Geometry of paint film

Progress in Organic Coatings 90 (2016) 200–221
Contents lists available at ScienceDirect
Progress in Organic Coatings
journal homepage: www.elsevier.com/locate/porgcoat
Geometry of a paint film: Basics revisited
Bernard Lestarquit
Independent consultant, France
a r t i c l e
i n f o
Article history:
Received 13 March 2015
Received in revised form
22 September 2015
Accepted 24 September 2015
Available online 1 November 2015
Keywords:
Binder Index
Oil Absorption
Pigment/extender packing
“Reduced” PVC
Latex Porosity Index
Binder Porosity Index
a b s t r a c t
“Pigment/binder geometry”: this is the title of a chapter in “Paint, flow and pigment dispersion”, the
well known textbook [1] by Temple C. Patton that contributed to the coatings education of generations
of paint chemists. The author had designed a graphical representation – the first one of this kind – of
the volume repartition of the main components of a dry paint film as a function of the Pigment Volume
Concentration, or PVC, of the paint. It was the source of inspiration for this article. His approach was
however focused upon systems where the binder phase was an oil and would need to be refined and
developed further when latex polymers are used instead. This is the object of the communication that
follows. It is also the opportunity to re-visit a number of basic concepts.
© 2015 Elsevier B.V. All rights reserved.
When formulating a paint above the critical PVC, one extra
“ingredient”, air, is introduced into the dry film, causing its porosity.
And the presence of air into the film has a strong effect on a number of key properties such as scrub resistance, opacity (dry hiding
effect), etc., and there is therefore a strong interest in controlling
it. The quantification, however, of the porosity level in a paint film
does remain a challenge. There was a attempt by Patton [1] through
a graphical representation of the “geometry” of a dry paint film,
describing and quantifying the various phases of the key ingredients in the film: pigment, binder and air. But it was limited to a
very special case: an oil binder and one pigment, that was not at all
adapted to a variable pigmentation (whose composition does vary
with the PVC) as well as to latex binders. At about the same time,
Stieg [2–5] introduced the concepts of Porosity Index (PI) and of
Latex Porosity Index (LPI), making the distinction between oil and
latex systems, an other way to quantify the film porosity. These concepts became very popular and are regularly described in articles,
textbooks. But there again, a variable pigmentation was not considered and, in addition, we found the LPI model to yield “impossible”
results, not representative of the real world. There was therefore
a need to re-visit these concepts. This is the objective of the work
presented here that resulted in the development of a set of unified models to characterize the dry film morphology below and
E-mail address: blestarquit@orange.fr
http://dx.doi.org/10.1016/j.porgcoat.2015.09.023
0300-9440/© 2015 Elsevier B.V. All rights reserved.
above CPVC, that are no longer differentiating between oil and latex
systems.
1. Morphology of a paint film
In an attempt to describe the morphology of a dry paint film,
a very simplified model will be considered, based on a mono dispersed pigment particle (could be a plastic bead) and a binder. In
the graphical representation (Fig. 1) that follows, each rectangle is
supposed to represent a volume unit of the dry film. The Pigment
Volume Concentration (PVC) of the film is increased from 0 (no
pigment) to 1 or 100% (no binder) from bottom to top (Appendix
A for the definition of PVC). Two different binder types are to be
considered, oil (on the left side) and latex (right side), as they are
representative of the two main classes of the binders used in decorative paints: solvent-based (or solubilized) and latex polymers.
At PVC = 0 (bottom of the picture), there is no pigment and only
the binder is involved, making the continuous phase of the paint
film. There are no visible differences between oil and latex systems.
Adding pigment to the binders (i.e. increasing the PVC) will not
immediately allow a differentiation of the behavior of these binder
types, at least up to a particular PVC, the critical PVC or CPVC. And,
as long as PVC ≤ CPVC the volume fraction of the pigment is equal
to its PVC in the film: ˚pig = PVC.
Adding more pigment to the system, the so-called Critical PVC
(CPVC) is reached. This is the point where there is just enough
binder to fill in the voids and wet the pigment particles: above the
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
201
during the film formation process: therefore CPVClatex ≤ CPVCoil as
more binder is needed with the latex and the pigment is not at its
maximum packing: ˚pig = PVC = CPVClatex .
When increasing the PVC above CPVC, there are fewer and
fewer latex particles that could interfere with the pigment particles
and the packing of the latter is increasing, the highest level being
reached at the maximum PVC (100%) which is common to every
system since there is no binder involved. Therefore:
CPVC latex ≤ ˚pig ≤ CPVC oil
(1)
Note: in real life, a dry paint film is not completely homogeneous
as it results from the random placement of the particles (pigment,
extenders and, eventually, polymer) that it contains. As a result,
on a microscopic scale, there are domains that could have a higher
local PVC than others and the dry film, even below CPVC, might
contain micro occlusions of air although there is no detectable film
porosity on a macroscopic scale.
The primary objective of this work is to establish the relationships between the volume fractions of the various film ingredients
(namely pigment, binder, as well as “air”) and the paint PVC, in
the form of a diagram similar to Patton’s. Before we reach that
point, however, we need to re-state a number of useful concepts
(the toolbox) that will be at the basis of the calculations to follow
later.
2. Toolbox
2.1. Volume fractions & paint PVC
Fig. 1. Morphology of a dry paint film.
CPVC, there is not enough binder, relative to the pigment, and air
becomes part of the film: the paint film is porous. This is also the
point where oil and latex binders do differentiate themselves.
1.1. Oil based system at PVC ≥ CPVC
In a volume unit of the dry paint film, the sum of the volume fractions of the key components (pigmentation, binder and, eventually,
air) is equal to 1:
˚pig + ˚bin + ˚air = 1
(2)
The subscript “pig” stands for pigmentation, i.e. the combination
of generally one pigment (like TiO2 ) with one or more extender.
Therefore:
We now follow the left branch of Fig. 1.
At PVC = CPVCoil , the pigment particles are in contact and, as they
are incompressible, it is impossible to get a higher packing for these
solid particles; besides, there is just enough oil to wet and fill in the
voids between the pigment particles. The pigment volume fraction
is still equal to the PVC: ˚pig = PVC = CPVCoil .
When increasing further the PVC (PVC > CPVCoil ), the oil is gradually replaced by air (the film becomes porous) whereas the pigment
remains at its maximum packing: ˚pig = CPVCoil .
Finally, at PVC = 1 (or 100%) air has completely replaced the
oil without bringing any change for the pigment volume fraction:
˚pig = CPVCoil . This particular situation is common to both oil and
latex systems as there is no binder involved.
˚pig = ˚tio2 +
1.2. Latex based system at PVC ≥ CPVC
expressing that the PVC of a paint is the sum of the individual PVCi
of pigment and extenders.
Later on in the text, a PVC ladder, i.e. from low to high values, will
be considered in order to evaluate its effect on some paint film properties. In general, unless otherwise specified, the PVC increase will
be made by increasing the extender(s) PVC while keeping constant
the TiO2 PVC whichever the level it has to begin with.
We saw earlier that the pigment volume fraction was perfectly
defined when the PVC was at and below CPVC and not so well (i.e.
latex system) above CPVC. We will therefore study both situations
separately. One should note that the above and following relationships do remain valid irrespectively of the binder that is used (oil
or latex). Therefore the subscript “bin” that is replacing the previous “oil” and “latex” ones. Although it will often be convenient
Latex binders are composed of discrete polymer particles of very
high molecular weight, generally dispersed in water. It is, in that
respect, very different from the generic oil system that includes
most solvent based polymers, i.e. low molecular weight and solubilized polymers. Latex polymer, in turn, are characterized by
their particle size and the polymer Tg that both affect their ability to deform under the action of the osmotic/capillary forces that
develop during film formation.
Therefore, when the latex system is reaching the CPVC (right
branch of Fig. 1) the pigment particles are not yet in contact but
are separated by a layer of coalesced latex particles (no longer in
the shape of discrete particles) that could not deform any further
˚ext i
˚tio2 represents the volume fraction of the pigment and
where
˚ext i that of all the extenders. In the cases that follow below,
one pigment only (TiO2 ) will be considered as it corresponds to the
situation in the vast majority of white paint formulations.
And, as per the definition of the PVC:
PVC =
˚pig
˚pig + ˚bin
=
˚pig
(3)
1 − ˚air
which can be re-written as:
PVC =
˚tio2 +
˚ext i
1 − ˚air
= PVC tio2 +
PVC ext i
202
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
to still use the “oil” subscript (more rarely “latex”) to signify the
singularity of that particular system.
Table 1
Oil Absorption of typical pigments & extenders.
Type
2.1.1. PVC ≤ CPVC
There is no porosity in the film below the CPVC: ˚air = 0. Therefore (from Eqs. (2) and (3)):
˚pig = PVC
(4)
˚bin = 1 − PVC
(5)
And the system is perfectly defined, as well at the CPVC where
PVC = CPVCbin , irrespectively of the nature of the binder.
2.1.2. PVC ≥ CPVC
We already know that the pigment volume fraction does vary
between two limits as described in Fig. 1 and formalized in Eq. (1)
(re-written by replacing the “latex” subscript by “bin”):
CPVC bin ≤ ˚pig ≤ CPVC oil
(6)
From Eq. (3) one gets the volume fraction of air:
˚air = 1 −
˚pig
(7)
PVC
which, by combining with Eq. (2), yields that of the binder:
˚bin = ˚pig
1
PVC
−1
(8)
Thus, both the volume fractions of the binder and of the air are
a function of that of the pigment, ˚pig and of the overall PVC.
2.2. Binding capacity of latex polymers
As illustrated in Fig. 1, more latex polymer is needed to bind
the pigment particles at CPVC than required with an oil binder in
the same situation. Based on this statement, it was Berardi [6] who
proposed the concept of a “binding power index” or Binder Index
(BI) defined as the “ratio of the minimum amount (volume) of oil
required to completely wet and fill the voids of a given amount
of pigment, to the minimum volume of latex solids required to
similarly bind the same amount of pigment”. This reminds us the
definition of the CPVC either in the oil or latex case and Berardi’s BI
can therefore be written as follows:
BI latex =
V oil
Vlatex
(9)
CPVC
where Voil and Vlatex are the volumes of oil and latex per unit volume of the same pigment, calculated at the CPVC of each system.
The Binder Index is a dimensionless number such that 0 ≤ BI ≤ 1,
assuming that the Binder Index of oil is the highest of all: 1.
For latex polymers [7], the Binder Index is very much influenced
by the latex particle size (the smaller, the higher) and the ability of
the latex particle to deform under the conditions of film formation
(the softer, the higher) either because the Tg of the polymer is lower
than the minimum temperature at which the film must form or
it is temporarily reduced by the use of volatile coalescing agents.
Thus the Tg alone is not enough to characterize the ability of a latex
to form a film: the MFFT (Minimum Film Forming Temperature),
generally lower than the Tg due to some hydroplasticization of the
latex polymer particles would be a better indicator.
Correctly determined, the Binder Index is a characteristic of the
latex polymer – whichever its composition, morphology – within a
predefined temperature range (generally the application temperature limits of a coating). Poor performers would have a BI as low
as.4 when high binding capacity latex could reach.9, i.e. very close
to the optimum of the oil (BI = 1). Years ago information on the BI
of latex binders used to be included in the product data sheets of
some latex suppliers. In most cases, it needs to be experimentally
TiO2 (enamel)
CaCO3 (natural)
CaCO3 (ppted)
China Clay
Aluminosilicate
Mean Ø ()
.22–.23
5
.3
2.5
.03
Density
OAwght
OAvol
CPVCoil
4.1
2.7
2.7
2.6
2.1
18
15
26
42
160
.782
.435
.753
1.19
3.61
56.1%
69.7%
57.0%
45.7%
21.7%
determined. The Binder Index of any kind of binder, including latex,
will be expressed as BIbin from now on (as long as there is no ambiguity on its meaning). See the first part of Appendix F for a different
expression of the Binder Index.
2.3. Oil Absorption (OA)
After the Binder Index that characterizes the binding capacity of a binder comes the Oil Absorption that applies to pigments
and extenders. This is the minimum amount of linseed oil that is
required to completely wet and fill the voids of a given amount of
pigment (sounds familiar: re. first part of Berardi’s BI concept). It is
expressed either in grams or in cubic centimeter (both do co-exist
in the literature and caution should be exercised when using the
data) of oil per hundred grams of pigment (and is expressed here
as OAwght by reference to the pigment weight). The spatula rub-out
test method used for the determination of OAwght is described in
ASTM D281. Several parameters will affect the OA such as particle
size, shape and size distribution, type of ingredient. It is often very
useful to express the Oil Absorption as the volume of oil required
to bind a unit volume of pigment: OAvol . Some examples of OA data
are presented in Table 1. The CPVCoil is related to OAvol through
Eq. (10) (with BIbin = 1) as it will be seen in Section 2.4. The Oil
Absorption determination can be very tricky when performed by
non-experienced people and a 15–20% error margin is often the
rule, i.e. not satisfactory. Titanium dioxide and mineral extenders
producers, on the other hand, are used to routinely determine the
Oil Absorption of their products in a quality control process, with
dedicated professionals, and the error margin, in that case, can be
as low as 2%. There is to note, however, that the OAvol and CPVCoil
data in Table 1 are to be considered as intermediate values within
a more complex calculation process and are therefore not rounded
to take into account the error margin.
When blending several pigments together (i.e. a TiO2 pigment
with an extender), the resulting OAvol is, in general, far from being
a linear combination of the individual OAvol of both ingredients,
particularly when there is a significant difference in the particle
sizes between the two. This is due to the so-called “packing” effect
as illustrated in Fig. 2.
It is obvious that by adding a small particle size pigment, small
enough to fit in the voids between the coarser ones, less binder
is needed to fill in all remaining voids. This “packing” effect will
result in a lower than expected Oil Absorption for the blend: Oil
Absorptions are not additive.
In practice, the Oil Absorption of binary blends of a TiO2 pigment
with an extender is experimentally determined and an example of
Fig. 2. Illustration of the “packing” effect.
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
203
Should it be expressed relatively to a volume unit of pigment,
then:
CPVC bin =
1
1 + (Vbin,CPVC /Vpig,CPVC )
Vbin,CPVC can be extracted from Eq. (9) by substituting “latex” for
“bin”: (Vbin,CPVC = Voil,CPVC /BIbin ), yielding:
CPVC bin =
1
1 + (Voil,CPVC /Vpig,CPVC )(1/BI bin )
and since (Voil,CPVC /Vpig,CPVC ) = OAvol , one gets a well known relationship:
CPVC bin =
Fig. 3. OAvol of a binary blend.
the graphical representation of OAvol is presented in Fig. 3. There
are numerous such examples in the literature.
In general, extenders have a much larger particle size than
the TiO2 pigment and similar curve shapes as in Fig. 3 for binary
blends are to be expected. However, the closer the particle size of
both ingredients in the blend and the smaller the observed “packing” effect. Thus, a blend of a TiO2 pigment with a precipitated
CaCO3 could display almost no “packing” effect and the resulting
Oil Absorption will be very close to a straight linear combination
of both Oil Absorptions for the pigment and the small particle size
extender.
In real life, a paint will contain more than one extender, particularly when formulated above CPVC and the determination of
the resulting Oil Absorption could soon become very complex. In a
series of articles, probably totally ignored nowadays, the Philadelphia Paint and Varnish Production Club [8,9] described a method
for estimating the Oil Absorption of ternary, quaternary, etc. blends
from the knowledge of that of the individual binary blends of each
extender involved in the blend with the same TiO2 pigment.
The bottom line is that, as soon as there is a pigment/extender
blend, the Oil Absorption of the blend will vary with its composition which can be, in a paint context, related to the PVCs of the
individual ingredients and, by extension, to the PVC of the system.
It will be designed as OAvol,PVC to express the Oil Absorption of a
pigment/extender blend composition at the considered PVC (note
that there is, theoretically, an infinity of compositions for a pigment/extender blend at the same overall PVC!). Obviously, for a
single ingredient or a fixed blend composition, the Oil Absorption
remains constant whichever the PVC. It will be seen later how to
transform the Oil Absorption data of a binary blend as in Fig. 3 into
a PVC related relationship (Fig. 6 and Appendix B).
BI bin
BI bin + OAvol
(10)
an important and very useful relationship since it is at the basis
of the experimental determination of the Binder Index of a binder
by experimentally determining the CPVC of a system with a set
pigmentation composition (generally a single extender) and a fixed
OAvol . The following relation, derived from the above Eq. (10) will
then be used:
BI bin = OAvol ·
CPVC bin
1 − CPVC bin
Eq. (10) needs, however, to be completed in order to take into
account the fact that OAvol is, most often, not constant, and will generally vary with the pigment/extender composition when the PVC
varies, thus replacing OAvol by OAvol,PVC (as seen before). For each
pigment/extender blend at a given PVC, it is possible to calculate
the CPVC that such a pigmentation composition would yield. And,
in general, this calculated CPVC would be different from the actual
PVC. Eq. (10) needs therefore to be re-written as follows, in order
to avoid any ambiguity:
CPVC bin,PVC =
BI bin
BI bin + OAvol,PVC
(11)
where OAvol,PVC is the Oil Absorption of the pigment/extender blend
that is present at the considered PVC and CPVCbin,PVC is the calculated CPVC for that binder (BIbin )/pigmentation system. For any PVC
below CPVC, CPVCbin,PVC > PVC and above CPVC, CPVCbin,PVC < PVC.
There is, obviously, one point where CPVCbin,PVC = PVC which is when
the PVC is at the CPVC of the system and Eq. (11) will then be
expressed as:
CPVC bin,CPVC =
BI bin
BI bin + OAvol,CPVC
(12)
which corresponds to the true CPVC of the system (see Appendix
B, Table 9 and Fig. 17).
We, now, should have all the tools that we need to proceed to
the next step: the calculation of ˚pig above CPVC and the graphical
representation of the “geometry” of a paint film.
3. Geometry of a paint film
2.4. CPVC relationships
The CPVC of a binder, whether oil or latex, is expressed as:
CPVC bin =
Vpig,CPVC
Vpig,CPVC + Vbin,CPVC
with Vpig,CPVC and Vbin,CPVC representing the volumes of pigment and
binder respectively, at the CPVC of the system.
We saw earlier that the volume fractions of the key film ingredients (pigmentation and binder) were perfectly defined in systems
at and below CPVC (Eqs. (4) and (5)) and there is no point to come
back to it here. Above CPVC, however, the presence of air is adding
complexity to the system and ˚pig becomes undefined. The volume
fractions of the binder and of the air can, nevertheless, be expressed
as functions of the pigment volume fraction like in Eqs. (7) and (8).
The point would now be to determine ˚pig .
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B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
Table 2
“Constitutive” equations for the volume fractions mapping.
3.1. Determination of ˚pig
In an oil and with a single pigment (i.e. a fixed Oil Absorption)
system, when the PVC of the paint is increased from the critical
PVC (as in Fig. 1) to its highest level at 100% (i.e. no binder present),
air is gradually replacing the oil whilst the volume fraction of the
pigment – already at its maximum packing – remains constant.
Air could be considered as a virtual “binder” which has the same
binding capacity as oil since:
CPVC oil = CPVC air = ˚pig
CPVC bin ≤ ˚pig ≤ CPVC oil
that, in order to better represent the reality (since at 100% PVC,
there are no binder nor oil to bind the pigment, only air), could be
re-written as:
CPVC bin ≤ ˚pig ≤ CPVC air
At each of these limits, ˚pig is at the maximum packing that is
compatible with the binder in presence, either the binder itself or
air. And there is no reason why, between these limits, it should no
longer be the case: the binder is now a “composite” binder formed
by the blend of the binder itself with air:
(13)
expressing that at any PVC such that: CPVCbin ≤ PVC ≤ CPVCair , ˚pig
is continuously at the maximum packing that can be achieved with
the “composite” binder (at the composition – or the binder/air ratio
– it has at the PVC, therefore the “PVC” subscript).
The objective is now to define the composition of the “composite” binder as well as its binding capacity BIcomp,PVC in order to be
able to calculate ˚pig . This will be done in the general case of a pigmentation with a variable Oil Absorption (expressed as OAvol,PVC
as seen previously. In this situation, CPVCoil and CPVCair are not
automatically identical since the pigmentation composition does
generally vary with the PVC and so does its Oil Absorption.
3.1.1. Binder Index of the composite binder: BIcomp,PVC
The “composite” binder composition and volume fraction are
given by the sum of the volume fractions of the regular binder and
that of the air at the considered PVC.
˚comp,PVC = (˚air + ˚bin ) = (1 − ˚pig )
and the Binder Index of that composite will be a linear combination of the individual BI times the volume fraction of the associated
binder (binder or air), divided by the total volume fraction of the
composite binder (it is therefore variable with the PVC):
BI comp,PVC =
˚bin · BI bin + ˚air · BI air
1 − ˚pig
(14)
By expressing ˚pig and using Eq. (11) (modified for BIcomp,PVC
and, now, OAvol,PVC since both terms do vary with the pigmentation
composition and the system PVC), one gets:
˚pig = CPVC comp,PVC =
˚pig
1 − ˚pig
· OAvol,PVC
Above CPVC:
˚pig :
CPVCbin,CPVC ≤ PVC ≤ 1
˚pig = PVC(BI +OA PVC
vol,PVC )+(1−BI bin )
bin
PVC = CPVCbin,CPVC ⇒ ˚pig = CPVCbin,CPVC
PVC = 1 ⇒ ˚pig = CPVC
air
1 − CPVC bin,CPVC ≥ ˚bin = ˚pig
˚bin :
0 ≤ ˚air = 1 −
˚pig
1
PVC
−1
≥0
≤ 1 − CPVC air
PVC
BI bin
CPVC bin,PVC = BI +OA
bin
vol,PVC
BI
CPVC bin,CPVC = BI +OAbin
bin
vol,CPVC
with:
OAvol,PVC = f(PVC =
PVCi )
We can now solve Eqs. (14) and (15) for ˚pig , replacing ˚air
and ˚pig with their expressions in Eqs. (7) and (8) respectively. and
setting BIair = 1:
˚pig =
PVC
PVC OAvol,PVC + BI bin + (1 − BI bin )
(16)
It can be verified that, by setting BIbin = 1 (oil system), one effectively gets ˚pig = 1/(1 + OAvol,PVC ) = CPVCoil,PVC which reflects the fact
that the CPVCoil is no longer constant, but will vary with the pigment composition at the considered PVC, unless there is a fixed
pigmentation and a constant Oil Absorption.
When the PVC is varied between the binder system CPVC
(CPVCbin,CPVC ) and the maximum PVC (PVC = 1 or 100%), the pigment volume fraction does vary between CPVCbin,CPVC and CPVCair
respectively, as the calculation does show. But it is difficult to predict which one of these limits will be the highest and ˚pig will vary
up or down when the PVC is increased, depending upon the system
as it will be illustrated later. On the other hand, ˚bin will exhibit
a continuous decrease and ˚air a continuous increase in the same
conditions.
It is possible to derive an equivalent expression (Appendix C)
for ˚pig that might be “better looking”:
˚pig =
PVC · CPVC oil,PVC (1 − CPVC bin,PVC )
PVC(1 − CPVC oil,PVC ) + (CPVC oil,PVC − CPVC bin,PVC )
(17)
but is less straightforward as it calls for computed CPVC data instead
of experimentally measured data such as Binder Index and Oil
Absorption as in Eq. (16).
All information that are needed to build the graph are now available and are summarized in Table 2 (in an allusion to Rheology, we
will call it the “constitutive” equations).
In order to facilitate the understanding upon the effects that
these models have on the distribution of the volume fractions of the
key ingredients in a dry paint film, a gradual approach was adopted,
starting with the basic case presented by Patton in his book (an oil
binder and a fixed Oil Absorption, i.e. very unrealistic in real life
situations)), then switching to any binder that has a lower binding capacity than oil, i.e. a latex polymer (but still with a fixed Oil
Absorption) and finally the general case: any binder, and variable
Oil Absorption.
3.2. Fixed Oil Absorption
BI comp,PVC
BI comp,PVC + OAvol,PVC
which yields a second expression of BIcomp,PVC :
BI comp,PVC =
0 ≤ PVC ≤ CPVCbin,CPVC
0 ≤ (˚pig = PVC) ≤ CPVCbin,CPVC
1 ≥ (˚bin = 1 − PVC) ≥ 1 − CPVCbin,CPVC
˚air :
Oil and air are therefore interchangeable as far as their effect on
the pigmentation packing is concerned.
With any binder different from oil, ˚pig is evolving between two
well defined limits (Eq. (6)):
CPVC bin ≤ (˚pig = CPVC comp,PVC ) ≤ CPVC air
Below CPVC:
˚pig :
˚bin :
(15)
3.2.1. Oil based system
The first example is initiated from Patton’s work: an oil based
system (BIbin = 1) and a single extender or a TiO2 /extender blend at
a fixed ratio with, therefore, a fixed OAvol all over the PVC range.
Whilst below CPVC, the relationship between the pigment and
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
205
Fig. 4. Volume fractions distribution in an oil based system.
Fig. 5. Volume fractions distribution for BIbin < 1, fixed OAvol .
binder with the PVC is immediate (Table 2), above CPVC, the expression of the pigment volume fraction is significantly simplified:
It is known that a number of film resistance properties such
as scrub resistance are directly related to the film porosity: same
porosity, similar scrub resistance (with binders at comparable Tg ):
higher binding capacity binders than the above model will allow
higher PVCs to be achieved whilst maintaining the film properties,
which could result in lower formulation costs.
All numerical data are summarized in Appendix C.
˚pig =
1
= CPVC oil = CPVC air
1 + OAvol
and ˚pig is constant since OAvol is constant as well. Thus, at the
system CPVC, ˚pig = CPVCoil and, at the maximum PVC (100%),
˚pig = CPVCair , both values being identical. In between these limits ˚pig = CPVCcomp,PVC which is constant since the oil/air composite
“binder” has a constant Binder Index as well.
The graph, constructed with an arbitrarily chosen Oil Absorption
((OAvol = 1) ⇒ (˚pig = .5)) (Fig. 4) is self explanatory. It is, in all aspect,
similar to Patton’s graph (Appendix C).
3.2.2. Any binder system
It is meant a binder that could have a lower binding capacity
than oil (BIbin ≤ 1). In this situation, the pigment volume fraction is
expressed as:
˚pig
PVC
=
PVC(BI bin + OAvol ) + (1 − BI bin )
and as BIbin and OAvol are constant, the pigment volume fraction
does vary with the PVC between two limits:
• at
the
system
CPVC,
PVC = CPVCbin
and
since
CPVCbin = (BIbin /(BIbin + OAvol )), then ˚pig = CPVCbin (obtained
after replacing the appropriate terms in the above equation),
• whilst when PVC = 1 (or 100%), then ˚pig = CPVCair (as seen previously).
And since the Binder Index of the composite “binder” is increasing from BIbin up to 1 with the PVC (air is gradually replacing the
binder), the pigment volume fraction will generally increase uniformly when the PVC is increased from CPVCbin to 1.
A latex polymer with a particularly low binding capacity
(BIbin = .4) was chosen in order to maximize the differences with
the standard oil system. The same pigmentation (OAvol = 1) as in
the previous case was used. The graphical results are presented in
Fig. 5. The dotted lines do correspond to the oil binder, as in the
previous case, in order to show the differences between both systems. When the PVC evolves from CPVCbin up to 100%, the paint film
is getting more and more porous and the pigment volume fraction
does go up from CPVCbin to CPVCair , as a result of the change in the
binding capacity of the binder/air composite.
This graph does clearly show the effect of a lower binding capacity binder versus the oil (dotted lines): the film porosity (and CPVC)
starts at a much lower PVC with that binder than with the oil.
3.3. Variable Oil Absorption
The OAvol curve in Fig. 3 does illustrate the fact that, in real life,
the Oil Absorption of a pigment/extender blend is far from being
constant when modifying the blend composition. There is therefore a need to model these Oil Absorption curves relatively to the
formulation parameters like the PVCs. The starting point will be the
OAvol curves of TiO2 /extender binary blends as illustrated in Fig. 3:
a mathematical model (Eq. (18)) in the form of:
OAvol = Ax3 + Bx2 + Cx + D
(18)
where x is the volume fraction of the extender in the pigment/extender blend and A, B, C and D are coefficients that depend
upon the nature of the blend components, was suggested to best
describe the variation of the Oil Absorption as a function of the
pigmentary composition (re. the Philadelphia Paint and Varnish
Production Club [8,9], previously cited, and Appendix B). These
mathematical models were calculated (regression) for a range of
TiO2 /extender couples (author’s data) of which two of these are
presented in this document as examples: Clay and Carbonate. Any
Oil Absorption curve data from the literature, once modeled, could
have been used instead. There is now just a need to relate this model
to the paint formulation parameters like the overall PVC, but not
only (Appendix B).
In a paint based on a binary TiO2 /extender mixture, the paint
PVC can be written as: PVC = PVC tio2 + PVC ext or PVC ext = PVC −
PVC tio2 and the volume fraction of the extender in the pigment/extender blend is:
x=
PVC − PVC tio2
PVC ext
=
PVC
PVC
Therefore, after replacing x by its value in Eq. (18), OAvol is now
linked to the paint PVC as well as to the TiO2 PVC through:
OAvol,PVC = A
+C
PVC − PVC tio2
PVC
PVC − PVC tio2
PVC
3
+B
+D
PVC − PVC tio2
PVC
2
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B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
Table 3
A, B, C and D coefficients.
A
B
C
D
Clay
Carbonate
.7622
.1305
−.4334
.7541
.4438
.0376
−.8330
.7569
Note that the coefficient D, corresponding to PVC = PVC tio2 , i.e.
when there is no extender, is the OAvol of the TiO2 pigment alone.
With the knowledge of the A, B, C and D coefficients for two
sets of data (Clay and Carbonate) represented in Table 3 it is now
possible to draw the OAvol,PVC curve (Fig. 6) as a function of the PVC
for any set TiO2 PVC. In the example presented below, the maximum
TiO2 PVC was arbitrarily set at 15% and remained at that value when
the total PVC was increased further with either a Clay or a Carbonate
extender. Obviously, any value for the TiO2 PVC would work and
combinations with the extender are infinite (although not always
realistic).
The very particular shapes as well as the significant OAvol variation over the PVC range are worth mentioning as well as the fact
that the above TiO2 /Carbonate curve and Fig. 3 are built from the
same dataset (with a fixed TiO2 PVC = 15%) (see Appendix B).
We can now draw the graphs displaying the volume fraction of
the key ingredients versus the paint PVC.
In this general case, the expression of the pigmentation volume
fraction is that one which was previously described in Table 2:
˚pig =
PVC
PVC(BI bin + OAvol,PVC ) + (1 − BI bin )
(Note the subscript for the Oil Absorption: OAvol,PVC , since it now
depends upon the composition of the pigmentation at the considered PVC)
And when the PVC is at the CPVC of the system
(PVC = CPVCbin,CPVC ), then ˚pig = CPVCbin,CPVC . For PVC = 1 (or 100%),
then ˚pig = CPVCair , i.e. the CPVC in air (or oil) for a pigmentation
composition at that PVC (see Eqs. (11) and (12) as well as Appendix
B for a reminder about the meaning of the subscripts).
3.3.1. TiO2 /Clay binary system
The first graph (Fig. 7) to come is about a TiO2 /Clay binary blend,
with a maximum (arbitrarily chosen) TiO2 PVC of 15%. It illustrates
the volume fractions distribution in the case of either an oil binder
(dotted lines) or a binder with BIbin = .4 (same as before).
• Oil-based (BIbin = 1) system (dotted lines graph)
Fig. 7. Volume fractions distribution for a TiO2 /Clay binary mixture.
With the oil system above CPVC, ˚pig is expressed as:
˚pig =
1
= CPVC oil,PVC
1 + OAvol,PVC
showing that, as OAvol,PVC does vary with the PVC, ˚pig is also varying with the PVC between two limits: CPVCoil,CPVC when the PVC
is at the CPVC of the system and CPVCair when the PVC is 100%.
The point is that, as the Oil Absorption is changing with the PVC,
it is not immediately obvious to predict which one of these limits
will be higher than the other. As a matter of fact, in this simple
case, if OAvol,CPVC ≤ OAvol,1 (the Oil Absorption for the pigmentation
composition at 100% PVC), then
CPVC oil,CPVC =
1
1 + OAvol,CPVC
≥
CPVC air =
1
1 + OAvol,1
which is the case in this example, and vice versa. The pigment volume fraction (˚pig ) would generally vary smoothly between these
limits (here, a constant decrease with the PVC), provided that the
Oil Absorption OAvol,PVC is varied uniformly with the PVC (here a
regular increase). The situation would be different if the minimum
Oil Absorption value calculated from the OAvol,PVC = f(PVC = PVCi )
graph (Fig. 6) does occur at a PVC that is in between the system
CPVC (CPVCoil,CPVC ) and the maximum PVC (100%). In the present
TiO2 /Clay case, the Oil Absorption reaches a minimum at around
a PVC of 25%, which is below the oil system CPVC (CPVCoil,CPVC ) at
about 56%. This seems trivial, but it is not, as in more complex systems (i.e. a TiO2 pigment combined with several extenders) the
minimum Oil Absorption could well occur at a PVC that is above
the system CPVC and then the evolution of ˚pig between its limits
might not be as smooth as anticipated. All data are summarized in
Appendix C.
• Any binder (BIbin ≤ 1) system (black lines graph)
In this case, ˚pig is expressed as:
˚pig =
Fig. 6. OAvol for binary pigment/extender blends.
BI comp,PVC
= CPVC comp,PVC
BI comp,PVC + OAvol,PVC
where BIcomp,PVC does evolve between BIbin and BIair = 1, i.e. a
constant increase with the PVC and OAvol,PVC is, in this case, continuously increasing with the PVC. In this example, with BIbin = .4,
˚pig is smoothly increasing with the PVC between its limits (it can
be verified – Appendix B – that the minimum Oil Absorption of the
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
207
Fig. 8. Volume fractions distribution for a TiO2 /Carbonate binary mixture.
TiO2 /Clay blends does occur at a lower PVC (around 25%) than the
system CPVC (CPVCbin,CPVC ) which is at about 36%).
3.3.2. TiO2 /Carbonate binary system
The TiO2 /Carbonate graph (Fig. 8) was built using the same
binders as above as well as with a maximum TiO2 PVC of 15%..
Similar remarks about the relative positioning of CPVCoil,CPVC or
CPVCbin,CPVC and CPVCair could be made. All data are summarized
in Appendix C. As previously, the oil system is represented by the
dotted lines.
A comparative examination of this graph with the previous one
(Fig. 7) shows that the Carbonate system does generate a lower
overall porosity than the Clay system. It can be observed, intuitively, that, for an equal film porosity level (˚air ), formulations
with Carbonate will have a higher PVC than the Clay ones and would
potentially be cheaper. This will be the basis of most formulation
optimization processes as described later.
It is very rare, however, that medium to high PVC paints do contain one extender only. For instance, a Clay/Carbonate combination
is very popular in some parts of the world and the determination of
the Oil Absorption of such ternary blends (or quaternary, etc.) can
be well estimated by using the method developed by the Philadelphia Paint and Varnish Production Club, mentioned earlier, from
the knowledge of the OA curves of the individual TiO2 /extender
binary blends. An example of a ternary blend will be described later
(Appendix G).
Fig. 9. Wet and dry film thickness.
systems (with BIbin = .4). The wet applied thickness was set at 100
and the Volume Solids was 35%. Data are available in Appendix D.
The graph shows that the dry film thickness is very much affected
by the film porosity in above CPVC paints.
3.4.2. Dry film density (Appendix D)
The density of the dry film (DFD) is simply expressed by summing the weights of the volume fractions of the film ingredients (as
the total volume is the volume unit):
DFD = ˚pig · Dpig + ˚bin · Dbin + ˚air · Dair
(with Dpig , Dbin and Dair being the densities of the pigmentation, the
binder and the air respectively), which, after replacing ˚bin by its
expressions in Eq. (8) and neglecting the density of air, yields an
expression also related to ˚pig :
DFD =
The mathematical models that were described earlier can be
used to perform some useful calculations:
3.4.1. Dry film thickness
Something to notice, which is common to all above cases, this
is the dry film thickness increase that is observed above the CPVC,
when comparing paints that are all formulated at the same Volume
Solids (VS) and applied at the same wet film thickness (WFT). This
is due to the presence of a new “ingredient”: air.
The dry film thickness (DFT) of a paint film is expressed by (calculation details in Appendix D):
PVC
· [PVC · Dpig + (1 − PVC) · Dbin ]
(20)
with ˚pig = PVC below CPVC.
And since the pigmentation is composed of several ingredients
(a TiO2 pigment and one or more extenders), its density is also
composite and is expressed as:
Dpig =
3.4. Some applications
˚pig
Dtio2 · PVC tio2 +
(Dext i · PVC ext i )
PVC
which is simply summing the density contribution to the overall
density of each individual ingredient. In the case of a binary blend,
the density can be written as:
Dpig =
Dtio2 · PVC tio2 + Dext · (PVC − PVC tio2 )
PVC
(19)
The density variation with the PVC of the previous TiO2 /Clay or
Carbonate cases is presented in Fig. 10.
In the Clay system, the high film porosity does contribute to
a significant density reduction when increasing the PVC (above
CPVC) whilst this effect is almost negligible in the Carbonate system as the potential density increase of a higher PVC is, in this case,
annihilated by a density reduction caused by the increased porosity.
Data are available in Appendix D.
thus linking the dry film thickness (DFT) to the previously calculated ˚pig (with ˚pig = PVC below CPVC). It is illustrated in Fig. 9
displaying the dry film thickness change with the PVC in both examples previously mentioned: the TiO2 /Clay and –/Carbonate binary
3.4.3. The “reduced PVC” concept
The PVC/CPVC ratio, also called the “reduced PVC” [10] and
defined by the greek letter lambda “” has the reputation of characterizing the performance level of a paint and is often used in paint
DFT = WFT · VS ·
PVC
˚pig
208
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
Fig. 10. Dry film density.
re-formulation processes when the new paint has to match the
properties of the model paint, particularly in above CPVC systems
where is claimed to be directly related to the film porosity:
PVC
=
= PVC
CPVC bin,PVC
BI bin + OAvol,PVC
BI bin
(21)
Obviously, the CPVC to consider is that one which corresponds
to the CPVC for the pigmentation at the considered PVC (CPVCbin,PVC )
and not at all that of the system (CPVCbin,CPVC ) which, most often,
has a different pigmentation composition from that at PVC. This
point is very often neglected and, when so, it might lead to wrong
conclusions.
As the “reduced PVC” is essentially used in above CPVC paints,
it would be useful to characterize its relationship with the porosity
level in the paint film.
˚air is expressed by: ˚air = 1 − (˚pig /PVC) (from Eq. (7)), thus:
˚air = 1 −
1
PVC(BI bin + OAvol,PVC ) + (1 − BI bin )
(22)
And, by combining both Eqs. (21) and (22) one gets two equivalent expressions that link ˚air and (Appendix E):
˚air =
BI bin ( − 1)
˚air
⇔ ( − 1) =
BI bin ( − 1) + 1
BI bin (1 − ˚air )
(23)
It clearly shows the interdependence of the film porosity with the
reduced PVC, as expected, but also with the binding capacity of the
binder. And these relations are valid irrespective of the pigmentation. Therefore, at a given Binder Index, the film porosity and the
“reduced” PVC are directly related, thus justifying the choice of as
a performance indicator when reformulating an above CPVC paint
with the same binder. Surprisingly enough, however, Eq. (23) does
show that a same value for does not correspond to the same
film porosity when comparing two paints with binders of different
Binder Index. Instead, a conversion must be made, should the film
porosity remains constant, when, i.e. switching from binder 1 to
binder 2:
˚air =
BI bin1 (1 − 1)
BI bin1 (1 − 1) + 1
=
BI bin2 (2 − 1)
BI bin2 (2 − 1) + 1
Fig. 11. Interdependence of , BIbin and ˚air .
which, after computation yields:
(2 − 1) = (1 − 1)
BI bin1
BI bin2
(24)
Obviously, these relations are only valid for above CPVC paints.
Data in Appendix E.
From Eq. (23) it is possible to draw a template illustrating the
interdependence of , BIbin and ˚air that could be used with any
paint formulation. This template is “universal”.
The lines do correspond to some arbitrarily chosen values of
: 1.1, 1.3, 1.7 and 2.5 and it can be seen that at a set porosity
(˚air ) is decreasing when the binding capacity of the binder is
increased. In other terms, comparing the performances of paint formulations based on binders with different BIbin but at the same is meaningless since the paints are not at the same porosity level
(Fig. 11).
It is now possible to superimpose to the above template
the graphics containing specific information (here the PVC lines)
related to the system under consideration.
This is illustrated in Fig. 12 displaying two graphs corresponding to two different situations: a TiO2 /Clay binary system in one
case (left graph) and a TiO2 /Carbonate system in the right graph.
Both these pigment/extender combinations were used to build
Figs. 7 and 8: a fixed TiO2 PVC (15%), arbitrarily chosen, and a variable extender PVC for the paint PVC adjustment. Both graphs have
a common background made from the above grid lines that is,
as said earlier, an invariant in this representation since it does not
depend upon the Oil Absorption of the pigmentation (re. Eq. (23)).
The variable part of these graphs is coming from the PVC grid lines
that are very much different when comparing Clay and Carbonate
(Appendix E). For each system, a few PVC lines topping at 100%
PVC were drawn. At 100% PVC, the volume fraction of air is at its
maximum (system depending), irrespectively of the binding capacity of the binder, therefore the horizontal line. The shaded area
(above 100% PVC) has no practical existence. As seen before, there
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
209
Fig. 12. Reduced PVC vs. Binder Index.
is a strong difference between the Clay and Carbonate systems, the
latter developing a much lower porosity.
On the Clay graph, we figured out what the switch from a low
(A) to a high (B) Binder Index polymer, at the same total or overall
film porosity, would change for some paint parameters:
Table 4 summarizes the formulation parameters change when
switching the binders with different binding capacity, at constant
film porosity.
binder, we have to deduct from it the amount of binder that is tied to
the pigment as if it were at the CPVC (calculated from the pigment
composition at PVC, that is to say: 1 − CPVCbin,PVC . But only a fraction
of it is involved (PVC/CPVCbin,PVC ), since the volume fraction of the
pigment at PVC is lower than that at CPVC. The volume fraction of
the “free” binder is therefore:+
˚FB = (1 − PVC) −
3.4.4. The “free binder” concept
Another concept had gained popularity some time ago as an
indicator of the paint performance level for below CPVC systems:
it is the “free binder” concept. It starts from the observation that,
at CPVC, all the binder in the formulation is engaged in binding the
pigment/extenders particles and there is none available for other
purpose. Below CPVC, there is an “excess” of binder and the idea
is to define the amount of free binder, i.e. the binder that is not
tied in just holding the pigment particles. It was assumed, with
good reason, that the more “free” binder and the better the film
performs.
At a given PVC (below CPVC), the total binder level in the dry film
is: ˚bin = 1 − PVC (as seen in Eq. (5)). In order to get the level of “free”
Table 4
Paint reformulation.
BIbin
Paint PVC
OAvol,PVC
˚air
Binder A
Binder B
.5
70.0%
.8639
.3126
1.9
.9
75.9%
.8843
.3126
1.5
= 1−
PVC
(1 − CPVC bin,PVC ) ⇒ ˚FB
CPVC bin,PVC
PVC
=1−
CPVC bin,PVC
(25)
a remarkable result that links directly the “reduced PVC” to the
amount of “free” binder.
Fig. 13 (still with a TiO2 PVC of 15%) illustrates the effect that
ingredients with different binder demand can have on the Free
Binder level. (data in Appendix E).
The concept of “free binder” is particularly interesting in the
formulation optimization process of elastomeric coatings, i.e. maximizing the system PVC (lower cost) at the desired elastomeric
performance, the latter being directly related to ˚FB for a given
polymer. It would be equally beneficial in the formulation optimization of conventional matt paints that are formulated just below
the CPVC.
3.4.5. The Binder Porosity Index
The total porosity (also called “overall” porosity) is represented
by the volume fraction of air, as seen in Eq. (7): ˚air = 1 − (˚pig /PVC).
There is also another concept, developed by Stieg and Patton, that
defines the porosity level of the binder phase and is expressed as
the ratio of the volume phase of air to that of (air + binder). It is
210
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
The division of (1 − BPI) – from Eq. (28) – by (1 − BPIoil ) – from
Eq. (29) – yields BIcomp,PVC , as expressed below:
1 − BPI
= BI comp,PVC
1 − BPI oil
(30)
from where a new expression for BPI can be written which, after
the replacement of (1 − BPIoil ) by its equivalent (Eq. (29)), leads to
Eq. (31)
BPI = 1 − BI comp,PVC (1 − BPI oil )
and
BPI = 1 − BI comp,PVC
1 − PVC
PVC · OAvol,PVC
(31)
OAvol,PVC can also be written differently, according to:
Fig. 13. Comparative Free Binder level in the (BIbin = .4) binder system
CPVC oil,PVC =
called either “Porosity Index [2]” (PI) in oil & solvent based system
or “Latex Porosity Index [3–5]” (LPI) for latex paints. As seen before,
we no longer have to make the distinction between oil and latex in
our approach and we will therefore define a Binder Porosity Index
(BPI) as:
˚air
˚air
BPI =
=
˚air + ˚bin
1 − ˚pig
PVC − ˚pig
PVC(1 − ˚pig )
(26)
Stieg and Patton, however, have designed another expression to
characterize the Latex Porosity Index (LPI):
LPI = 1 − BI latex ·
CPVC oil (1 − PVC)
PVC(1 − CPVC oil )
and in order to be able to compare it to BPI, we need to express the
latter in a different way:
BPI =
(˚air + ˚bin ) − ˚bin
˚bin
=1−
˚air + ˚bin
1 − ˚pig
Which, after the replacement of ˚bin by its expression in Eq. (8)
could be re-written as:
BPI = 1 −
˚pig (1 − PVC)
PVC(1 − ˚pig )
(27)
It can be verified that Eqs. (27) and (26) are totally interchangeable.
Should we remember that ˚pig = CPVCcomp,PVC (Eq. (13)), i.e. it
is at the CPVC of the composite binder for any pigment/extender
composition at the considered PVC, then:
BPI = 1 −
CPVC comp,PVC (1 − PVC)
PVC(1 − CPVC comp,PVC )
And since CPVCcomp,PVC = (BIcomp,PVC /(BIcomp,PVC + OAvol,PVC )), as we
have to use the Binder Index and Oil Absorption variables at the
value they have at the considered PVC, BPI becomes, after substitution and simplification:
BPI = 1 −
BI comp,PVC (1 − PVC)
PVC · OAvol,PVC
(28)
In the particular case of an oil binder, since BIcomp,PVC = BIoil = 1:
BPI oil = 1 −
1 − PVC
PVC · OAvol,PVC
⇒
OAvol,PVC =
1 − CPVC oil,PVC
CPVC oil,PVC
which after substitution in Eq. (31) finally yields the expected
expression for the Binder Porosity Index:
BPI = 1 − BI comp,PVC ·
CPVC oil,PVC (1 − PVC)
PVC(1 − CPVC oil,PVC )
(32)
an expression that is very similar, in its form, to that of the Latex
Porosity Index (LPI) described by Stieg and Patton:
Note that PI and LPI would have the same basic definition as the
above BPI. After replacing ˚air by its above expression in Eq. (7), it
yields:
BPI =
1
1 + OAvol,PVC
(29)
LPI = 1 − BI latex ·
CPVC oil (1 − PVC)
PVC(1 − CPVC oil )
(33)
At first glance, both porosity models (BPI in Eq. (32) and LPI
in Eq. (33)) do look very similar. We can increase that similarity
by recollecting that Patton/Stieg only looked at a fixed pigmentation composition, i.e. a fixed Oil Absorption. With that constraint
in mind, CPVCoil,PVC is no longer a variable and the “PVC” subscript
can be dropped. And by doing so, the only difference that remains
between BPI (at a fixed Oil Absorption) and LPI are the factors
BIcomp,PVC in BPI versus BIlatex in LPI. A fundamental difference, however, since the former term (the Binder Index of the “composite”
latex/air binder) is variable with the PVC whilst the latter one has
a fixed value.
The consequences of this divergence are quite significant as
it can be demonstrated through a back calculation of ˚pig (see
Appendix F) from either BPI or LPI when the latter do vary from
0 (no porosity) to 1 (at 100% PVC): ˚pig varies like CPVCcomp,PVC
when it is calculated from BPI, as expected, and remains constant
and equal to CPVClatex when it is calculated from LPI. And it was
shown earlier that ˚pig cannot remain constant in these conditions
as the packing of the pigmentation does evolve with the PVC. Therefore, the expression of LPI, as it stands, was wrongly established by
its authors and would deliver incorrect values for the estimation of
the porosity, except in one particular situation: an oil binder, where
BIcomp,PVC = BIoil = 1.
The graph (Fig. 14), drawn in the very simple case of a fixed Oil
Absorption and a Binder Index of.4 (i.e. the example used previously) is perfectly illustrative of the contradiction:
While BPI or LPI do vary from 0 to 1, the PVC of the paint evolves
between the CPVC of the binder or latex and 100%. There are more
data of interest in the table of Appendix F.
The LPI concept was invented by the early 1970s and quickly
became very popular. It was the object of multiple publications –
as such – in coatings related textbooks, the last one no later than in
2013. The fact that it remained unchallenged for such a long time
is very surprising!
A number of film properties are related to the binder porosity,
such as the film opacity (dry hiding) as it affects strongly the refractive index of the binder/air composite that surrounds the pigment.
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
211
with yi being the volume fraction of the extender i in the extender
blend ( yi = 1). In the case that is considered here, as there are
only two extenders, Clay and Carbonate and a constant TiO2 PVC
(PVC tio2 = 15%), the Oil Absorption of the Pigment/extenders blend
can be expressed as (Appendix G):
OAvol,PVC = yOAclay,PVC + (1 − y)OAcarb,PVC
Fig. 14. Back-calculation of ˚pig from BPI or LPI.
An average refractive index of the binder/air system can be easily
calculated:
ncomp = BPI · nair + (1 − BPI) · nbin
ncomp , nbin being the refractive index of the composite and of the
binder respectively (nair = 1). Similarly, an average refractive index
can be calculated for the pigmentation (npig ) and, since the opacity
of a paint film does depend upon the difference in the refractive
index between the pigment and the binder (here the “composite” binder) phases, it is possible to anticipate the evolution of the
opacity of a film when modifying the formulation.
The above model to estimate the refractive index of the “composite” binder is the simplest ever model and might appear very
crude. A number of authors did spend time defining more sophisticated models to characterize the optical properties of nano porous
materials and their findings are summarized in a document published by the UCLA on the web [10]. These models, and their
designation, are reproduced – ad re-written with BPI as a variable
– in Appendix F (the model described above is called the “Parallel”
model). The calculation performed with these models for 0 ≤ BPI ≤ 1
shows that, apart from the “Series” type, all other models did yield
very similar results: the above “Parallel” model does appear to be
adequate in a coatings context (data in Appendix F).
4. Paint formulation optimization
The models described in Table 2 can be used, as such, to reformulate/optimize paint formulations, provided that a prior knowledge
of the binder indexes of the potential binder candidates as well as
of the Oil Absorption profile (mathematical model) of the binary
blends of the main TiO2 pigment with the various extenders that
are to be used, be known.
It is indeed very rare that, beyond semi-gloss paints, one
extender only be used to formulate a paint. Instead, a combination
of extenders is the rule. As an example, we will focus on a paint
formulated above CPVC, using a popular (in some areas) combination of extenders (besides TiO2 ): Clay and Carbonate, the very same
ingredients that were studied earlier.
The first point to determine, in the case of an extender blend, is
the resulting Oil Absorption of that blend. The Philadelphia Paint
and Varnish Production Club, previously cited, does propose that
a linear combination of the Oil Absorption curves (as in Fig. 6) be
made in such a way that the resulting OAvol,PVC be equal to
OAvol,PVC =
yi OAext i ,PVC
where OAclay,PVC and OAcarb,PVC are represented by the curves in Fig. 6
which parameters are listed in Table 3. There is to notice that the
PVC that is mentioned in the subscripts is the overall paint PVC, not
the PVC of the individual ingredients.
As stated before, the TiO2 PVC was maintained constant in this
exercise, for the sake of simplicity, but the models that were developed are valid for any pigment/extender blend.
Our starting point formulation has a 70% PVC (above the CPVC),
a fixed TiO2 PVC (15%), a Clay/Carbonate PVC ratio of 1/2 and a
binder with a moderate binding capacity (BIbin = .7). There will be
one guideline to stick to: that the dry film overall porosity (˚air )
remains equal to that of the control paint: a number of properties,
those mostly affected by the film porosity (scrub resistance, dry
hiding, etc.), should then remain unchanged, or very close, when
modifying the formulation. The effect of a switch to a higher binding
capacity binder (BIbin = .9) will be considered as well. But before we
start the simulation, we need to define another concept:
4.1. The Water Demand*
Paint rheology (we are now dealing with the wet paint, not the
dry film) adjustment is an important challenge that the paint formulator does meet in a formulation optimization process. Indeed,
when replacing high absorbing extenders by less absorbing ones,
the paint PVC can certainly be increased, sometimes significantly,
but the viscosity will more likely drop, to a point that the cost of the
extra thickener needed for the rheology adjustment might become
excessive. An “indicator” is therefore needed to avoid that it goes
too far: this will be the Water Demand*.
Besides the Oil Absorption that characterizes a powdery material such as pigments and extenders, there is another parameter
that is more or less linked to it, this is the Water Demand of these
powders. It is sometimes, very empirically set at a value that is proportional to the Oil Absorption of the powder under consideration.
Intuitively, when comparing the viscosity of suspensions, at equal
concentration, of different powders in water, the highest viscosities
are reached with the powders having the highest water absorption.
And in the case there is a blend of pigment and extenders, the overall water demand will simply be the addition of the individual water
demands of these ingredients as there is no “packing” effect in the
diluted water phase. We therefore defined a Water Demand* (WD* )
concept that, indeed, does not quantify the real water demand of
the powder (therefore the *) but that is an indicator of the level of
water absorption by the powder, in the wet paint:
WD∗ = VS ·
PVC i · OAvol,i
The Volume Solids is part of the equation since a higher VS
means a higher water demand and WD* is the water demand
per unit volume of wet paint. In general, however, paint reformulations are performed at equal VS.
As said earlier, this is purely an indicator that already proved to
be very useful in many practical re-formulation work: the number
by itself is useless, but its variation, when modifying the pigment/extender blend composition, is meaningful!. Thus, when, i.e.
WD* is reduced then the corresponding low shear viscosity of the
paint is likely to get lower as well and will need to be adjusted with
212
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
Table 5
Reformulation of a high PVC paint at equal dry film porosity.
Binder Index
Paint PVC
Volume Solids
Clay PVC
Water Demand*
Dry binder volume
Fig. 15. Reformulation options at equal dry film porosity.
some additional thickener. It will be used in the re-formulation
example below.
4.2. Paint reformulation options
As a general rule, the re-formulation of a paint is made at equal
Volume Solids (VS) to that of the control paint, this will be the
case in this example. In some rare situations, however, it might be
economically interesting to slightly reduce the VS of the new formulation, provided that all the key properties are matching those
of control paint. There are, indeed, many reasons why one would
like to re-formulate an existing paint, like improving its properties, reduce its cost, raw material standardization, etc. Obviously,
the models that we will use are fairly limited as it does not include
any sub-models to represent the various paint properties. We will
therefore assume, with good reasons, that, if the total porosity of
the dry film is maintained constant, then the properties that depend
upon the film porosity would also remain relatively unchanged.
The control paint is using a blend of two extenders: Clay and
Carbonate. The Clay has a much higher Oil Absorption than the
Carbonate and replacing some or all of the Clay by the Carbonate
will contribute to a reduction of the overall Oil Absorption that will
allow a PVC increase in order to match the initial film porosity.
This is one re-formulation option, using the same binder as in
the control paint (BIbin = .7), that is illustrated in Fig. 15. This graph
was built in such a way that any point on the grayish part does correspond to a paint having the same overall porosity (˚air ) as that of
control paint. And, following the line corresponding to BIbin = .7, one
can see that, starting at the “control” level, the paint PVC (X axis)
can be increased while reducing the Clay PVC. And this PVC increase
could be quite significant! But, watching at the Water Demand* (Y
axis), one would observe a reduction that would result in a lowering
of the low shear viscosity of the paint that will need to be compensated by using more thickener. Moreover, besides the viscosity
reduction, the rheological behavior of the paint might be seriously
affected by the reduction of the Clay PVC as it does contribute to
a thixotropic effect that is probably highly desired. Therefore, in
the present case, the useful re-formulation domain, using the same
binder, would be very narrow.
We can increase the size of that domain, however. This would
be by using a binder with a higher binding capacity than that of
control paint. In this example, the Binder Index of the new binder
is.9. It corresponds to a small particle size, mono dispersed latex
polymer which is probably more expensive than the control binder.
The line BIbin = .9 line represents all the possible options with that
Control paint
Same WD*
Same Clay PVC
.7
70.0%
35.0%
18.3%
16.9
105
.9
76.3%
35.0%
15.2%
16.9
82.9 (−21.0%)
.9
74.7%
35.0%
18.3%
17.6
88.5 (−15.7%)
binder. One point on this line has an equal Water Demand* to control: it correspond to a paint (New) at 76.3% PVC and the Clay PVC
is 15.2% i.e. not too far away from the 18.3% of the control (i.e.
the thixotropic effect should not be too much affected). With the
same Clay PVC as control, the paint PVC would be 74.7%, but, as the
Water Demand* is higher, less thickener would be needed in the
new formulation for a similar low shear viscosity. The optimum
formulation lies most probably in between both these candidates.
The results are summarized in Table 5:
Thus, one option (same WD* ) should yield a paint with similar low shear viscosity to control, a slightly lower thixotropy and
will use 21% less dry binder volume (per liter of wet paint) while
the other option (same Clay PVC) will give a paint with a similar
thixotropy to control, about 16% in dry binder volume reduction
and will use less thickener for the low shear viscosity adjustment.
Given the fact that all paints have equal dry film porosity, they also
should have similar scrub resistance (should the binders have the
same Tg ) as well as the same dry hiding and opacity. Would the
binder volume reduction be large enough to compensate for the
higher new binder nominal cost ?
This was one example of a paint re-formulation using the above
models. Although fairly theoretical, these models are also based on
sound, practical data such as Oil Absorption and Binder Index and
can therefore be personalized by anyone’s own data. These models
can be very useful, for instance, in determining the boundaries of
an experimental domain (i.e. all below or all above CPVC) when
setting an experimental design (particularly the “Mixture Design”
type). Furthermore, it is also possible to link these models with
more practical ones such those obtained from these experimental
designs, i.e. by setting some of the data obtained from our models
(porosity, free binder, reduced PVC, etc.) as responses in the design.
It certainly will prove very useful.
5. Conclusion
The notion of paint film “geometry”, initially described by Patton, was generalized to any binder, and modeled in order to
take into account the effect of the pigment/extender packing that
is induced by a paint PVC variation. The mathematical models
that were described are “universal”, i.e. no longer differentiating
between oil and latex systems and can all be adjusted to individual experimental data (Oil Absorption curves of binary components
systems, Binder Index of the binder) that are very basic and easy
to determine. They are, therefore, very representative of the real
world. A number of well-known concepts were re-visited and,
sometimes, corrected, on the basis of these new models: the link
between the “reduced” PVC () and the film porosity was formally established and quantified as well as its relationship with
the binding capacity (BIbin ) of the binder; the concepts of Porosity
Index (PI) – for oil – and Latex Porosity Index (LPI) were amended
and unified under the new concept of Binder Porosity Index (BPI).
These models, as such, could already be used as a tool to help reformulating/optimizing a paint formulation. But they would be much
more powerful in predicting the paint film properties once they are
combined with the responses of experimental designs that cover
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
the experimental domain under consideration. Incidentally, these
models are already very well suited for the determination of the
boundaries of an experimental domain (i.e. all below or above CPVC
such as there is no discontinuity within the domain). Hopefully, the
paint formulator will find an immediate interest in applying these
models.
Table 6
Formulation calculation.
Volume of ingredients
Relation
TiO2 pigment
Extender(s)
Dry binder
Water phase
Total wet volume
Vtio2 = WV · VS · PVC tio2
Vext i = WV · VS · PVC ext i
Vbin = WV · VS · (1 − PVC)
Vwϕ = WV · (1
− VS)
WV = Vtio2 +
Vext i + Vbin + Vwϕ
Acknowledgement
Grateful thanks to Dr. Andrew Trapani, John Haigh and JeanLuc Blaise, from Dow Chemical, for their helpful comments and
suggestions.
composed of several ingredients (TiO2 and extenders) then the
PVC is the sum of the individual PVCs:
Vpig = Vtio2 +
Appendix A.
A.1. Glossary of the acronyms & technical terms
• Binder: This is the polymer that “binds” the pigment/extender
particles together in a paint. Under the generic term binder (subscript “bin”) are understood latex polymers (subscript “latex”)
and oil (subscript “oil”).
• BI: Binder Index. A number that characterize the binding capacity
of a polymer. BIbin (generic term) and BIlatex are interchangeable
in the above calculations; and BIoil is a particular case as BIoil = 1.
• BPI: Binder Porosity Index. It is the ratio between the volume
fraction of air to that of (air + binder) in above CPVC paints:
BPI = (˚air /(˚air + ˚bin )).
• CPVC: Critical Pigment Volume Concentration. The point, in a
PVC ladder, where there is just enough binder to fill in the
interstitial voids between – and coat – the pigment/extenders
particles. Above CPVC, the dry paint film becomes porous. It
is related to the binding capacity or Binder Index (BIbin ) of
the binder and the binder demand or Oil Absorption (OAvol ) of
the pigment/extender through the following generic relation:
CPVCbin = (BIbin /(BIbin + OAvol )).
• CPVCbin,PVC : For each PVC, an associated CPVC can be calculated
from the Binder Index of the binder (BIbin ) and the Oil Absorption of the pigmentation that was used at the PVC (OAvol,PVC ) as
per CPVCbin,PVC = (BIbin /(BIbin + OAvol,PVC )). CPVCoil,PVC is a particular
case when the binder is an oil (BIoil = 1) and the subscript “bin” or
“latex” are interchangeable.
• CPVCbin,CPVC : It corresponds to the special case when the PVC
and the calculated CPVC from the same pigmentation are equal:
CPVCbin,CPVC = (BIbin /(BIbin + OAvol,CPVC )). In practice, it corresponds
to the CPVC that is determined from a PVC ladder. CPVCoil,CPVC is a
particular case when the binder is an oil (BIoil = 1). The subscript
“bin” or “latex” for the Binder Index are interchangeable.
• DFD and DFT: Dry Film Density and Dry Film Thickness respectively.
• : Reduced PVC expressed as: = (PVC/CPVCbin,PVC ).
• OA: Oil Absorption. Expressed as a volume of oil per unit volume
of pigment or extender: OAvol . When the Oil Absorption varies
with the pigment/extender ratio in a formulation, and therefore
the PVC, it is designated as OAvol,PVC or OAvol,CPVC when the PVC is
at the CPVC of the system.
• PI & LPI: Porosity Index and Latex Porosity Index. Same definition
as BPI. PI does apply for oil systems while LPI is designed for latex
systems.
• Pigment: Unless otherwise specified, it represents both the TiO2
pigment and the extenders. Subscript: “pig”.
• PVC:
Pigment
Volume
Concentration.
PVC = (Vpig /(Vpig + Vbin )) = (˚pig /(˚pig + ˚bin )) with Vpig , Vbin
(alternatively ˚pig , ˚bin ) being the volume (fraction) of the pigmentation and dry binder respectively. Expressed as a decimal
number (.xx) or as a percentage (%). When the pigmentation is
213
Vext i ⇒ PVC = PVC tio2 +
PVC ext i
• ˚pig , ˚bin and ˚air are the volume fractions of the pigment, the
binder and the air in the dry paint film. ˚pig + ˚bin = ˚air = 1
• ˚FB : Volume fraction of the amount of “Free Binder” (below CPVC
paints). ˚FB = 1 − • VS: Volume Solids. There are two definitions for the Volume
Solids. The first one is the ratio between the volume of the dry
pigment and binder to the total wet volume (WV) of the paint:
VS = ((Vpig + Vbin )/WV). And the second one, denoted as VS* , is the
ratio between the volume of all non-volatiles (pigment, binder
and additives) and the total wet volume of the paint. The difference (although generally small) between the two is increasing as
Volume Solids decreases since more thickening agent has to be
used for viscosity control. VS* , generally experimentally determined, has to be used in every calculation involving the film
thickness (Spreading Rate, Scattering Power determination, etc.).
VS, on the other hand, is very useful in the calculation of formulations as the volume of each key ingredient can be easily
determined (Table 6).
Appendix B.
B.1. Oil Absorption models
The Oil Absorption of binary blends of TiO2 with an extender
were determined for a Clay and a Carbonate extender and modeled
according to the following mathematical model:
OAvol = Ax3 + Bx2 + Cx + D
where x represents the volume fraction of the extender in the blend
and A, B, C and D being constants that are specific to the extender
in the blend. Thus, for the TiO2 /Clay and the TiO2 /Carbonate
binary mixtures (the sale TiO2 pigment was used), the models are
represented by Tables 7–9 and the corresponding graphical representation is given in Fig. 16.
The TiO2 /Carbonate curve corresponds to that one in Fig. 3.
B.2. Oil Absorption and PVC
The OA curves in above figure must undergo a transformation before they can be exploited as they must be related to
the PVC of the paint. This is done by defining the variable x as:
x = ((PVC − PVCtio )/PVC), which is equal to the volume fraction of the
extender in the pigment/extender blend. The maximum value for
x is x = 1 − PVCtio when the maximum PVC (1 or 100%) is reached.
Therefore, only a part of the above graph (that one corresponding
Table 7
OAvol models.
TiO2 /Clay: OAvol = .7622 · x3 + .1305 · x2 − .4334 · x + .7541
TiO2 /Carbonate: OAvol = .4438 · x3 + .0376 · x2 − .8330 · x + .7569
214
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
Table 8
Correlation between the extender volume fraction and the PVC.
x
PVC
0
.15
.250
.20
.500
.30
.625
.40
.700
.50
.750
.60
.786
.70
.813
.80
.833
.90
.850
1.0
Table 9
PVC versus CPVCbin,PVC .
PVC
15.0%
17.5%
20.0%
22.5%
25.0%
27.5%
30.0%
36.4%
40.0%
50.0%
53.9%
55.5%
60.0%
70.0%
74.5%
80.0%
90.0%
100.0
TiO2 /Clay
TiO2 /Carbonate
OAvol,PVC
CPVCbin,PVC
CPVCoil,PVC
OAvol,PVC
CPVCbin,PVC
CPVCoil,PVC
.7541
.6971
.6659
.6524
.6505
.6557
.6654
.6993
.7204
.7762
–
.8034
.8241
.8640
–
.8970
.9248
.9482
34.7%
36.5%
37.5%
38.0%
38.1%
37.9%
37.5%
36.4%
35.7%
34.0%
–
–
32.7%
31.6%
–
30.8%
30.2%
29.7%
57.0%
58.9%
60.0%
60.5%
60.6%
60.4%
60.0%
–
58.1%
56.3%
–
55.5%
54.8%
53.7%
–
52.7%
52.0%
51.3%
.7570
.6400
.5580
.5000
.4581
.4277
.4053
–
.3593
.3444
.3421
–
.3405
.3405
.3417
.3429
.3457
.3485
34.6%
38.5%
41.8%
44.5%
46.6%
48.3%
49.7%
–
52.7%
53.7%
53.9%
–
54.0%
54.0%
–
53.8%
53.6%
53.4%
56.9%
61.0%
64.2%
66.7%
68.6%
70.0%
71.2%
–
73.6%
74.4%
–
–
74.6%
74.6%
74.5%
74.5%
74.3%
74.2%
to: 0 ≤ x ≤ 1 − PVCtio ) will be used for the transformation of the
graph into that of Fig. 6. In the example of this paper, the TiO2 PVC
is set at 15% and the relation between x and the PVC, is illustrated in
Table 8, which explains the distortions that were observed between
both sets of curves.
B.3. PVC and CPVCbin,PVC
Based on the above Oil Absorption data and the Binder Index of
the binder, it is now possible to calculate the corresponding Critical
PVC or CPVCbin,PVC , for both TiO2 /Clay and TiO2 /Carbonate binary
systems (fixed TiO2 PVC at 15%) as in Table 9.
Reminder: CPVCbin,PVC is the CPVC that is calculated from the pigment/extender blend at the considered PVC, i.e. 37.5% at PVC = 30%.
The OAvol,PVC data of each system were used to draw the graph
in Fig. 6. The CPVC data were used to draw the graph in Fig. 17.
The straight line running across each graph is simply visualizing
the points on the same slope, where the paint PVC is at the CPVC of
the system CPVCbin,CPVC .
It compares the Clay and the Carbonate system (note the data
shift on the Y axis). These graphs also do illustrate the relation between CPVCbin,PVC and the PVC: On the low PVC side,
PVC < CPVCbin,PVC and it is the reverse for the high PVCs. There is
one point where both PVC and CPVCbin,PVC are equal: this is at the
critical PVC of the system, i.e. when CPVC = CPVCbin,CPVC . On each
graph, there is a comparison with an oil based system.
Appendix C.
C.1. Geometry of a paint film
The volume fraction of ˚pig can be expressed in two different
ways:
• as per Eq. (16):
˚pig =
PVC
PVC(BI bin + OAvol,PVC ) + (1 − BI bin )
• or as per Eq. (17):
˚pig =
Fig. 16. OAvol of TiO2 /extenders binary mixtures
PVC · CPVC oil,PVC (1 − CPVC bin,PVC )
PVC(1 − CPVC oil,PVC ) + (CPVC oil,PVC − CPVC bin,PVC )
Eq. (17) is derived from Eq. (16) in the following way:
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
215
Fig. 17. CPVC as a function of the PVC.
CPVCbin,PVC is expressed as: CPVCbin,PVC = (BIbin /(BIbin + OAvol,PVC ))
we can
from where
get an expression of BIbin : BI bin =
CPVC bin,PVC
1−CPVC bin,PVC
OAvol,PVC
.
In
a
similar
way,
CPVCoil,PVC
is
expressed
as:
CPVCoil,PVC = (1/(1 + OAvol,PVC )) which yields an expression of
OAvol,PVC :
OAvol,PVC =
D.1. Dry film thickness
The volume of the dry paint film (Vdf ) is the sum of the volumes of
the film components, included the entrapped air when formulated
above the CPVC:
Vdf = Vpig + Vbin + Vair
On the other hand, the Volume Solids of the paint is defined by
1 − CPVC oil,PVC
CPVC oil,PVC
VS =
Vpig + Vbin
and, therefore of BIbin :
BI bin =
1 − CPVC oil,PVC
CPVC oil,PVC
CPVC bin,PVC
1 − CPVC bin,PVC
And by plugging these expressions of BIbin and OAvol,PVC into Eq.
(16), one gets the equivalent expression of ˚pig in Eq. (17).
with, Vwf the volume of the corresponding wet applied film. By
combining both equations, this can be re-written as:
VS · Vwf = Vpig + Vbin = Vdf − Vair
And by dividing all terms by Vdf , one gets:
VS ·
C.1.1. Construction of Figs. 4 and 5:
Below CPVC, the construction of all the graphs is very simple
since ˚pig = PVC and ˚bin = 1 − PVC, since there is no film porosity
(˚air = 0). It is above CPVC that the calculation is a bit more complex
and details are reported in following tables (there are more data on
the tables than needed, i.e. the Binder Index of the “composite”
binder is indicated for reference) (Tables 10 and 11).
Vwf
Vwf
Vdf
=1−
Vair
= 1 − ˚air
Vdf
since (Vair /Vdf ) = ˚air by definition. If an equal surface is considered,
then the ratio of the volumes becomes a ratio between film thicknesses, wet (WFT) and dry (DFT) and replacing ˚air by its expression
in Eq. (7), one obtains the equation of the dry film thickness, which
is valid over the whole PVC spectrum (below and above CPVC).
DFT = WFT · VS ·
C.1.2. Construction of Fig. 7
See Tables 12 and 13.
C.1.3. Construction of Fig. 8
See Tables 14 and 15.
Appendix D.
Preamble: The calculations below do neglect the presence of
ingredients other than those of the pigmentation and the binder, i.e.
the additives of a formulation. Had they been included, their effect
on modifying the data described below would have been relatively
low, and do not impact the conclusions that could be made.
PVC
˚pig
The evolution of the dry film thickness over the whole PVC range
was calculated, as an example, in both situations that involve binary
blends of TiO2 either with a Clay or a Carbonate as illustrated in
Figs. 7 and 8 for BIbin = 0.4. The calculation below was made for a
wet applied film thickness of 100 and a set Volume Solids of 35%.
On the tables below, CPVC stands for CPVCbin,CPVC and, obviously,
˚pig is calculated from Eq. (16) (Table 16).
And for the TiO2 /Carbonate system (Table 17).
D.2. Dry film density
• Eq. (20): DFD = (˚pig /PVC) · [PVC · Dpig + (1 − PVC) · Dbin ] gives the
density of a dry film over the whole PVC range (with ˚pig = PVC
216
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
Table 10
Data for Fig. 5.
BIbin = .4
.3
.4
.5
.6
.7
.8
.9
1.0
˚pig
˚bin
˚air
Paint PVC
.2857
.2857
.7143
0
.2941
.6863
.0196
.3448
.5172
.1379
.3846
.3846
.2308
.4167
.2778
.3056
.4430
.1899
.3671
.4651
.1163
.4186
.4839
.0538
.4624
.5
0
.5
OAvol,PVC
CPVCoil,PVC
CPVCbin,PVC
BIcomp,PVC
CPVCcomp,PVC
1.0
.5
.2857
.4
.2857
1.0
.5
.2857
.4167
.2941
1.0
.5
.2857
.5263
.3448
1.0
.5
.2857
.6250
.3846
1.0
.5
.2857
.7143
.4167
1.0
.5
.2857
.7955
.4430
1.0
.5
.2857
.8696
.4651
1.0
.5
.2857
.9375
.4839
1.0
.5
.2857
1.0
.5
Table 11
Data for Fig. 4.
BIbin = 1.0
Paint PVC
.5
˚pig
˚bin
˚air
.5
.5
0
OAvol,PVC
CPVCoil,PVC
CPVCbin,PVC
BIcomp,PVC
CPVCcomp,PVC
1.0
.5
.5
1.0
.5
.6
.7
.8
.9
1.0
.5
.3333
.1667
.5
.2143
.2857
.5
.1250
.3750
.5
.0556
.4444
.5
0
.5
1.0
.5
.5
1.0
.5
1.0
.5
.5
1.0
.5
1.0
.5
.5
1.0
.5
1.0
.5
.5
1.0
.5
1.0
.5
.5
1.0
.5
Table 12
Data for Fig. 7.
BIbin = .4
Paint PVC
˚pig
˚bin
˚air
Clay PVC
OAvol,PVC
CPVCoil,PVC
CPVCbin,PVC
BIcomp,PVC
CPVCcomp,PVC
.3639
.4
.5
.6
.7
.8
.9
1.0
.3639
.6361
0
.3816
.5724
.0459
.4208
.4208
.1583
.4496
.2998
.2506
.4715
.2021
.3265
.4885
.1221
.3894
.5022
.0558
.4421
.5133
0
.4867
.2139
.6993
.5885
.3639
.4
.3639
.25
.7204
.5813
.3570
.4446
.3816
.35
.7762
.5630
.3401
.5640
.4208
.45
.8241
.5482
.3268
.6732
.4496
.55
.8640
.5365
.3165
.7706
.4715
.65
.8971
.5271
.3084
.8567
.4885
.75
.9248
.5196
.3019
.9328
.5022
.85
.9482
.5133
.2967
1.0
.5133
Table 13
Data for Fig. 7 (oil).
BIbin = 1.0
Paint PVC
.5545
.6
.7
.8
.9
1.0
˚pig
˚bin
˚air
.5545
.4455
0
.5482
.3655
.0863
.5365
.2299
.2336
.5271
.1318
.3411
.5196
.0577
.4227
.5133
0
.4867
Clay PVC
OAvol,PVC
CPVCoil,PVC
CPVCbin,PVC
BIcomp,PVC
CPVCcomp,PVC
.4045
.8034
.5545
.5545
1.0
.5545
.45
.8241
.5482
.5482
1.0
.5482
.55
.8640
.5365
.5365
1.0
.5365
.65
.8971
.5271
.5271
1.0
.5271
.75
.9248
.5196
.5196
1.0
.5196
.85
.9482
.5133
.5133
1.0
.5133
Table 14
Data for Fig. 8.
BIbin = .4
Paint PVC
.5390
.6
.7
.8
.9
1.0
˚pig
˚bin
˚air
.5390
.4610
0
.5745
.3830
.0424
.6258
.2682
.1060
.6698
.1675
.1627
.7081
.0787
.2133
.7416
0
.2585
Carbonate PVC
OAvol,PVC
CPVCoil,PVC
CPVCbin,PVC
BIcomp,PVC
CPVCcomp,PVC
.3890
.3421
.7451
.5390
.4
.5390
.45
.3405
.7460
.5402
.4599
.5745
.55
.3409
.7458
.5400
.5700
.6258
.65
.3429
7446
.5384
.6957
.6698
.75
.3457
.7431
.5365
.8383
.7081
.85
.3485
.7416
.5344
1.0
.7416
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
217
Table 15
Data for Fig. 8 (oil).
BIbin = 1.0
Paint PVC
.7453
.8
.9
1.0
˚pig
˚bin
˚air
.7453
.2547
0
.7446
.1862
.0692
.7431
.0826
.1743
.7416
0
.2585
Carbonate PVC
OAvol,PVC
CPVCoil,PVC
CPVCbin,PVC
BIcomp,PVC
CPVCcomp,PVC
.5953
.3417
.7453
.7453
1.0
.7453
.65
.3429
7446
7446
1.0
7446
.75
.3457
.7431
.7431
1.0
.7431
.85
.3485
.7416
.7416
1.0
.7416
Table 16
Dry film thickness for the TiO2 /Clay system.
<CPVC
CPVC
>CPVC
PVC
VS
<.3639
.35
.3639
.35
.4
.35
.5
.35
.6
.35
.7
.35
.8
.35
.9
.35
1.0
.35
OAvol,PVC
˚pig
˚air
–
= PVC
0
.6993
.3639
0
.7204
.3816
.0459
.7762
.4208
.1583
.8241
.4496
.2506
.8640
.4714
.3265
.8970
.4885
.3894
.9247
.5021
.4421
.9482
.5133
.4867
WFT()
DFT()
100
35
100
35
100
36.7
100
41.6
100
46.7
100
52.0
100
57.3
100
62.7
100
68.2
Table 17
Dry film thickness for the TiO2 /Carbonate system.
<CPVC
CPVC
>CPVC
PVC
VS
<.5390
.35
.5390
.35
.6
.35
.7
.35
.8
.35
.9
.35
1.0
.35
OAvol,PVC
˚pig
˚air
–
= PVC
0
.3421
.5390
0
.3405
.5745
.0424
.3409
.6258
.1060
.3429
.6698
.1627
.3456
.7081
.2133
.3485
.7415
.2585
WFT()
DFT()
100
35
100
35
100
36.6
100
39.2
100
41.8
100
44.5
100
47.2
Table 18
Dry film density.
Clay
PVC
.2
.3
.3639
.4
.5
–
.6
.7
.8
.9
1.0
Carbonate
Dpig
DFD
PVC
Dpig
DFD
3.72
3.35
3.21
3.16
3.05
–
2.98
2.92
2.88
2.85
2.83
1.62
1.78
1.87
1.84
1.75
–
1.67
1.6
1.54
1.49
1.45
.2
.3
3.75
3.40
–
3.23
3.12
3.09
3.05
3.00
2.96
2.93
2.91
1.63
1.79
–
1.95
2.11
2.17
2.17
2.17
2.17
2.16
2.16
below the CPVC). In the case of a binary pigment/extender blend,
the density of the pigment is given by:
–
.4
.5
.5390
.6
.7
.8
.9
1.0
Appendix E.
E.1. “Reduced PVC” and the film porosity ˚air
Dpig =
Dtio2 · PVC tio2 + Dext · (PVC − PVC tio2 )
PVC
with Dtio2 and Dext the densities of the TiO2 pigment (4.1) and of
the extender (2.6 for the Clay and 2.7 for the Carbonate) respectively. The binder density, Dbin , is obviously that of the dry binder
(1.1 in this example) (Table 18).
The “reduced PVC”, , is expressed as:
=
BI bin + OAvol,PVC
PVC
⇒ PVC ·
CPVC bin,PVC
BI bin
Therefore PVC · (BIbin + OAvol,PVC ) = · BIbin which plugged into
˚pig (Eq. (16)) yields:
˚pig =
PVC
1 + BI bin ( − 1)
218
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
Table 19
/˚air invariant.
= 1.1
= 1.3
= 1.7
= 2.5
BIbin
˚air
BIbin
˚air
BIbin
˚air
BIbin
˚air
.4
.5
.6
.7
.8
.9
1.0
.0385
.0476
.0566
.0654
.0741
.0826
.0909
.4
.5
.6
.7
.8
.9
1.0
.1071
.1304
.1525
.1736
.1936
.2126
.2308
.4
.5
.6
.7
.8
.9
1.0
.2188
.2593
.2958
.3289
.3590
.3865
.4118
.4
.5
.6
.7
.8
.9
1.0
.3750
.4286
.4737
.5122
.5455
.5745
.6000
Table 20
TiO2 /Clay PVC curves.
PVC = 50%
PVC = 60%
PVC = 70%
PVC = 80%
PVC = 100%
BIbin
˚air
BIbin
˚air
BIbin
˚air
BIbin
˚air
BIbin
˚air
.4
.5
.6
.7
.7762
–
–
.1583
.1213
.0810
.0367
0
–
–
.4
.5
.6
.7
.8
.9
1.0
.2506
.2275
.2028
.1766
.1485
.1185
.0863
.4
.5
.6
.7
.8
.9
1.0
.3265
.3126
.2981
.2830
.2673
.2508
.2336
.4
.5
.6
.7
.8
.9
1.0
.3894
.3818
.3741
.3661
.3580
.3497
.3411
.4
.5
.6
.7
.8
.9
1.0
.4867
.4867
.4867
.4867
.4867
.4867
.4867
Table 21
TiO2 /Carbonate PVC curves.
PVC = 60%
PVC = 80%
BIbin
˚air
.4
.5
.5108
–
–
–
–
.1583
.1213
0
–
–
–
–
PVC = 100%
BIbin
˚air
BIbin
˚air
.4
.5
.6
.7
.8
.9
1.0
.1627
.1485
.1337
.1184
.1026
.0862
.0692
.4
.5
.6
.7
.8
.9
1.0
.2585
.2585
.2585
.2585
.2585
.2585
.2585
Table 22
Free binder of TiO2 /extenders systems.
PVC
TiO2 /Clay
OAvol,PVC
15.0%
17.5%
20.0%
22.5%
25.0%
27.5%
30.0%
36.4%
40.0%
50.0%
53.9%
.7541
.6971
.6659
.6524
.6505
.6557
.6654
.6993
–
–
–
TiO2 /Carbonate
CPVCbin,PVC
34.7%
36.5%
37.5%
38.0%
38.1%
37.9%
37.5%
36.4%
–
–
–
and since ˚air = 1 − (˚pig /PVC), one gets, after rearrangement:
−1=
˚air
BI bin (1 − ˚air )
FB
.5657
.5205
.4647
.4079
.3438
.2744
.2000
0
–
–
–
OAvol,PVC
CPVCbin,PVC
FB
.7570
.6400
.5580
.5000
.4581
.4277
.4053
–
.3593
.3444
.3421
34.6%
38.5%
41.8%
44.5%
46.6%
48.3%
49.7%
–
52.7%
53.7%
53.9%
.5665
.5455
.5215
.4944
.4635
.4306
.3964
–
.2410
.0689
0
systems described earlier will be considered in alternate order
(Tables 20 and 21).
E.2. Free binder
See Table 22.
i.e. it depends on both the porosity of the dry film and the binding
capacity of the binder. It is therefore independent of the pigmentation and constitutes an “invariant” or a template that would apply
to any system. Here are the data that were computed to construct
this graph (Table 19).
On top of this graph will come the PVC graphs that are
specific to the pigmentation: the TiO2 /Clay and TiO2 /Carbonate
Appendix F.
F.1. Binding capacity index of a latex polymer (BIlatex )
We saw in Eq. (9) that the binding capacity of a latex polymer
was equal to the ratio of the volume of oil (Voil ) needed to bind a
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
defined volume of pigment at the CPVCoil by the volume of latex
polymer (Vlatex ) needed to bind the same volume of pigment at the
CPVClatex :
CPVC oil =
Vpig
Vpig + Voil
⇒ Voil = Vpig
1 − CPVC and, in a similar way,
CPVC latex =
Vpig
Vpig + Vlatex
⇒ Vlatex = Vpig
1 − CPVC
latex
CPVC latex
yielding a new expression of the Binder Index for a latex polymer:
BI latex =
(1 − CPVC oil )CPVC latex
CPVC oil (1 − CPVC latex )
(34)
Obviously, the subscript “latex” could be replaced by “bin” as a
matter of generalizing the concept to any binder type. But we will
keep it that way for the moment.
F.2. Porosity index, latex porosity & Binder Porosity Index
The Binder Porosity Index (BPI) – that applies to above CPVC systems – is defined as the ratio between the volume of air entrapped
into the dry film (re. the film porosity) to the total volume or
air + binder, i.e. the volume of the “composite” binder defined earlier: BPI = (˚air /(˚air + ˚bin )). Knowing that ˚air + ˚bin + ˚pig = 1 and
that ˚air = 1 − (˚pig /PVC) the Binder Porosity Index can be rewritten
as:
BPI =
PVC − ˚pig
PVC(1 − ˚pig )
(35)
It is the same concept that Stieg and Patton used to define the
Porosity Index (PI) of a binder:
PI =
˚air
˚air + ˚bin
But it was applying to very specific conditions: an oil binder
and a pigmentation with a fixed Oil Absorption, i.e. a case that
was met in Fig. 4, where ˚pig remains constant above the CPVC:
˚pig = CPVCoil .
PI could be re-written as:
PI =
(˚air + ˚bin ) − ˚bin
˚bin
=1−
˚air + ˚bin
1 − ˚pig
CPVC oil (1 − PVC)
PVC(1 − CPVC oil )
(36)
It is well understood that CPVCoil has a fixed value and is not affected
by a PVC change since the Oil Absorption is constant.
By analogy with PI, the authors did define a Latex Porosity Index
(LPI) that applies to latex polymers:
LPI = 1 −
CPVC latex (1 − PVC)
PVC(1 − CPVC latex )
(37)
The ratio between the volume fraction of the latex polymer
(1 − LPI) to that of the oil (1 − PI) was set to represent the variable
e (Patton):
CPVC latex (1 − PVC) PVC(1 − CPVC oil )
1 − LPI
e=
=
·
1 − PI
PVC(1 − CPVC latex ) CPVC oil (1 − PVC)
e=
LPI = 1 − e ·
CPVC latex (1 − CPVC oil )
CPVC oil
CPVC latex
= e·
⇒
1 − CPVC latex
1 − CPVC oil
CPVC oil (1 − CPVC latex )
CPVC oil (1 − PVC)
PVC(1 − CPVC oil )
(40)
where the CPVClatex term has disappeared. It can already be seen
in the expression of e in the first part of Eq. (39) that the variable
e would be a constant if CPVClatex is also constant, i.e. that the pigmentation has a fixed OA (already the case in the oil system). It
will, otherwise, remain a variable that depends upon the PVC. In
order to define e, the authors did remark that at the latex CPVC
(PVC = CPVClatex ), there is no porosity and LPI = 0. The variable e can
then be expressed as:
e=
CPVC latex (1 − CPVC oil )
CPVC oil (1 − CPVC latex )
(41)
which is identical to Eq. (34). Therefore e = BIlatex when
PVC = CPVClatex . Does-it mean that it should remain constant
at any PVC ?. There is nothing to prove it but the authors decided
that it was the case (true when the pigmentation has a constant
OA). Therefore, according to Stieg, Patton;
LPI = 1 − BI latex ·
CPVC oil (1 − PVC)
PVC(1 − CPVC oil )
(42)
which will become equal to PI when setting BIlatex = 1. By comparison, BPI is expressed as (Eq. (32)):
BPI = 1 − BI comp,PVC ·
CPVC oil,PVC (1 − PVC)
PVC(1 − CPVC oil,PVC )
(43)
and when there is a constant OA for the pigmentation, BPI and LPI
will only differ by one point: BIcomp,PVC for BPI and BIlatex for LPI. Even
more, when the PVC is at the CPVC of the binder, then there is no air
in the binder/air composite and BIcomp,CPVC = BIbin and LPI = BPI (only
on those specific conditions). The fact that PI and LPI were defined
under very specific conditions does not plead for their usage in a
general situation and the generic Eq. (35) seen earlier could be used
to illustrate the difference between BPI and LPI:
(B | L)PI =
PVC − ˚pig
PVC(1 − ˚pig )
⇒ ˚pig =
PVC(1 − (B | L)PI)
1 − PVC · (B | L)PI
(44)
with (B | L)PI being either BPI or LPI. It shows that it is possible to
calculate ˚pig from the knowledge of the latter.
• For BPI: ˚pig = (PVC(1 − BPI)/(1 − PVC · BPI)):
where (1 − PI) represents the binder volume fraction in the
binder/air composite. After replacement of ˚bin by its expression
in Eq. (8), PI becomes:
PI = 1 −
and by replacing CPVClatex /(1 − CPVClatex ) by its above expression
(Eq. (39)) in the LPI equation (37), one gets a new expression of LPI:
oil
CPVC oil
219
The numerator PVC(1 − BPI) can be expressed as follows:
1 − BPI = BI comp,PVC ·
= BI comp,PVC ·
CPVC oil,PVC (1 − PVC)
PVC(1 − CPVC oil,PVC )
1 − PVC
OAvol,PVC · PVC
after replacing CPVCoil,PVC by: CPVCoil,PVC = (1/(1 + OAvol,PVC )). It
yields:
PVC(1 − BPI) = BI comp,PVC ·
1 − PVC
OAvol,PVC
– And for the denominator (still replacing CPVCoil,PVC by its
expression):
1 − PVC.BPI = (1 − PVC) + BI comp,PVC ·
(38)
(39)
1 − PVC
OAvol,PVC
Therefore:
˚pig = (BIcomp,PVC · (1 − PVC)/((BIcomp,PVC + OAvol,PVC )
(1 − PVC))) = CPVCcomp,PVC , i.e. a result which varies with the PVC
and is in agreement with our expectation.
220
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
Table 23
BPI versus LPI in a latex (BIbin = .4) system and fixed OAvol .
CPVC
PVC
>CPVC
.2857
.3
.4
.5
.6
.7
.8
.9
1.0
.7143
.2857
.0196
.6863
.2941
.1379
.5172
.3448
.2308
.3846
.3846
.3056
.2778
.4167
.3671
.1899
.4430
.4186
.1163
.4651
.4624
.0538
.4839
.5
0
.5
.4167
1.0
.5
.5263
1.0
.5
.6250
1.0
.5
.7143
1.0
.5
.7955
1.0
.5
.8696
1.0
.5
.9375
1.0
.5
1.0
1.0
.5
.2857
.0278
.2941
.2105
.3448
.3750
.3846
.5238
.4167
.6591
.4430
.7826
.4651
.8958
.4839
1.0
.5
.2857
.0667
.2857
.4000
.2857
.6000
.2857
.7333
.2857
.8286
.2857
.9000
.2857
.9556
.2857
1.0
.2857
˚air
˚bin
˚pig
0
BIcomp,PVC
OAvol,PVC
CPVCoil,PVC
.4
1.0
.5
BPI
˚pig from BPI
0
LPI
˚pig from LPI
0
Table 24
Refractive Index of the “composite” binder.
BPI
Parallel
Drude
Maxwell–Garnett
Bruggeman
Lorentz
Series
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.500
1.450
1.400
1.350
1.300
1.250
1.200
1.150
1.100
1.050
1.000
1.500
1.458
1.414
1.369
1.323
1.275
1.225
1.173
1.118
1.061
1.000
1.500
1.449
1.400
1.349
1.299
1.249
1.200
1.150
1.100
1.051
1.000
1.500
1.449
1.397
1.345
1.293
1.242
1.191
1.141
1.092
1.045
1.000
1.500
1.442
1.387
1.333
1.282
1.232
1.183
1.136
1.090
1.044
1.000
1.5
1.429
1.364
1.304
1.250
1.200
1.154
1.111
1.071
1.034
1.000
• For LPI: ˚pig = (PVC(1 − LPI)/(1 − PVC · LPI)) and following a similar
calculation path (noting that OA is constant), one gets:
˚pig =
BI latex
= CPVC latex
BI latex + OAvol
As CPVClatex is constant, this result is in contradiction with
the fact that, above CPVC, ˚pig is variable in a latex system:
CPVClatex ≤ ˚pig ≤ CPVCoil . Therefore, the expression of LPI does not
appear to be right.
These results are illustrated in Table 23, corresponding to the
data of Fig. 5, i.e. with a fixed Oil Absorption.
Apart from the “Series” model that seems to be undervaluing the
Refractive Index of the “composite”, all other models are in good
agreement. The largest discrepancy between those is reached at
BPI = 0.5 where the average value for the six models involved is
1.241 for a standard deviation of 0.025: the “Parallel” model appears
to be well suited in a coatings context.
Appendix G.
G.1. Oil Absorption of a pigment/extenders ternary blend
The Oil Absorption of a TiO2 pigment and of a blend of extenders
such as Clay and Carbonate (in our example) is expressed by:
F.3. Refractive index of the “composite” binder
OAvol,PVC = yOAclay,PVC + (1 − y)OAcarb,PVC
The model that we are proposing to calculate the Refractive
Index of the “composite” binder might appear very crude: a simple linear relationship (called the “Parallel” model). A number of
authors, on the other hand have developed more sophisticated
models that are expressed in the table below. These were extracted
from a publication of the UCLA on the web and actualized with the
variable BPI.
where y is the volume fraction of the Clay in the Clay/Carbonate
blend. It is the result of a linear combination of the Oil Absorption of the individual TiO2 /extender binary blend. And, by author’s
experience, it works! (although data are not available here). It, definitely, works well when the difference in particle size between the
extenders and the TiO2 is large, which is almost always the case
(even with the so-called fine or very fine extenders). In the present
example, the Clay/Carbonate volume (or PVC) ratio is 1/2. The oil
Absorption of this blend is represented in the graph (Fig. 18).
Models
Equation
Parallel
Drude
ncomp = BPI · nair + (1 − BPI) · nbin
n2comp = BPI · n2air + (1 − BPI) · n2bin
Maxwell–Garnett
n2comp = n2bin 1 −
Bruggeman
BPI ·
n2 −n2
comp
air
n2 +2n2
comp
air
Lorentz–Lorentz
Series
n2
−1
comp
= BPI ·
n2
+2
comp
1
= 1−BPI
ncomp
nbin
G.2. Re-formulation of an above CPVC paint
3BPI(n2 −n2 )
bin
air
2n2 +n2 +BPI(n2 −n2 )
bin
air
+ (1 − BPI) ·
n2 −1
air
n2 +2
air
+ nBPI
air
bin
air
n2 −n2
comp
bin
n2 +2n2
comp
bin
+ (1 − BPI) ·
=0
n2 −1
bin
n2 +2
bin
Which, after computation, yielded following data (Table 24).
displaying the Refractive Index of the “composite” binder,
according to the model that was used, as a function of BPI.
Here are the data that did allow the construction of the graph
in Fig. 15. There is to note that a math software, capable of solving simultaneous equations, is absolutely needed to perform the
calculation as it allows to set a constraint while performing the calculation. This constraint is that the dry film porosity (˚air ) is kept
constant and equal to that of the control formulation, with, in some
case, an additional constraint: that the Water Demand* remains
constant too. The control paint has a PVC of 70%, a TiO2 PVC of 15%
B. Lestarquit / Progress in Organic Coatings 90 (2016) 200–221
221
Table 25
BI curves at constant dry film porosity.
BIbin = .7
BIbin = .9
WD*
PVC
16.95
15.72
14.56
13.40
17.62
16.30
15.06
13.82
70.0%
73.9%
77.7%
81.5%
74.7%
78.0%
81.2%
84.5%
Clay PVC
18.3%
12.0%
6.0%
0
18.3%
12.0%
6.0%
0
OAvol,PVC
.5152
.4507
.3948
.3493
.5071
.4469
.3938
.3441
1.22
1.22
1.22
1.22
1.17
1.17
1.17
1.17
Table 26
Clay PVC curves at constant dry film porosity
Clay PVC
18.3%
12.0%
*
WD
PVC
16.95
17.31
17.62
17.75
–
70.0%
72.5%
74.7%
75.6%
–
WD
*
15.54
15.72
16.03
16.30
16.42
6.0%
*
PVC
WD
72.7%
73.9%
76.1%
78.0%
78.8%
14.40
14.56
14.83
15.06
15.16
0
PVC
WD*
PVC
76.6%
77.7%
79.6%
81.2%
81.9%
13.27
13.40
13.63
13.82
13.90
80.6%
81.5%
83.1%
84.5%
85.0%
Appendix H. Supplementary Data
Supplementary data associated with this article can be found,
in the online version, at http://dx.doi.org/10.1016/j.porgcoat.2015.
09.023.
References
Fig. 18. OAvol of ternary blends.
and a Clay PVC of 18.3% for a Volume Solids that is set at 35%. The
TiO2 PVC is not affected (nor the VS) by the re-formulation process.
The PVC is represented on the X axis of the graph and the Water
Demand* is on the Y axis. There are two types of lines that were
drawn: the lines corresponding to a fixed Binder Index (BIbin = 0.7
for the control, and 0.9) and those corresponding to a set Clay PVC
(18.3 – 12.0 – 6.0 and 0% PVC respectively). Here are the data that
were used (Tables 25 and 26).
[1] T.C. Patton, Paint, Flow and Pigment Dispersion, 2nd ed., Wiley Interscience,
1979.
[2] F.B. Stieg, R.I. Ensminger, Offic. Dig. 33 (1961) 792.
[3] F.B. Stieg, J. Paint Technol. 39 (1967) 701.
[4] F.B. Stieg, J. Paint Technol. 41 (1969) 243.
[5] F.B. Stieg, J. Paint Technol. 42 (1970) 329.
[6] P. Berardi, J. Paint Technol. 27 (1963) 24.
[7] E.J. Schaller, J. Paint Technol 40 (1968) 433.
[8] Philadelphia Paint and Varnish Production Club, Offic. Dig. 31 (1959) 1490.
[9] Philadelphia Paint and Varnish Production Club, Offic. Dig. 33 (1961) 1437.
[10] G.P. Bierwagen, T.K. Hay, Prog. Org. Coat. 3 (1975) 281.
[10] https://www.seas.ucla.edu/∼pilon/OpticsNanoporous.html.
Bernard Lestarquit graduated from the School of Chemistry (ENSCS) of the
University of Strasbourg (France). After a first coatings experience with a TiO2 manufacturer, Tioxide (British Titan Products), he joined the European Laboratories of
Rohm and Haas, then located in Zürich (Switzerland), later to get transferred to
Sophia Antipolis (France), and spent several years as well at the Springhouse, PA
(USA), Research Center of the company where he occupied various positions. A Rohm
and Haas Research Fellow, he was heading the European Section of the Emerging
Technologies Department at the time of the acquisition of the company by Dow
Chemical. He is currently acting as a consultant and teaches Coatings Technology.