Progress
in Organic
Coatings,
21 (1993) 387403
387
Pigment inhomogeneity and void formation in organic coatings R. S. Fishman and D. A. Kurtxe Physics
Department,
North Dakota
State
University,
Fargo, ND581055566 (USA)
G. P. Bierwagen Department of Polymers and Coatings, Fargo, ND581055566 (USA]
North Dakota
State
University,
(Eeceived August 4, 1992)
Abstract The conventional theory of organic coatings assumes that the volume density of pigment particles is uniform throughout the sample. The coating is then described in terms of the pigment volume concentration .P, which equals the volume of pigment divided by the volume of pigment and polymer. Voids form in the coating when the pigment particles are randomly densepacked, which implies that 9 exceeds the critical pigment volume concentration @= or that A
[email protected]=> 1. Due to fluctuations in the local density of pigment however, some regions in the coating may become randomly densepacked even below Qc. Hence, voids may form in the denselypacked islands even when A< 1. Our model for void formation contains two fitting parameters: N,, is the smallest number of pigment particles in a denselypacked cluster that may contain a void; C, is the coarseness of the polymer spacefilling in the volume of the sample not occupied by pigment. When C, = 0 and A < 1, the polymer completely fills the interstitial volume and the void concentration vanishes. But when C, > 0, voids may form in the denselypacked regions of the coating even below Gc. The coarseness parameter C, depends on the sample preparation, on the properties of the pigment and polymer, and on the pigment volume concentration 9. For any nonzero C,, we conclude that optical measurements will systematically underestimate Qc. On the other hand, since the density of polymer is less than half the density of pigment, the peak in the mass density p(.F) of the coating will overestimate @=. Unless C, is abnormally large, the void percolation threshold value QV is larger than @= and is a decreasing function of the coarseness C,. The predictions of this simple model are in good agreement with experiment, and are relevant to the general class of random concentrated composites which include organic coatings, ceramic polymer slips, composite solid polymer electrodes and some forms of battery separators.
1. Introduction
Many properties of an organic coating change when the concentration of pigment particles exceeds the critical pigment volume concentration (CPVC) [ 1, 21. Above the CPVC, the pigment particles have reached their random, densest packing [3, 41 and voids form between the particles. The presence of voids profoundly changes the properties of the coating. For example, voids alter the blistering, porosity and gloss of the coating [ 1, 2, 51. Although these properties are expected to change only at a,, experimentalists actually 00330655/93/$6.00
0 1993  Elsevier Sequoia. All rights reserved
388
observe a more gradual transformation in the coating’s properties starting bslow the CD,value [l, 61. To explain this behavior, we have studied the effects of density fluctuations on the formation of voids. The global theory of void formation assumes that the volume fraction of pigment particles (pl is tmiform throughout the coating. If qz is the volume fraction of polymer, then (1) (Px+‘pz
defines the pigment volume concentration. (Below the CPVC, 9= pr, but above the CPVC, this is no longer true  see eqn. (2) below.) Because the pigments are randomly packed, the maximum value of ‘pl is somewhat lower than the packing fraction for a closepacked lattice. It is well known [3, 71 that the random densestpacking fraction for monosized spheres is 0.64. For a distribution [4] of pigment sizes, with a possible absorbed layer 121 of polymer, the random densestpacking value @, may be larger or smaller than 0.64 f7J. When A=$Y/ajc < 1, the pigment particles are not densely packed and the void volume fraction 93 vanishes. But when A > 1, there is not enough polymer to fill all the interstitial regions between the denselypacked pigment particles. Therefore, the void density 4p3becomes nonzero only when 9 exceeds @, which is also called the CPVC. Above the CPVC, the pigment particles remain densely packed and ‘pl = @ is a constant. If we let cp3be the void volume fraction, then [8], ~lf~2+(P3=1
@>
and the void density above the CPVC is given by qP3=1 $;
[email protected]>QG
(3)
So for 9 > @c or A > 1, the polymer concentration decreases and the void concentration increases but their sum remains constant at (1  @,). Of course, 503=0 when S>@, or A>l. Both the optical and transport properties of a coating depend sensitively on the void concentration [9]. The global model correctly predicts that those properties change above ds,. However, this model also has several shortcomings. First, it fails to explain why many properties of a coating do not change discont~uously at aS,but rather change more gradually [ 1 I, starting below f5] @,. Similarly, the global model cannot explain why the slopes of the plot of void volume fraction and of the coating’s mass density p versus 9 change continuously at &. Second, the global model of void formation fails to explain why different properties of the coating seem to have different critical values [lo] of 9. In particular, transport properties such as porosity seem to have a different critical value of 9 than optical properties such as gloss. Third, the global model cannot explain why the properties of a coating, including measurement of the CPVC, depend so sensitively on sample preparation, film formation and ~eatment [S, 1113 J.
389
In this paper we propose that local fluctuations in the density of pigment particles have a profound effect on the properties of the coating. Because the pigment particles are distributed randomly, the local value of the pigment volume fraction cpr may vary from region to region. Therefore, some local regions in the coating may become densely packed before the sample as a whole has exceeded the CPVC. Because the polymer spaceWing is disrupted by the fluctuations in the pigment density, the polymer may not completely fill the interstitial volume in the denselypacked islands of the coating. Hence, voids may form in the denselypacked islands even when 9 < @, (both global values). As a result, the physical properties of the coating affected by void formation will change gradually below the CPVC, rather than with the abruptness required by the global theory. Clearly, the formation of voids in a coating is a very complicated process. As the coating is prepared, the polymer may be insuiiicient or too viscous to penetrate regions with a large density of pigment. In the denselypacked islands of pigment in the coating, there may be insufllcient polymer locally to illl the interstitial volume. Hence, the void concentration depends on the size distribution and masses of the pigment particles, on the nature of the polymer, and on exactly how the coating ilhn is applied. Our model contains two phenomenological parameters which govern the formation of voids in the coating. The volume fraction of c of denselypacked islands depends on the volume V,, of the smallest island which may contain a void. As we show below, our conclusions are relatively insensitive to the precise value of N,,. For a more complete discussion of this issue, see Fishman et al. [14]. The second variable, C,, describes the fluctuations of cpzin the composite system. When C,= 0, the polymer completely illls the volume not occupied by pigment, and the global and local q2 values are everywhere equal. In this limit, the global film description applies and (p3=0 below @,. But when C, is nonzero, the polymer does not completely fill all the interstitial volume in the denselypacked islands of pigment. This implies that local values of A exceed 1 because of the local variations in ‘pl and rp2.The parameter C, depends on the size distribution and masses of the pigment, as well as on sample preparation (especially degree of pigment dispersion in the coating [ 11 I), conditions of film formation [ 81, and the form of the polymer in the ilhn [ 12 ]. If the pigment particles are well dispersed and nonflocculated, or if the coating is calendered or annealed after original film formation, then C, may be negligible [ 10 1. But if the pigment particles are poorly dispersed or flocculated, the coarseness of the polymer spacefilling may be appreciable and voids will form below the CPVC. Further, with highly viscous polymer systems, dispersion polymers, very rapid solvent evaporation or poor paint preparation, intimate contact between the film matrix polymer and pigment may fail to occur and voids may form below 9 = Gc. The theory proposed here assumes that voids only appear in the regions of the coating where A locally exceeds 1 and that the pigment is random densely packed in these regions. With perfect film formation [6, 11, 121,
voids will not appear outside of the denselypacked pigment regions. In future work, we may consider the formation of voids in latex coatings due to imperfect 6lm formation caused by imperfect latex particle coalescence. In this paper, however, we treat latex and solution polymer coatings as equivalent in terms of the 6lrt1 quality, even though there are obvious differences in terms of the form of the polymer. Different kinds of coatings will only be distinguished by different phenomenological parameters C, and iV,, which govern the formation of voids in the denselypacked islands. Of course, if voids do appear in the sparcelypacked regions of a coating, then our model would have to be modifled. The goal of this work is to predict the volume fraction of voids (p3 as a function 9, @,, N,,, and C,. Our model is described in Section 2. In Section 3, we discuss the results of this model for optical properties and the mass density. Then, in Section 4, we calculate the threshold for voids to percolate through the coating. This is not to be confused with the volume spanning contact percolation threshold for pigment particles, which occurs at [ 151 ‘pr 2: 0.15. This latter is the conductivity percolation threshold for metallic or conductive particles in a low conductivity matrix [IS]. We are addressing here the spanning connectivity of the voids themselves in the coating, an issue of obvious importance for the transport properties of the coating. FinaIly, Section 5 contains our conclusions and recommendations for further work.
2. Formalism
The local fluctuations of the pigment and polymer packing can be described mathematically by generalizing the work of Lu and Torquato [ 171 who studied density fluctuations in a medium with two phases. In organic coatings, three different phases are present: pigment, polymer and void. Following Torquato and Lu, we define the quantities Q(X) to be the local volume fraction of pigment (i = l), polymer (i = 2) or void (i = 3) in a sphere of volume V, centered around X. For any observation volume V,, and any position X, the sum of the #) terms equals 1: 71(X) + 72(X) +
73(x)
=
1
(4)
As V, increases, the local variables 7&) tend to their bulk values cpi.Because
the coating contains three rather than two phases, the pigment and polymer densities may both fluctuate simultaneously. In terms of the local densities, the local value of 9 is defined by 9(x)
=
71(x) c(x) + %(X1
(5)
Even if 9 < Gc globally, 9(x) may exceed @, in local regions of the coating. If both 9(r) > @, and am = Gc, then the CPVC is locally exceeded and the pigment particles are locally densepacked. Because of the ubiquitous nature
391
of random dense packing in composites and dispersions [ 1S], the driving forces for random dense packing seem to occur independently of the local polymer concentration, Q(X). Then the local volume fraction of voids T&X) is given by:
(6) We emphasize that voids can form only in locally densepacked islands where T*(X)= @=. In Fig. 1, we schematically represent a coating which is below the CPVC but contains a denselypacked cluster. As shown, the denselypacked cluster with 9(x) > @, contains a void. Of course, a realistic coating will contain many such denselypacked clusters even below the CPVC, especially considering the statistics of site occupation in coatings [ 191. The predictions of this model depend sensitively on the observation volume V,,. In the limit as V, ) ~0,9(x) + 9 and voids would not appear below the CPVC. Voids are possible below the CPVC because small regions in the sample may become denselypacked before ‘pl reaches its maximum value of @,. Physically, the observation volume should be large enough to include many pigment particles and allow the development of random dense packing, but small enough to include only a single denselypacked cluster. In many other systems, the averaging volwne also plays a crucial role. For example, the energies of the molecules in a gas fluctuate on a microscopic scale. Properties like temperature and pressure can only be formulated by averaging over large observation volumes containing many molecules. If a temperature gradient is applied across a sample, then a local value of the temperature T(X) can be dehned by averaging over a volume large enough to contain many molecules, but small enough to reflect the changes in temperature for different regions. When the temperature gradient is removed, the temperature across the sample will become uniform due to the thermal motion of molecules from one region to another. In organic coatings, the
Fig. 1. A schematic representation of a coating with 9 < c?$ which contains a denselypacked cluster with Y”(x)> GC.In this denselypacked cluster, which is shown to contain a void, the polymer molecules are represented by squiggly lines.
392
fluctuations in the local pigment and polymer densities are frozen in when the sample is prepared. But annealing the coating may reduce the fluctuations in the pigment and polymer densities due to the movement of pigment and polymer particles between different regions. Because the pigment packing is random [ 201, the volume fraction D of denselypacked islands is nonzero for any nonzero value of ‘pi. If every pigment particle has the same volume VP (i.e., the pigment particle size distribution is monodispeme), then V, may contain as many as N,, = VJV, randomly densepacked particles. The probability that V, is densely packed is that same as the probability that each of the N, available sites in V,, is occupied by a pigment particle. But the probability of one site being occupied is N,/N, where Ni is the total number of pigment particles and N is the number of particles required to densely pack the whole sample of volume V. Since the volume actually occupied by pigment is ql V and the maximum volume which can be occupied is GCV, we conclude that Ni = qplV/V, and N= QCV/Vp. Thus, the probability that a single site in V, is occupied is
[email protected], and the probability that all N, sites in V, are occupied is (rpi/@p. If the average size of a denselypacked island is much larger than V,, then the volume fraction of denselypacked islands is also given by
p=
[email protected] I
\ N,
Pl
(7)
t
If cpl< Gc, the homogeneous limit is recovered when N,, ~0. But if qp,= Qc, a= 1 is independent of the observation volume V,. This argument can be readily extended to treat a distribution of pigment sizes. For example, suppose that pigments come in two sizes, with particle volumes VP(‘) and VP(2). Also assume that the denselypacked observation volume contains N,(i) and N,@) of each type. If the distribution of sizes in V, is the same as in V, then the probability that the Noto vacancies in V,, are occupied is (~,/@~)N”“‘. Therefore, the probability that the observation volume is densely packed by the two types of pigment particles is +~l)
[email protected])Z($
(8)
where N,=x
i
N,(i)
is the total number of pigment particles that can be densely packed into V,,. Clearly, this argument can be extended to handle any distribution of pigment sizes. Therefore, the volume jimtion of denselypachzd islands depends mlv on cpl, @,, ad NO. However, this argument assumes that the observation region is sufficiently large that the distribution of pigment sizes in every denselypacked volume V, is the same. More generally, we might expect that the composition of the denselypacked regions will vary throughout the sample. In that case,
393
both N, and @= will become functions of position. While that possibility may be examined in future work, we assume for the remainder of this paper that N, and @= are constant throughout the coating. If there are also local versus global differences in the individual concentrations of pigments, the arguments just given would have to be modified to include local versus global @= values, because the @=values are tiected by particle size and volume concentrations of the ~~~du~ packing entities 171. However, the dependence of @c values on ~omposi~on is weak, and to ilrst order we will ignore this effect relative to the much stronger pz fluctuations which can occur. Thus, the fluctuations in the cpivalues are at least an order of magnitude greater than the fluctuations in &. We further assume that the average volume of a denselypacked island V, is much larger than the observation region V,. If V+ is of the same order as VO, then b will underestimate the true volume fraction of denselypacked pigment. So our result for P should become more accurate in the vicinity of the CPVC, where large portions of the sample are randomly densepacked. Also near the CPVC, c does not depend very sensitively on the number N,, of particles which can be packed into V,. Since the coordination number for a spherical particle in a randomly densepacked cluster of identical particles is close [20] to 8, N,,= 9 for a coating of monosized pigments. But if the size distribution of pigment particles is broad, then N, may be quite a bit larger than 9. For example, imagine again that the coating contains only two sizes of pigment particles with size ratio 17=V (‘)/V (‘I. As q increases, the smaller pigments will become more effective in ia&& the interstitial volume of the larger particles. Therefore, the number of pigments N, in a denselypacked volume V,, will increase as 77 increases. More generally, N, may be regarded as a fitting parameter which describes the size distribution of the pigment particles. If the size distribution of pigments is fixed, then N, must be independent of B and the density of voids in the sample. So for a fixed type of pigment, N, can be regarded as a constant. In the rest of this paper, N,, is set at 10. Unlike the local fluctuations in the pigment density, which are determined entirely by the statistics of hardcore particles [ 211, the local fluctuations in the polymer volume fraction depend on many variables, including the sizes and masses of the pigment particles, the type of polymer, wetting of the pigment by the polymer, polymer viscosity and sample preparation. Because it is difficult to describe the fluctuations in the polymer density with a microscopic theory, we prefer to describe the coarseness of the polymer spacefilling with a phenomenological parameter. To parametrize the coarseness of the polymer spacefilling, we separate the total volume V of the ssmple into two regions: the denselypacked islands occupy region b; the rest of the coating is labeled as region a. Similarly, the volume V’ = (1  cpl)V not occupied by pigment is also separated into regions a and b. At any point x in V’, the volume fraction of polymer is given by:
Since Q(X) vanishes in region a but is nonzero in region b, q(x) = 1 in region a but q(x) < 1 in region b. Hence, the fluctuations in the local polymer density q(x) are a measure of the void density p3. The coarseness [ 171 of the polymer spacefilling can be parametrized by:
c,= $
(11)
where the standard deviation of q(x)
is defined by
(12)
a: = (q2>  W2
The parameter C, measures the local fluctuations of the polymer density in the volume V’ not occupied by pigment. When a coating is annealed or calendered [ 111, C, tends to zero and (p3 is nonnegligible only for 9> @,. But if the coarseness C, is nonzero, voids will form even when 9 < @=. If the volume fraction of void is (p3, then the values of q(x) in regions a and b are (13)
4a(x) = I and
q,(x) =
u(
[email protected])G(
[email protected]
(14)
c
Of the volume V’ = (1  cp,)V, region b occupies the volume ~(1  @dV while region a occupies the rest. So the expectation value of q is given by: lcp10(
[email protected]
(a> =
(15)
Similarly, the expectation value of q2 is given by:
(q2>= =p
lq1a(
[email protected],)
4:+
I9, 1
1
(1  Vbl)a(I  @J
a(I  @J 4: 1R
{(I  CPA0  R)  .%3”(I  @J + &I
Expressing the coarseness C, in terms of D, 9, 19(1~3)u(
[email protected]
cq=
(1 rp3$
9)
(
@(I  W
@,, and (p3, we find
1’2 1
(16)
(17)
Together with eqn. (7) for a, eqn. (17) defines our model. If the polymer was able to completely fill the interstitial volume between the pigment particles and completely wet all particles in the composite film,
then C, would vanish and voids could not form below the CPVC. But the ~uctuatio~ in the pigment packing may induce ~u~~atio~ in the underlying polymer matrix. The coarseness of the polymer spacefilling may depend on the relative mass densities of the pigment and polymer, on the size distribution of pigment particles and on sample preparation. In particular, a sample that has been annealed or calendered may have fewer voids below the CPVC than an untreated sample. Also, larger and more massive pigment particles may retard the flow of the polymer in film formation and thus generate voids. Because of the many and complex mechanisms which may cause fluctuations in the polymer density, C, depends very sensitively on the material properties and preparation of the coating. Indeed, C, must be a function of the pigment volume concentration 9. In the limit 9+ 0, the pigment concentration vanishes and C, must equal zero. Similarly, C, must also tend to zero as 9 3 1, when the polymer ~oncen~tion vanishes. A microscopic theory of void formation would specify N,, and the functional form of C,. Hopefully, future work may enable us to calculate these functions a priori. But even in the absence of a microscopic theory for C, and N,, the model developed in this section allows us to reach some useful results, which are discussed in the following two sections. In the particle packing in coatings, the pigment particles pack with an adsorbed layer of polymer. Often, (ol is defined to exclude the adsorbed polymer density, which is included in y,a. However, the adsorbed layer of polymer is not subject to the same ~u~ua~o~ as the interstitial polymer. Except in very coarse coatings f 18 ], the adsorbed polymer remains bound to the pigment particles. Therefore, we define ‘pl to include the adsorbed layer. It is straightforward to modify our results to exclude the adsorbed layer of polymer.
3. Optical properties and mass density Using the model developed in the previous section, we have studied the effects of fluctuations on the optical properties and mass density of a coating both below and above the CPVC. To present our results, we have taken (p, =0.6667 and Nb = 10. For comparison, for monodispersed spheres, at, = 0.64 [ 141. But other values of @, and N,, can also be used to demonst~te the results of our model. Because of their complexity, eqns. (7) and (17) cannot be solved analytically for’ the void fraction rp3 as a function of 9. However, these equations are quite simple to solve numerically. In Fig. 2, we plot (p3versus 9 for four values of the coarseness. In the homogeneous limit when C, =~i 0, the void density vanishes below the CPVC. Regardless of the coarseness, 50~ reaches 1  @== 0.333 when 9 = 1. At this point, the polymer density 402 vanishes and the voids completely occupy the interstitial volume. When the coarseness is nonzero, voids may form below the CPVC. As expected, rp3 increases as the coarseness C, of the polymer spacefiIli.ng increases. Also,
396
Fig. 2. The volume density of void cp3versus 9 for Gc=0.6667, 0.05 (long dash), 0.10 (medium dash) or 0.20 (short dash).
1.0
N,,= 10 and C,=O (solid),
,A /
’ / I ;#
0.8 
0.6

F3 0.4 
0.2 
O%.o
0.;
Fig. 3. The volume fraction of denselypacked islands u versus 9 asinFxg.2.
for the same parameters
there is now a gradual, rather than abrupt, onset of void formation in the film, more typical of most experimental observations. The effect of fluctuations on the local densepacking of the coating is shown in Fig. 3 for C. Even in the homogeneous limit when C,= 0, P is nonzero below the CPVC. Setting C, = 0 in eqn. (17), we find:
(18)
397
while @= 1 when 9 > a,. Because the void density (p3 increases with C,, rpl=E9( 1  pa) must decrease with increasing C,. Hence, the volume fraction u of denselypacked islands actually decreases as the coarseness of the polymer increases. For any nonzero C,, the coating is completely densepacked with *= 1 only when 9 = 1. Hence, there are alwuys regions in the ma&g which are not beak packed, ewn above the CPVC The volume fraction of pigment cpl is plotted in Fig. 4. In the homogeneous limit, q1 = 5a below @$ and ql = @, for 9> (ip,. As expected, for a fixed value of 9, the pigment vohnne fraction decreases as the coarseness C, increases. However, the dependence of ‘pi on C, is actually much weaker than the dependence of fi on C,. Our model can also be used to predict the effect of fluctuations on the mass density p of the sample. If the mass density of pigment is p1 and the mass density of polymer is p2, then the total density of the coating is: WV
P=PIQl+P2Q2 ignoring
as
negligible the density of the air that fills the voids. Hence, p/p1
is given by P Pl
Gw
==Q1+PQ2
where y =p2/pI. Using p =0.3, i.e. a film pigmented mainly with TiOz, this scion is plotted in Fig. 5 for the same values of the coarseness used in Figs. 2, 3 and 4. In the homogeneous limit, we find that
fL s+jL(l9q; Pl
s
cm
and 0.0
r
0.6
<
0.4
0.2
L
0a.O Fig. 4. The volume density of pigment q, versus 9 for the same parameters as in Fig. 2.
398
E‘ig.5.  The _mass density p/p1 versus 9 for p=pz/p, 0.3. ia other as
P

Pl
in Fig.
are the same
2.
[email protected][email protected] $1 (
; 1
s>d&
(22)
So when C, =0, p([email protected]) peaks at @= and the slope dp/@ is ~cont~uous at cp,* When the coarseness is nonzero, however, the slope of p(p) becomes continuous across di, and the peak in p(p) is shifted from dp,.Figure 5 also indicates that the maximum value of p(g) which is achieved in the homogeneous theory is never reached when the coarseness is nonzero. In fact, the experimental observation [22] that the slope of the mass density is continuous across G$ proves that the local pigment and polymer densities must fhrctuate in such coatings. When p < 112, the mass density peaks above ec. But when p > 112, the mass density peaks below the CPVC. The massdensity peaks at Cp,independent of C, only when ,u= l/2. Since p1 is usually two to four times larger than pz, the peak in the mass density will overestimate the CPVC. On the other hand, Fig. 2 indicates that the optical properties of a coating begin to change below a,. So estimates of @, based on measurements of hiding power will underestimate the CPVC. While the void vohune fraction sp3below @c depends quite sensitively on the parameter N,, the mass density p is relatively insensitive to changes in N,. For example, with N,= 10, ~=0.3 and C,=O.l, the mass density p/ p1 peaks at 9= 0.698 with a value of 0.7408. On the other hand, if N, is increased to 20, then p/p1 peaks at a value of 0.7455 when 9 = 0.689. Unlike the mass density, which depends only indirectly on the density of voids and which peaks above the CPVC, the optical properties of the coating below the CPVC depend rather sensitively on N,.
4. Transport properties Other estimates of the CPVC are based on transport properties, such as porosity. A coating becomes porous when voids on the surface of the coating admit a liquid such as mercury. The threshold value of 9 above which the coating becomes porous has been dubbed the point of critical surface porosity (CSP) [ lo]. The void percolation threshold [ 14, 151, on the other hand, is the point at which chains of voids will span the coating from one end to another. While the CSP may not exactly coincide with the void percolation threshold, they are probably very close. Although some experiments indicate that the CSP is larger [lo] than @_ other work suggests that the CSP may be smaller [23] than Gc. Two conditions are required for the percolation of voids through the coating. First, the denselypacked islands themselves must percolate, so that a chain of denselypacked islands spans the coating. Second, the voids must percolate so that they form continuous chains across each island. If one condition is met without the other, then voids will not percolate across the sample. In the global theory, voids only appear above GJ~.Therefore, the void percolation threshold @” must exceed @=. However, fluctuations in the density of pigment and polymer may lower the void percolation threshold @“. Of course, the pigment particles percolate across the coating much below @Jo,,as discussed above. Regardless of the shape of the percolating objects, the percolation (volume spanning) threshold in three dimensions [ 14, 151 is approximately given by cp,= 0.15. This implies, for example, that the pigment particles will percolate [ 241 across the coating when ‘pl > rp,.The denselypacked islands will percolate when (23)
“‘% or when 49 > @&JYNo
(24)
Since VP is the volwne occupied by the islands and tp3Vis the volume occupied by the voids inside the islands, the voids within each island wilI percolate when
If C, = 0, then (p3= 0 below Gc and c = 1 above @,. So in the homogeneous limit, the void percolation threshold is given by q$=
2
Qi
l4%
(26)
Since @” is larger than @,, the i.slands percolate befwe the voids. If rp,= 0.15 and Qc=0.6667, then @“= 0.784. As the coarseness increases, @” decreases
15 .2
Fig.6. The voidpercolation threshold Cp,versus coarseness C.+for~~==O.SSS? and N,,10 (solid), 20 (dash) or 40 (short dash).
gradually, as shown in Fig. 6. But for values of C, less than 0.25, @” remains quite a bit larger than @,. For values of the coarseness smaller than 0.25, the void percolation threshold always exceeds the CPVC. A&bough it seems highly unlikely, our model cannot rule out the possibly that very coarse coatings may percolate below the CPVC, if C, z+0.25. In Fig. 6, we also display the dependence of @, on the parameter N,. Like the peak in the mass density discussed in the previous section, the void percolation threshold above the CPVC is relatively insensitive to changes in N,. As shown, Zp, increases only slightly when N,, increases from 10 to 40. In the lit N, + 03, cPvtends towards the value of eqn. (277, independent of C,. The most extensive experimental me~~ernen~ of the CSP have been made by Ramig and Floyd [ 10 ] on films containing plastic pigment particles. These authors flnd that the CSP is usually CCL.lO20% larger than ec. Identifying the point of critical surface porosity with the percolation threshold, we would find that the results of Ramig and Floyd agree with our conclusions. Inuigumgly, they also found that the CSP increases as the size distribution of the pigment particles becomes broader, in agreement with our prediction that a?, increases with N,. However, for very broad distributions of particles, they found that the CSP decreases with the width of the d~~bution. For very broad distributions of pigment particles, the smallest randomlypacked cluster which contains voids may not contain a representative number of particles of every size. Hence, N,, may be a nonmonotonic function of the width of the particle distribution. Ramig and Floyd [ 101 also found that the CSP is a nonmonotonic function of the pigment size, with a mlnlmum at a characteristic size of 3.5 run. This result suggests #at the coarseness of the polymer spacefilling may
401
also be a nonrnono~~~ function of the pigment sizes. Further work is required to compare the experimental results with the theoretical predictions for the coarseness C,. Rasenberg and Huisman [ 111 have found that the CSP of magnetic tape coatings is increased by calendering. Because the calendered coating probably has a lower value of the coarseness C,, their result agrees with the conclusions of thii section. Rasenberg and Huisman also found that the calendered coating had the same value for @= as the engendered coating, which suggests that the size d~t~bution of pigment particles is not altered by the calendering process. 6. Conclusions
In this paper, we have developed a theory for the formation of voids in organic coatings. We find that fluctuations in the pigment density allow voids to form below the CPVC of the coating as a whole. Due to mutation, some portions of the coating may become denselypacked even when 9 < @=. Because of the presence of voids below Gc, the optical properties of the coating begin to change gradually below the CPVC, rather than suddenly at the CPVC as predicted by the global theory of void formation. On the other hand, the mass density will peak above the CPVC when p2/pI < l/2. So the conventional techniques for measuring (p, may systematically underestimate or overestimate the point of critical pigment volume concentration. We have also shown that the void percolation threshold Ca,is probably above Gc. Because the void concentration increases with the coarseness parameter C,, fluctuations in the polymer space6lling lower the percolation threshold. Our theory of void formation contains two phenomenological parameters: C, and iV,. For a given distribution of pigment sizes, N, is independent of 9. However, as discussed above, C,(9) must vanish as 9 + 0 and as 9 + 1. The fictions form of C&9) can be dete~ed experimentally by measuring, for example, the mass density at a fixed value of 9. Our model would then predict the void concentration (p3as a function of 9. If other properties of the coating, such as hiding power or permeability, depend on 50~in some known fashion, then those properties would also be predicted by the model. Since particle packings occur quite frequently in many fields of chemistry and physics, this work has widespread applications. This work is applicable to magnetic tape f 111, zincrich [25f and latex architectural [S] coatings, as well as to composite structures in dentistry [ZSl! The proposed theory may reveal how compositional changes alter the properties of many types of composite materials. Aclcuowledgements One of the authors (RSF) would like to acknowledge support from the EPSCOR (Experimental Program to Stimulate Cooperative Research) program,
402
administered by ASEND in North Dakota and from the donors of the Petrolem Research Fund, administered by the American Chemical Society. Nomenclature
pigment vohune concentra~on (PVC) critical pigment volme concen~ation (CPVC) 9’/Gc = reduced pigment volume concentration smallest number of pigment particles in a denselypacked cluster that may contain a void coarseness of polymer spacefilling in the volume of the sample not occupied by pigment mass density of coating pigment mass density polymer mass density void percolation threshold vahre volume fraction of pigment particles volume fraction of polymer (thence 9 = (pI/(4pl+ &) void volume fraction particle spanning threshold for particulate composites (conductivity percolation threshold) volume fraction of denselypacked islands of pigment in coating volume of the smallest island which may contain a void position vector in coating local volume fractions of pigment (i = l), polymer (i= 2) or void (i= 3) in a sphere of volume V, centered around x T~(x)/(T~(x) + 7&x)) = local value of pigment volume concentration about x local value of temperature pigment particle volume total number of pigment particles number of particles required to densely pack the whole sample of volume V TV/& + TV) = local value of polymer volume fraction regions in coating ~occupied and occupied by densely packed islands of pigment, respectively values of q(x) in a, b respectively V (‘?Vo(z) = pigment particle size ratio [see eqn. (lo)] pl/pI = mass density ratio between polymer and pigment
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