Geometry of a paint film: Basics revisited

Progress in Organic Coatings 90 (2016) 200–221

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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: [email protected] 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

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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

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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.

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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

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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

PVC − PVC tio2 PVC

+C

3

PVC − PVC tio2 PVC



+B

 +D

PVC − PVC tio2 PVC

2

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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

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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

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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)

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

which, after computation yields:



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 −

Fig. 11. Interdependence of , BIbin and ˚air .

(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

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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

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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



Fig. 13. Comparative Free Binder level in the (BIbin = .4) binder system

˚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

(31)

1 1 + 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 =



OAvol,PVC can also be written differently, according to: 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:

1 − PVC PVC · 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.

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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

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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

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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

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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

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

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)

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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 Lorentz–Lorentz Series



3BPI(n2 −n2 ) bin air 2n2 +n2 +BPI(n2 −n2 ) bin air bin air n2 −n2 n2 −n2 BPI · 2air comp + (1 − BPI) · 2bin comp = 2 n +2n2 n +2n comp comp air bin n2 −1 n2 −1 n2 −1 comp = BPI · air + (1 − BPI) · bin n2 +2 n2 +2 n2 +2 comp air bin 1 ncomp

=

1−BPI nbin

+

G.2. Re-formulation of an above CPVC paint 0

BPI nair

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.