor: Lee61: E =

(4)

ratio Ien,gth/diameter

(z-5+ 1.92q?JC7.73@‘)

E,(l--q5)-

(vitreous

(f S 1).

state)

(5)

Lande16’:

E=Eo 4,

(

l--

4

E=

(6)

&II>

being the maximum Sato3 %

-2s packing

fraction. i

Eo(l+[~~P(l -Y)I} (1 -W

- b,‘W(l -Y>

y31

(7)

where: 4 =y3 and $ = b3/3) (1 +y-y2)/(1 --y+y’). 5 is the adhesion parameter (= 0 for perfect adhesion, = 1 for no adhesion), the surface effect being omitted. As can be seen, these formula are dependent only on 4 and take no account either of the particle dimensions or of the existence of an adsorbed polymer layer. V one accepts the existence of a polymer layer tightly adhering to the solid particles, +

248 has to be replaced written37:

by the true volumic fraction

4e,

and a general equation

E = E,(l-&)-”

can be

(8)

being a function of the particle distribution and shape. For a statistical distribution, n = 2.5, for the spherical shaped pigments and 1 for the lamellar ones. c$, is related to 4 by: 72

where AR = adsorbed layer thickness and D = mean particle diameter. According to Ziege15 % +, = @U+AWW where AR/R,-, is the relative increase of the particle diameter in the spherical case. This term can be estimated by viscosimetric measurements or by comparison of loss moduli of filled and uncalled samples using the expression: 3

E;;/E”= l-4 In a recent study on the creep of alkyd resins and using a model obeying the following conditions: spherical shaped pigments uniformly distributed in a cubic arrangement, perfect adhesion between pigment and binder, absence of changes in the polymer viscoelastic properties due to the solid particles incorporation, Heertjes36 showed that: E=E,,

TUT’d2 [ 4(1 -nd)

+1---

m2d2 4

1

(9)

with nd = (6+/x) li3, d being the particle diameter and n3 the number of particles/cm3 in the film. Heertjes finds a good correlation between theory and experimental results, at least in the case of alkyd resins pigmented with Ti02 and BaSO, (Table 1). It must be emphasised that these equations and the following for the torsion modulus are valid only when small deformations are involved and when the nonlinear viscoelastic behaviour may be neglected, and of course in identical experimental conditions (relative humidity, temperature, test rates, weathering) and in equilibrium conditions. Zapp and Guth 56 showed that the modulus increase in a rubbery polymer may be divided into two contributions: one of these is entropic in nature and proportional to the specXc area S of the pigments.

249 TABLE

1

THEORETICAL AND EXPERIMENTAL FOR A Ti02 PIGMENTED ALKYD

GIlth

0 0.08 0.16 0.24 0.32

Nielsen (eqn. 3)

Eilers (eqn. 2)

(eqn. I)

VALUES

Sat0 (ev.

OF THE

7)

RATIO

Heer tjes k-v. 9)

E/E,

Exptl. ualue (ref. 36, Table 3)

1

1

1

1

1

1

0.77

0.81

0.57

0.86

0.78

0.77

0.57 0.41 0.31

0.64 0.48 0.35

0.46 0.36 0.32

0.76 0.66 0.58

0.58 0.39 0.24

0.57 0.39 0.24

0.11

0.12

0.40

Taking this into account, to give good results: E=EE,+-

Bueche”

proposed another equation

which is claimed

KTSX

(10)

u

where: K = Boltzmann constant, T = absolute temperature, S = specific surface area of the pigment particle, X = pigment volume to 1 cm3 of rubber and U = surface area of an adsorption site. Finally Noharass, taking account of the viscoelasticity, gives a general formula: IgE=(l-$)IgE,+

lg Q---lg l+--

’

E,

+ ti lg E,

(11)

k

wr,

where I&, E,, EC are the modulus of the polymer in the glassy state, the rubbery state, and of the pigment, w is the angular frequency, r, the most probable relaxation time and k the relaxation time distribution factor. He also showed that the viscoelastic behaviour of such a heterogeneous system is mainly due to the polymer fraction and that the loss angle (ttn S) of the system increases proportionally to the polymer fraction (l-4), whereas the glass transition temperature shows no changes. More complex equations are given by Japanese authors taking account of volume effects, surface effects, pigment cavities58. Droste also decomposes the reinforcement into two contributions with the hypothesis that the polymer phase in contact with the particle has different properties from those of the polymer bulk; he writes: 4% =

mechanical effect due to the fillers

phase modifications due to the idlers >

(12)

250

where qr = relative viscosity, q, = viscosity of the composite, qmo= viscosity of the polymer matrix exposed to the filler, and qmD= viscosity of the pure polymer. Moreover, postulating that the relative variation of the ratio ~J~~O is entirely due to the change in T. and that the viscoelastic properties dependence at temperatures above T, follows the W.L.F. equation, one may write:

]nh

4o(T-

=

52+(T-

tfT

TJ

(13)

TJ

If T’, and TgOare, respectively, the glass transition temperature of the composite and the pure polymer, then

1n’Imc= tlm.

208CT,,- Tgo)

(14)

(52 + Z-- T,,) (92 + Tg- Tgo)

By the fact that one postulates that the reference viscosity at the transition temperature T, is the same for both the filled and unfilled polymer, it follows that: qr

=

a-exptl mc

208(Tgc- TgJ

(15)

(52 + T- T,,) (52 + Tg- T,J

This relation allows us to determine the mechanical reinforcement alone. In the case of solids of shapes other than spherical or in a unknown state of agglomeration it would be better to use the Kerner equation motied by Brodnyan63

for an ellipsoidal shape 1 c L/D c 15. Concerning the torsion modulus, mention must be made of the Mooney equation, applicable to a wide range of concentrations38:

when the moduli are small and the Poisson ratio = 0.5. G and G,, are, respectively, the torsion modulus of the Hled and un.fXled polymer, k is the Einstein coefficient

(2.5 for spheres, higher for cylindrical or flat shapes or aggregates) and S is the relative sedimentation volume of the Hoer or the reciprocal of the maximum packing fraction (1.35 for spheres). One prefers, however, the Kerner equation3’ as more general, because it takes account of the Poisson ratio; it neglects on the other hand the factor S: G

l+AB$

G,=

l-B&

A=

7--5v 8-10~’

B = (G&J--1 (G/GJ + A ’

(18)

251 Equation (18) can be modified to fake into account fraction f.$, G

the effects of maximum

l+AB@

-= G

packing

(18 bis)

1 -Btw

The function I++is not a unique function of 4. A number of functions proper boundary conditions; two such functions are

can fu@ll the

Zor11’5*16, considering a model consisting of cubic particles dispersed in a cubic arrangement (Gittermodel), showed that: G -=1-lGO =l+-

X3

in the glassy state

X - X2 + (GO/G,) X2

1-x

in the rubbery state

where X= +lJ3, Gr = modulus of the pigment, G,, = modulus of the polymer. Brinkman5g -

G

(20)

G = modulus of the composite,

and

showed that

= (l__#p2

(20

Go 3.2 Changes in the ultimate properties The response of a polymer stretched competitive processes:

at constant

speed depends

upon two

(a) deformation of the chains, resulting in an increase of the stress; (b) continuous relaxation which tends to reduce this stress. The rupture of the sample (macroscopic rupture) is preceded by various disrupture and energy dissipation processes appearing in certain localised regions of the sample. The rupture process generally involves three steps: - initiation: characterised by the formation of a cavity or a crack that begins to increase in size; - slow propagation or growth of a crack until an instability is reached; - rupture: high speed propagation of the crack. In fact many microcracks develop within a stressed specimen, but is it customary to focus attention among them on a particular crack which becomes unstable and propagates. These effects are diflicult to observe; they happen on a microscopic scale. The modes of crack formation and the propagation of the crack are affected by the stress state and the dissipation rate of the stored elastic energy near the crack as well

252

as by the material characteristics. The growth rate may be affected by the formation and enlargement of other cracks; it can be reduced and even stopped. The same happens when heterogeneities are present; for example, a rubbery phase in a polymer in its glassy state stops the crack propagation. In the phase of the rupture process cracks may develop from induced stress, or sometimes preexisting cavities_ All materials contain, or develop under stress, heterogeneities that give regions of stress concentration. In brittle material cracks almost always grow from surface defects; in softer material crack growth can occur in regions of triaxial tension within the sample. The stress distribution in the vicinity of the cracks depends not only upon their size and the applied load but also upon their number and their distribution (homogeneous or localised). It must be said that cracks or microcavities, once formed, propagate always in a direction normal to the applied force. The crack will grow when the stress near the crack tip reaches the cohesive strength of the materials; generally slowly at the beginning but at a progressively increasing rate until a constant high velocity develops. Usually the transition between slow and high speed propagation occurs rather suddenly. The existing failure theories for rubbers are based on two different failure criteria: the critical energy criterion and the critical stress criterion. The first set of theories is based on the Griffith crack theory, which is based on the concept that the sample will fail when the increase in surface energy resulting from the growth of the crack is exactly equal to the !oss in stored elastic energy as the crack grows. For an ideally elastic body, it may be demonstrated according to the Griffith criterion that the tensile strength F, is related to the elastic modulus E, the specific surface energy y (energy required to obtain a unit area of new solid surface) and the crack length 2a by the equation: 112 2 EY F,= --(22) ar a ( 1 In reality our materials are not ideally elastic and the energy terms must be enlarged in their concept so as to include other energy demands that are involved during crack propagation. Since the materials (binders) used in the paint field are viscoelastic in nature, their properties will be time-temperature dependent, and in particular we must take account of the various time-dependent processes that reduce the stored elastic energy when a polymer is deformed. The Griffith criterion can be written in the form: WC=KG

in which W is the stored energy density, C the half-length of the crack, K a dimensionless geometrical factor and G the fracture surface energy. The catastrophic failure is considered to occur when

253 The factor K depends somewhat on the magnitude of the deformation near the crack, and G is again the energy required to create a unit area; it depends on thenature of the dissipative process that accompanies the crack growth. It ensues that G will be a function not only of the properties of the material but also of the stress state and the rate of crack growth. In other words G will also be a function of the experimental conditions, i.e. time- and temperature-dependent. On the other hand, the Bueche-Halpin rupture theory starts from a stress criterion. Crack propagation occurs when the stress at the crack tip is high enough to break chain bonds along the fracture plane. The rate of crack growth during the slow step is controlled by the crack tip. With the supplementary assumption that the crack in a sample submitted to a constant load grows at a constant rate until the criterion for high speed propagation is satisfied, Bueche-Halpin establish the foilowing semiquantitative relation: (24) in which cb(tb/aT) is the rupture stress when the time to break is &/a,, K is a constant close to unity, D(t,/qa,) is the creep compliance at time t,/a,, and aT is the time factor. The constant q is introduced to relate the time scale for macroscopic creep to the scale for the creep processes at the crack tip; q is several orders of magnitude greater than the unit. I’t follows that to obtain maximum fracture resistance, the structural properties of the materials must bc such that when cracks are created under the action of a stress, there immediately appear mechanisms that thwart their growth. One of these is due to structural heterogeneities such as fillers. These serve to blunt cracks or to dissipate the strain enetgay and hence postpone fracture to higher stresses and/or longer time. Bueche43V44 has given a picture of the reinforcement. Supposing that rupture occurs by successive bond rupture steps and that the rupture of a segment causes the rupture of an adjacent one, and so on. The presence of solid particles means as a consequence that the stresses are more uniformly redistributed among all the chain segments in so far as the polymer chains are tightly bonded onto the particles. 3.2.1 Iizjuence of the experimental conditions upon the t&mate properties We know that the ultimate properties of amorphous polymers depend on their nature (chemical and physical) and the experimental conditions. The shape of curves showing “ultimate properties” of detached pigmented films uerSuS either Ig T(absolute temperature) for a given strain rate or Ig strain rate for a given temperature suggest that, often, the time-temperature equivalence principle can be applied. Let us remember that time-temperature superposition is based on the fact that for many polymers a change in temperature produces the sam, p effect as a change in time scale or strain rate at a given temperature and that a decrease in temperature produces the same effects as an increase in strain rate. This principle has been well established by Williams, Landel and Ferry64 on the ground of the linear viscoelastic behaviour, the only one well known, and which can be represented mechanically by a Maxwell model

254 of springs and dashpots with the assumptions that the modulus of the springs is directly proportional to the absolute temperature (to satisfy the kinetic theory of rubberlike elasticity) and that all relaxation times have the same temperature dependance.

-c-.-c

Ti

tli

T,

rio

90

T

aT

since pi = ?i/Ei

The subscript “i ” refers to the i’th element, “0” to some reference temperature TO. ‘c is the relaxation time and q the viscosity. aT thus gives the temperature dependence of the viscosity q (and more generally of any viscoelastic property) or of the internal viscosity for crosslinked polymers: it is also the mobility of the polymeric segment at the reference temperature TOdivided by its mobility at temperature T. Experimentally, it was found that isothermal viscoelastic properties (modulus for example) versus time at different temperatures T’, T,, T2 . . . have a similar shape

when expressed in a semi-logarithmic (X-lg t) or logarithmic (lg X-lg t) diagram. When shifting all those curves but one (that at TO) along the log I scale in such a manner that portions of them all superpose, it was found that one single continuous curve, called the master curve, was obtained. The master curve thus obtained represents the viscoelastic behaviour of the material at temperature To over many decades of time and is expected to represent the complete isotherm covering the whole viscoelastic field of the material under examination. This experimental manner of proceeding in the construction of a master curve can be expressed in a mathematical form. Let us consider for example a set of isothermal relaxation curves Ig E,-ig t for tempe_rature To, Tl, . . . T; , To being the reference temperature. The time t required by the polymer to relax to a tied value of the mqdulus being only temperature dependent, one may write: lg tl = lg t,+lg

a(TJ

lg t2 = lg z,+lg

a(T2)

Ii f,,= Ig f,+lg

a(TJ

or, generally speaking, lg t = lg to+ lg a(T), or t = toaT. The curves being superposable, Ig aT is only dependent on T and TO. log aT is the numerical Vahe of the vector by which the curve at temperature To must be shifted to he superposable on the one at temperature T. aT, being the ratio of two times, is dimensionless and is called the time factor.

255 It has been shown by Williams, Landel and Ferry that for most amorphous polymers the experimental values for aT fitted well with the empirical equation known as the W.L.F. universal equation: lga(T)

=

-

CI(T-- T-J)

(25)

C2+ (T- To)

which applies to a temperature range between Tg and TB+ 100. The values C, and C, depend on the reference temperature; when T, = T’., Cl = 17.44 and C 2 = 51.6. In fact C1 and C, also depend slightly on the nature of the polymer. This leads us to a very important result known as the time temperature superposition principle, which can be expressed as follows: the viscoelastic propel ties of a material obtained at one temperature

may be transformed to another temperature by a single multiplication of the time scale. This principle has been shown to be applicable to any mechanical properties which are functions of both time and temperature; the W.L.F. equation empirically established has been the subject of much theoretical work and found to have a molecular basis. In particular Bueche, using the notion of jump frequency of chain

segments lgaT

and of free volume, =

----

B

2.3WK) in

showed that: T- To

KCJh + CT- TJ .

which B is a constant,_f(T’) the fractional free volume at reference temperature and 4 the expansion coefficient of the free volume.

(26) T,

Fig. 8. Plot of stresses acd strains at vari&s strain rates and temperatures for a polyurethane coating pigmented with TiOz (PVC 20%) (ref.:21).

256 In particular, as long as non-linear effects are not excessive and are confined largely to the crack tip region, the time temperature dependence of tensile strength and of ultimate elongation can often be reduced by the log scale shifting procedure_ Thus, data obtained at different extension rates and temperatures can be superposed to yield master curves that are functions of the temperature-reduced extension rate R.a, or the temperature-reduced time-to-break &Jar (Figs. 8, 9).

Fig. 9. Master curves from results plotted in Fig. 8, at temperature been determined experimentally.

T = To. The time factor cl= has

It must be emphasized that the reduction of the data works well only if no structural changes occur with time, temperatures, or extension and when all the relaxation times have the same temperature dependence. Under these conditions the time effects can be eliminated by eliminating the time scale itself. Thus by plotting reduced tensile strength versus ultimate strain on a log-log plot, Smith has developed the so-called failure envelope (Fig. IO). The fact that the experimentally determined time factors aT obtained by superposition of the ultimate properties satisfy the general W.L.F. equation allows us to conclude that the ultimate properties of the pigmented films (as do unpigmented films) vary because there is a change of the interna viscosity with temperature; the ultimate properties are dependent on the deformation rate only because the viscous resistance due to the chain rearrangement under the action of the applied force is proportional to their relative rate; it follows also that the rupture rate of the primary valence bonds and the dependence of this rate versus temperature is not the rate-controlling step

257

‘r j-

i-

Fig.

8.84

Butyl gum vulconizate (resin cured)

Temp OC

\-.. +t .d

140

em

100 70

t

40 25 IO -5 -20

+I P

-45 -35

/

-‘b

.34

% P

0

q

10.

in the rupture process. At small elongation rates, the chains of the macromolecular network are slowly solicited. The viscous forces should then be small. Moreover, the chains will at any instant be practically at their equilibrium elongation. The stress will be constant at small rates so far as the sample is at a temperature such that the chair,s or segments are free to move. If the rate increases, or if the temperature decreases, it will become more and more difficult for the chain segments to elongate as rapidly as the test requires and to be in equilibrium. An increase in the stress follows_ The W.L.F. relation is thus applicable at low and moderate rates only. Now according to Smith, the incorporation of fillers such as carbon black or silica modses the stress relaxation mechanism by reducing the number of short relaxation times and increasing the number of long ones, and also gives an increase in the modulus and activation energy for the relaxation process. This is reflected in a shift of position and a change in the shape of the middle part of the failure envelope. Mechanically, the Ciler particles alter or prevent the proliferation and growth rate of cracks and increase the enerm dissipation associated with crack growth. In the Bueche-Halpin theory, effects of fillers are seen in the parameter q, which is supposed to indicate the number of chains that rupture in the slow growth step of the crack. The authors suggest that this may indicate an increase in tortuosity of the crack because it must pass around the particles. It also seems likely that many short chain segments that bridge neighbouring particles may break, but the ab%,ty of the particles to redistribute the load over the

258

set of chain segments adsorbed to their surfaces tends to stabilise the local situation since the stress is more uniformly redistributed. The same phenomena also exist in two-phase polymers, one of which is in a glassy state, the other in a rubbery state, when there appears a strain-induced crystallisation. It must be said that sometimes fillers change the shift factor ur dependence on temperature from the W.L.F. type to the Arrhenius type. The reason for this behaviour is not well understood, since W.L.F. dependence occurs when viscosity is controlled by free volume and this changes with temperature. There appears no obvious reason

why fillers should prevent free volume in a material above its TBfrom being temperature dependent, although the free volume may undergo changes as we saw previously. Then we must admit that the temperature and tune dependence of the filled polymers is controlled by the viscoelastic characteristics (molecular motions) of the material in the vicinity of the particles rather than by an average viscoelastic response of the bulk sample. And it is logical to believe that the molecular motions are well hindered. It has been seen that an increase of 10°C in T’ might result in an increase in the modulus and the tensile strength and in a decrease in elasticity. 3.2.2 Whimate strengths or tensile strengths In the case of perfect adhesion between solid particles and the polymer, the tensile strength of rigid polymers which have Hookean stress-strain curves is given by: where E is the Young’s modulus of the filled polymer; the impact strength (area under the curve) is assumed to be a,~,/2 = E&g/2. The predicted results are illustrated in Fig. 10 on the basis of various theories for modulus. In any case theories predict a beneficial increase in tensile strength with PVC, at least from a given value of this latter, if good adhesion can be achieved between filler and polymer and if the solid particles are well dispersed without weak aggregates being formed. If there is no adhesion between polymer and Sllers, all the load must be supported by the polymer alone. Then in any cross-section the fraction occupied by the polymer is equal to the volume fraction of polymer, and it might be expected that the tensile strength cB would be equal to the product of the tensile strength ai of the unfilled polymer and the volume fraction $ of the polymer. The filler particles distorting the stress when the specimen breaks, the fracture travels from one particle to another. With the same model as previously mentioned, Nielsen34 predicts that (27) where S is a stress concentration function, which can have a maximum value of 1.0 when there is no stress concentration. However, in most cases S is expected to have a value of the order of 0.5. The tensile strength decreases rapidly as filler content increases. As stated by Nielsen, at volume fractions below 15% the no-adhesion curve, surprisingly enough, is higher than the curves for perfect adhesion. Undoubtedly this is due in part to the effect of stress concentration. If this latter could be evaluated, the curve for the case of no adhesion would be shifted to below the curve

a&

= (1 -+2’3) s

259

Fig. 11. Theoretical values of the relative tensile strength calculated from various theories against PVC. l-case of good adhesion (combination of eqn. (37) with eqn. (2) for modulus); a-same but using eqn. (18) instead of (2); 3-n. (27) with S = 1; d-eqn. (28); 5-eqn. (27) with S= 1 and corrected for 4 (4 is multiplied by 1.76); 6-eqn. (28) corrected for 4 (4 is multiplied by 1.76); 7-eqn. (27) with S = 0.5.

for perfect adhesion. The elongation to break and the impact strength can be calculated if eqn. (27) is incorporated in those of the modulus for Hookean polymers. Also in the case of no adhesion, Smith4’ proposes, on the other hand, the following expression: % = 0;;(1 - 1.21 qP3)

(28)

In the case of spherical particles, this becomes, taking account of the time-temperature superposition principle,

oB = (/IB h &+) (I-

1.21 $2’3)

(2%

where A and B are constants which depend on the reference temperature and the specific polymer. Smith shows that when dewetting occurs under stress, the Cller

260 only dilutes the binder and has no reinforcing effect. In that case the modulus after yield E, is suggested to be proportional to (&,-0.26). & being the volume fraction of the binder, the factor 0.26 is introduced, since if the spheres were monodisperse, the maximum volume fraction of pigment would be 0.74. The fact that E, is proportional to &- 0.26 means that E, = 0 when &, = 0.26, which seems reasonable. It follows that: E, = 1.35E,(+,-0.26)

(30)

Asbeck4’ proposes a simple hypothesis to explain changes in the ultimate properties of pigmented polymers. It includes the time factor, the effect that strain rate is known to have on the ultimate properties of polymers, and the notion of PVC. It is obvious that the addition of pigment to a coating is equivalent to diminishing the amount of resin contained in the composition, and that, consequently, the actual rate imposed upon the resin is increased signilicantly at a given testing rate as the PVC increases and approaches the CPVC. Making the assumption of a statistical distribution of the pigment and that the modulus of the pigment is much higher than that of the polymer, and consequently that all deformation takes place within the polymer phase, and taking account of a shape factor, he shows that the total tensile force at break may be considered to consist of two contributions consisting of (a) the area occupied within the tubular space between two pigment particles (as viewed from the direction of strain) x the strength due to the strain rate at these pointif A,, and (b) the area occupied by the remaining space x the strength of the unpigmented vehicle at the measured strain rate of the instrumentf(R,) A,, where A r and A, are the respective areas of high and low strain. Mathematically this is expressed as:

t~wc = f U-W-(gy'3+f(R,)-(l

-

c-y’3

(31)

R1 =d[(&y/3-l] At low PVC the first term is low, whereas at PVC approaching the CPVC it is the second term that vanishes. R, and R2 are detied in terms of resin properties to account for the changes in tensile properties at any PVC. By making a series of plausible assumptions, Bueche gives the following quantitative relationship between tensile strength G and times to break for polymers:

where K is the Boltzmann constant, Tis absolute temperature, V the volume of a polymer segment, I,+the jump rate of the segments of the molecule (number of times/set

261 a molecule where:

changes

its equilibrium

position

to a new one), i, the time to break,

and

In (2 F,2/7rc7’a4) F, is the load which each segment of the molecule can support and a is the diameter of the molecular segment. @ is practically a constant, so that the major changes in G come from the term involving tb. It follows that a plot of the tensile strength ZEXSUS lg t, (or the reciprocal of the strain rate l/R) will give a straight line with slope -KT/V. This line must deviate from linearity at extremely high rates due to the nonconstancy of @ and because the material behaves like a glass. One can also take lg R (equivalent to log l/t,,) instead of lg (l/R). The tensile strengths obtained at different temperatures and strain rates may be superposed when plotted as Ig [l/R *a,]. By making appropriate substitution in eqn. (31), Asbeck4’

showed that: (32)

where K1 = @, K2 = -KT/V, d is the jaws separation, and s is the speed of jaws separation. This Asbeck4’ formula shows that the true strain rate due to the presence of pigmentation can readily account for the increase in tensile strength observed. The values are solely dependent upon the binder properties and pigment content. Pigment particle sizes and shape are not taken into account. But an exact evaluation of the CPVC is required and that is another prob!em. It must be said that the proposed equation is valid only if the interfacial interaction binder/pigment is sufficient to prevent interfacial failure; fortunately this seems to happen in the great majority of cases for most coating compositions. One inconsistency appears in this equation, however, as pointed out by Asbeck himself, since it predicts an infinite value for G at the CPVC. Two factors mitigate against this situation. One is the fact that at extremely high strain rates or very low temperatures the W.L.F. equation is no longer applicable and that cold drawing of the polymers occurs and @ is not a true constant. The other is the fact that in practical packing of pigment particles, each particle does not necessarily touch all its nearest neighbours, so that zero distance of separation of the particles does not in fact occur. Thus a definite maximum of the tensile strength curve is found at the CPVC. Because of its surface tension, the vehicle occupies the spaces where the pigment particles have their closest approach, and if it is assumed that all the load is supported by the vehicle perpendicular to the direction of the strain, the tensile strength above the CPVC is to a first approximation given by: OPVC =

l

5

(

cpvc_cpvc _

213

>

’ %Pvc

(33)

262

Here the particle shape is expected to influence to some extent the value of the theoretical tensile strength. 3.2.3 Ultimate strains It is evident that strain magnification occurs when a sample of a deformable matrix containing essentially non-deformable particles is subjected to an elongation (or compression), since only the binder can deform. Thus, barring the formation of voids, the total sample deformation has to be accommodated by the binder alone, and hence the strain in the binder is larger than the overall strain. It is necessary to distinguish between the maximum strain which would prevail between the closest points of approach of two neighbouring particles and the mean strain which is obtained by averaging the strain over the entire range of particle separation distances. The former value will be relevant to failure properties, whereas the latter can be used for modulus, and energy storage. Although the particles are most likely to be randomly distributed in the binder, and may be aggregated, a number of authors have found more convenient theoretically to assume a particular spatial arrangement. Smith4’, using a model of spheres in a close-packed array, predicts a strain magnification: &‘I&.= l/(1 - l.105~“3)

(34)

where E is the measured overall strain AL/L,, AL is the change in length, L,, is the initial specimen length, E’ is the magnified strain in the binder. This equation gives the strain in *he binder along the line between centers of adjacent spheres, i.e. the maximum strain rather than the mean one. Thus for a given elongation E of the sample, the actual elongation E.’experienced by the binder must be greater. If one assumes that the polymer breaks at the same elongation at break in the filled system as in the bulk unfilled, the elongation at break of the filled syste& eB relative to the unfilled binder E: is: &a/&; f

l-

1.105f#P3

(35)

Bueche, using a model of cubical particles in a cubic array, finds that the maximum strain ma&cation

is given by:

&‘/E A l/(1 - $P3)

(36)

and hence &a/&i = (1 - 4,““)

(33

Nielscn34 proposed the same expression (eqn. 37) for the case of perfect adhesion betwren filler and polymer. Ziegel et aLs5 showed that the mean strain magnification, obtained when considering a model of spheres whose centers are randomly distributed, is given by: E’/&= l/(1 - cp) neglecting adsorption

(38) of polymer at the interphase.

263

This can be shown to hold not only for spheres but also for particles of any shape or orientation. In order to take into account the effect of bonded polymer Ziegel showed that eqn. (38) must be written: B’/E = l/[l -+(I

+Ar/R)3]

at the interface,

(39)

where Ar is the increase in particle radius R of the spheres.

It was found that (1 +Ar/R)3 eqn. (39) becomes:

is of the order of 1.75 for many systems, thus

&‘I&= I+ 1.75@+(l.75&)2+...

(40)

if expanded. The same equation must hold for yield strains. These equations are plotted in Fig. 12 for comparison. If the binder gives rise to microcracks, the elongation of the unfilled polymer is smaller than when filled. It is known that cracks propagate in a direction perpendicular to the stress and that a rubbery phase acts to stop their growth. Solid particles in a thermoplastic in its glassy state should have similar effects. The knee yield point which appears in the stress-strain curves should be an indication of the value initially required to induce or provoke microcrack formation. The linearity which often

10

20

30

40

50

60

70

I 80

(doIaw

Fig. 12. Theoretical values of the relative elongation at break from various equations against PVC. l-calculated values in the case of no adhesion by incorporation of eqn. (27) into the theory of Sato for modulus of fiUed polymers; 2-eqn. (37) (Bueche, Nielsen), case of perfect adhesion; 3-eqn. (35) (Smith); 4-eqn. (37) correctedfor Q (being m~~plied by 1.76); S-eqn. (35) corrected for 4 (being multiplied by 1.76).

264

0 Fig. 13.

appears after this knee may be explained if one supposes that formation and propagation of such defects arise in a similar manner as postulated by Halpin for the growth of microcracks in viscoelastic materials. If during their growth the microcracks encounter a solid occlusion to which the binder does not adhere very well, the rupture

of the interfacial bonds Gil set up ulterior propagation. With rigid spherical occlusions, the stress value at the knee of the curve diminishes linearly with temperature and is independent of the solid volumic fraction. At low temperatures, Le. in the glassy state, the materials behave in a brittle manner an6 there are no more yield points in the curves. This happens because the minimum value of the stress required for microcrack formation is nearly equal to the material’s resistance to fracture. The ultimate strains are small and may satisfy one of the above equations. On the other hand, if the binder gives rise to microcracks even in the absence of stress, the elongation at break is higher than given by the equations. It is therefore very difficult to predict the ultimate strain. It has been supposed in the above that interactions behveen pigment and binder are negligible. If strong interactions occur, the deformation mechanisms are more complex and are still under investigation. 4. THE

MULLINS

EFFECT

Rubbers containing reinforcing fillers sometimes display a curious effect which can give information on the reinforcing action of the filler: this is the Mullins effect. If a filled vulcanised rubber is slowly stretched for the first time, it gives a stressstrain curve somewhat like curve OABC (Fig. 13). If this rubber is first stretched to B and then allowed to complete recovery (relaxation), the rubber will follow the curve ODBC in the next stretching test. Although the curves do not differ very much at low elongation, the rubber has been softened by the first stretching process and the intermediate portion of the second stress-strain curve is not very different from what might be expected for an unfilled rubber. A possible explanation of this Mullins softening effect is that among the chains connecting the surfaces of two adjacent filler or pigment particles, some of these will be rather elongated while others will be very loosely coiled since they must conform to the gaussian distribution of chain and lengths. Obviously, then, when the specimen is stretched, those that are rather elongated must break at a relatively small elongation, but before breaking, these chains will hold an

265 enormous load and will give rise to a high modulus. On the second stretch, these chains, being broken, will not be holding any load and consequently the modulus will be lower. 5. MULTIAXL4

L TENSIONS

All that has been written above concerning

the ultimate properties

has been derived from uniaxial experiments. Obviously, a paint on its substrate is subjected to multiaxial tensions and at least biaxial. Unfortunately very little experimental and theoretical work exists for this case. It is difficult, therefore, to draw general conclusions about the dependence of fracture processes on the state of combined stress. Nevertheless it seems that with rubber gum vulcanisates under triaxial tension, the internal fracture occurs at a critical

stress that increases linearly with Young’s modulus. Dickie and Smith46 determined the ultimate properties of an SBR vulcanisate in equibiaxial tension (EBT) with circular sheets of rubber inflated with gas pressure. Quite surprisingly, the ultimate elongation in equibiaxial test is considerably greater than in simple tension (ST) and moreover is essentially independent of extension rate and temperature. In addition, the tensile strength in EBT is smaller than in ST and is also considerably less dependent on extension rate and temperature, although intuitively these two quantities would be expected to be less in EBT than in ST. This behaviour may be due to the fact that, although cavities formed or enlarged significantly at smaller deformation in EBT than in ST, it is difficult for a crack to propagate. According to Smith, there is no doubt that such stability results because a crack tends to propagate perpendicularly to the maximum principal stress. As there is no preferred direction for crack growth in a specimen under EBT, a cavity is probably relatively stable. However, it is found that under biaxial tension conditions that yield a pure shear deformation, fracture appears to develop in the same manner as in simple tension.

But, in general, due to the complex possible modes of crack growth which are influenced by the nature and the state of the macromolecule and the type of imposed stress, it is difficult if not impossible to predict the conditions for fracture under multiaxially applied loads from the knowledge of the condition in fracture in simple tension. CONCLUSIONS

While apparently quite theoretical, these considerations on the effect of the pigmentation of a binder have considerably technological importances, particularly in the assessment of adherence and impact strength as well as other properties already mentioned. We all know that adherence, whatever the method used for its measurement, as well as the impact resistance, depend on the viscoelastic response of the

266 material under test, and we have just seen that these properties are not only a function of temperature and rate of test but also of the pigmentation level and nature. It seems at first sight that a correlation does exist between tensile strength and falling weight energy absorption whatever theoretical sigmficance the correlated factors, if they are of practical utility, seem to have for the moment. It seems logica! to attribute the increase in tensile strength of a pigmented coating with increasing PVC (below the CPVC) largely to the changes resulting from the actual strain imposed on the polymer matrix due to increased pigment loading. Perhaps eqn. (32), which shows that the magnitude of the tensile strength strongly depends on the speed of testing and test temperature on one hand, and on clearly definable parameters like PVC and CPVC on the other hand, gives the most satisfactory description of these interdependencies. With these defined parameters we implicitly take into account the pigment characteristics: particle size, size distribution, as well as degree of dispersion and true volume of the pigment (pigment + absorbed layer). Any deviation from the predicted values could be related to some modification in the composite characteristics, for example, the occurrence of pigment de-wetting (rupture of bonds between pigments and binder) under the influence of stresses, existence of f?hn defects (voids) etc. From a more theoretical point of view, the study of the ultimate properties of filled binders should allow the characterization of the property often called polymeriiller interaction, especially if tests are carried out at different temperatures and rates; the determination of the nature of the inter-facial forces and the thickness of the polymer absorbed layers. In other words, it should allow the characterization of the interphase pigment-binder, the properties and dimensions of which affect the composite properties. REFERENCES 1 G. Calbeck, Ind. Eng. Chem., 18 (1925) 1220. 2 W. K. Asbeck, Ind. Eng. Chem., 41 (1949) 1470,47 (1955) 1472. 3 W. Bosch, Ofl Dig., 26 (1954) 1291. 4 P. Berardi, Point Technol., 27 (I 963) 1219. 5 D. S. Newton, J. Oil Colour Chemists’ ASSOC., 45 (1962) 180. 6 P. Nylen and E. Sunderland, Modern Surface Coatings, Interscience, New York, 1965, p. 382. 7 F. B. Stieg, J. Paint Technol., 39 (1967) 701; 41 (1969) 243. 8 W. K. Asbeck, M. Van Loo and D. D. Laiderman, Ofi Dig., 24 (1952) 156. 9 W. B. Wiegand, Ind. Eng. Ckem., 17 (1925) 939. 10 A. C. Elm, Ofi Dig., 25 (1953) 751. 11 W. Bosch, Ofl Dig., 27 (1955) 999. 12 J. C. Becker and D. D. Howell, Off. Dig., 28 (1956) 775. 13 P. Bernardi, 1. Paint Technaf., 27 (7) (1963) 24. 14 T. F. Mika, Ofi Dig., 31 (1959) 521. 15 H. L. Jaffe and J. H. Fickensher, Ofi Dig. Fed. Sot. Paint Technol., (33) (1961) 331-344. 16 U. Zorll, Proc. ZXth Congr. FATIPEC, Brussels, 1968, Section 3; Farbe Lack, 73 (1967) 200. 17 H. A. Oosterhof, 3. Oil Colour Chemists’ Assoc., 48 (1965) 526. 18 N. P. Suryanarama, Paint Man& 32 (1968) 26. 19 E. J. Schaller, J. Paint Technol., 40 (1968) 433. 20 I. Vansco and E. Vos, &~mro_@-e,58 (12) (1968) 970.

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