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Fate of a coalescing aid after latex paint application.

Abstract Water-based coatings require coalescing aids to achieve properties equivalent to solvent-borne paints. A common coalescing aid in latex paints is 2,2,4-trimethyl-1,3-pentanediol monoisobutyrate (TMPD-MIB). The relatively large quantity of TMPD-MIB used in latex paints has raised concerns regarding its emissions to both indoor and outdoor atmospheres. In this study, a one-dimensional dual (paint and material) layer diffusion model was developed to estimate emissions of TMPD-MIB from two latex paints applied to gypsum board. The paints contained different pigment volume concentrations (PVC) and different amounts of TMPD-MIB. Different modeling approaches were used depending on the PVC of the paint. The proposed model for paint drying and TMPD-MIB emissions was tested with data from previous chamber experiments. Experimental data were first used for purposes of parameter estimation, and the model was then compared against an independent experimental dataset. The diffusion coefficient of the paint layer was adjusted as a function of the water content remaining in the wet paint film. The effective diffusion coefficient of TMPD-MIB in the paint layer was found to be dependent on the PVC and water content of the paint.

Keywords Coalescing aid, Emissions, Gypsum board, Paint, 2,2,4-Trimethyl-1,3-pentanediol monoisobutyrate
Nomenclature

[j.sub.A]         Rate of mass transfer per unit area (mg [m.sup.-2]
                  [h.sup.-1])

[c.sub.A]         Concentration of TMPD-MIB in paint film or gypsum
                  board (g [m.sup.-3])

z                 Space coordinate measured normal to the section
                  (m)

[D.sub.AL1]       Diffusion coefficient for TMPD-MIB in paint film
                  ([m.sup.2] [h.sup.-1])

[D.sub.AL2]       Diffusion coefficient for TMPD-MIB in gypsum board
                  ([m.sup.2] [h.sup.-1])

[R.sub.A]         Kinetic retardation factor for transport in paint
                  film or gypsum board

[c.sub.L1]        Concentration of TMPD-MIB in layer 1 (paint) (mg
                  [m.sup.-3])

[c.sub.L2]        Concentration of TMPD-MIB in layer 2 (material) (mg
                  [m.sup.-3])

[D.sub.eff1]      Effective diffusion coefficient of TMPD-MIB in
                  layer 1 ([m.sup.2] [h.sup.-1])

[D.sub.eff1]      Effective diffusion coefficient of TMPD-MIB in
                  layer 2 ([m.sup.2] [h.sup.-1])

[R.sub.1]         Kinetic retardation factor for transport of
                  TMPD-MIB in layer 1

[R.sub.2]         Kinetic retardation factor for transport of
                  TMPD-MIB in layer 2

[k.sub.a]         TMPD-MIB mass transfer coefficient (m [h.sup.-1])

[k.sub.c]         Water mass transfer coefficient (m [h.sup.-1])

a                 Material thickness (m)

b                 Location of the paint-air boundary (m)

[M.sub.T]         Initial mass of TMPD-MIB applied to gypsum board
                  (mg)

[f.sub.T]         Fraction of TMPD-MIB present in the paint

[DELTA][z.sub.1]  Distance between nodes in layer 1 (m)

A                 Area over which the paint is applied ([m.sup.2])

[DELTA][delta]    Variation of the paint thickness (m)

[C.sub.w]         Saturation vapor density of the water (kg
                  [m.sup.-3])

[rho]             Density of water (kg [m.sup.-3])

RH                Relative humidity (%)

[t.sub.evap]      Time for water to completely evaporate from the
                  film (h)


Introduction

The U.S. Census Bureau estimated that more than 440 million gallons of interior water-type architectural coatings were used in the U.S. in 2007. (1) For indoor applications, latex coatings were introduced because they are easy to use and lacked the health hazards and odors of their solvent-borne counterparts. However, to achieve the equivalent properties of solvent-borne paints, the addition of rheology modifiers and coalescing aids is required. (2) A common coalescing aid used in latex paints is 2,2,4-trimethyl-1,3-pentanediol monoisobutyrate (TMPD-MIB), which softens latex particles, allowing them to fuse together to form a continuous film free from distinct physical boundaries. (3) Despite the prevalence of TMPD-MIB in latex paints, exposure of painters and building occupants to TMPD-MIB and associated health effects, if any, are still not well understood.

Several researchers have modeled emissions of volatile organic compounds (VOCs) from architectural coatings. Pioneering work by Hansen (4) explained the process of solvent evaporation based on the diffusion equation for a lacquer film. First-order decay models including sink effects for VOCs were used by Clausen. (5) Double-exponential decay models were used by Chang and Guo to describe the process of mass transfer for stain applied on a wood material. (6) A model to capture the wet, dry, and intermediate stages of the paint drying process was developed by Sparks et al. (7) However, most attempts to model VOC emissions have been largely empirical, e.g., 1st- or 2nd-order decay models. Although these models capture the trend of VOC emissions for a specific source, they fail to capture the mechanistic behavior of such emissions.

Some of the more mechanistic emission models are specifically for dry building materials, (8-13) while others focus on wet materials. (14-16) All of these models are mechanistically similar, and could work well for either the initial paint drying stage or for the dry stage. However, these models lack the connection between wet and dry phases for predicting emissions of TMPD-MIB and other VOCs from latex paints.

Extensive research has also been completed to better understand paint drying, but associated models do not account for the mass transfer mechanisms of the coalescing aid. Some of these models for normal film drying are characterized by an initial constant rate of water loss followed by a decreasing rate until complete water evaporation takes place. (17-19)

A model that can predict the fate and transport of TMPD-MIB and the effects of paint properties such as pigment volume concentration (PVC) does not currently exist. The objective of this research was to develop a semimechanistic model to predict TMPD-MIB emissions from latex paints, accounting for the paint drying phase (water evaporation from the film) and specific paint characteristics. The model should be applicable to other volatile paint additives given sufficient experimental data for parameter estimation.

Model development

As a latex paint dries, the aqueous solution is transformed from a dispersion of polymer particles in a continuous water phase to a dry polymer film. (20) Several factors are experimentally known to influence latex film formation: the ambient conditions; the presence of surfactants, plasticizers, and pigments; and the latex particle structure. (21) Although polymer latex film formation is an area that has been studied for over 50 years without universal agreement being reached as to the mechanisms involved, (22) three different stages of the film formation process are recognized: (1) evaporation of water and particle ordering; (2) particle deformation; and (3) interdiffusion of polymers across particle-particle boundaries. For the simplified model presented herein, the film formation process will be assumed to be a normal drying two-stage consistent with Eckersley and Rudin, (20) in which polymer particles come close together and the water evaporates at a uniform rate from the aqueous phase. As the majority of the water evaporates, the mechanisms for paint coalescence transition from evaporation to diffusion-dominated processes, where the shrinking of the film stops and polymer particles arrange in a honeycomblike structure.

Soon after paint application, a dynamic mass transfer process for TMPD-MIB occurs concurrently with water loss. TMPD-MIB desorbs from paint resins, volatilizes to air, diffuses into and out of the pores of the underlying substrate, and diffuses through a "skin" of coalesced latex formed at the paint-air interface soon after paint application. Gas-solid adsorption can occur within the pores of the substrate.

Model assumptions

A simplified mathematical model of the fate of TMPD-MIB after paint application on a porous substrate can be described based on the following assumptions:

1. The model is applicable for thermoplastic paints.

2. Paint drying is normal to the surface and no lateral drying effects are present.

3. Water is lost from the paint with a constant rate of evaporation.

4. The dried skin layer formed shortly after paint application is assumed to be porous enough to allow for transport of water or other chemicals.

5. Fick's law in one dimension applies to mass transfer in both material film and substrate.

6. The material is homogeneous and the diffusion coefficient for TMPD-MIB in the material is constant.

7. There are no chemical reactions inside the paint or material that generate or consume TMPD-MIB.

8. Temperature effects are negligible.

9. No swelling occurs in the latex.

Model description

The coating layer is referred to here as Layer 1 (L1). The material (substrate) layer is referred to as Layer 2 (L2). The key transport phenomena considered in the model are: transport of TMPD-MIB between the interface of paint and bulk air, transport from paint to the underlying substrate, and transport out of the substrate.

Considering Fick's law for diffusion in one dimension, positive into the material, equation (1) describes the mechanics of TMPD-MIB transport for a homogeneous and isotropic medium in the z direction:

[j.sub.A] = [[D.sub.A]/[R.sub.A]] [[[partial derivative][c.sub.A]]/[[partial derivative].sub.z]] (1)

where [j.sub.A] is the rate of TMPD-MIB mass transfer per unit area (mg [m.sup.-2] [h.sup.-1]), [D.sub.A] is the diffusion coefficient of TMPD-MIB ([m.sup.2] [h.sup.-1]), [c.sub.A] is the concentration of TMPD-MIB (mg [m.sup.-3]), [R.sub.A] is a kinetic retardation factor that accounts for the transport delay of TMPD-MIB through the latex particles, and z is the space coordinate measured normal to the section (m). The ratio [D.sub.A] to [R.sub.A] is hereafter referred to as an effective diffusion coefficient [D.sub.eff] ([m.sup.2] [h.sup.-1]).

Applying Fick's second law to each layer yields equations (2) and (3).

[[[partial derivative][c.sub.L1]]/[[partial derivative]t]] = [[D.sub.AL1]/[R.sub.2]] [[partial derivative].sup.2][c.sub.L1]/[partial derivative][Z.sup.2]] (2)

[[[partial derivative][c.sub.L2]]/[[partial derivative]t]] = [[D.sub.AL2 ]/[R.sub.2]] [[partial derivative].sup.2][c.sub.L2]/[[partial derivative][Z.sup.2]]] (3)

At the air-paint film interface, a flux matching boundary condition in conjunction with a mass transfer model was used to represent transport at the boundary (equation 4). A flux matching boundary condition between paint and material layers is assumed at the material-paint boundary (equation 5), while at the bottom of the material layer a no flux boundary condition was assumed (equation 6). Initial conditions are presented as equations (7)-(9).

[D.sub.eff1] [[[partial derivative][c.sub.L1]]/[[partial derivative]t]] [|.sub.z=b] + [k.sub.a] * [c.sub.L1][|.sub.z=b] = 0, t > 0 (4)

[D.sub.eff1] [[[partial derivative][c.sub.L1]]/[[partial derivative]t]][|.sub.z=a] = [D.sub.eff2][[[partial derivative][c.sub.L2]]/[[partial derivative]t]][|.sub.z=a], t > 0 (5)

[[[partial derivative][c.sub.L2]]/[[partial derivative]t]][|.sub.z=0] = 0, t>0 (6)

[c.sub.L1] (z, t)[|.sub.t=0] = [c.sub.L1.sup.0] (z), z [member of] [0, a) (7)

[c.sub.L2] (z, t)[|.sub.t=0] = [c.sub.L2.sup.0] (z), z [member of] (a, b] (8)

[c.sub.L1.sup.0] (z) = [c.sub.L2.sup.0] (z), z = a (9)

For equations (4)-(9), [c.sub.L1] and [c.sub.L2] are the mobile concentrations of TMPD-MIB in layers 1 and 2, respectively (mg [m.sup.-3]), [D.sub.eff1] and [D.sub.eff2] are the effective diffusion coefficients of TMPD-MIB in layers 1 and 2 ([m.sup.2] [h.sup.-1]), respectively, [k.sub.a] is the TMPD-MIB mass transfer coefficient (m [h.sup.-1]), z is the space coordinate measured normal to the section (m), a is the material thickness (m), and b is the location of the paint-air boundary (m). Figure 1 depicts the two-layer schematic representation, as well as the grid used for the numerical solution. The grid includes P nodes for the paint layer and M nodes for the material layer. A greater number of nodes were used for the paint layer to obtain a node independent solution. An implicit finite volume method was employed to solve equations (2)-(9).

[FIGURE 1 OMITTED]

A series of three coefficients per node can be calculated when discretizing equations (2)-(6), obtaining the coefficients that correspond to the terms in brackets in equations (10)-(15). The [C.sub.iLi] terms are the TMPD-MIB concentration values from each node at time t. The superscript 0 denotes the values from the previous time step. The equations for the interfaces, paint-air (S1) and paint-material (S2), can be substituted into neighboring nodes, affecting nodes 1L1, PL1, and 1L2, leading to equations (10)-(13). Inner layer nodes for paint and material have the same coefficients as equations (11) and (14), respectively. The implicit solution scheme of equations (10)-(15) was solved using a tridiagonal matrix algorithm (Thomas algorithm) programmed in Matlab[R]. The 2LM was previously validated with an analytical solution that is available for a case of constant film and diffusion coefficients. (23)

[C.sub.1L1] [[-3[[alpha].sub.1] - 1 + ([2[[alpha].sub.1] * [k.sub.cs])] + [C.sub.2L1] [[alpha].sub.1] = -[C.sub.1L1.sup.0] (10)

[C.sub.1L1] [[[alpha].sub.1]] + [C.sub.2L1] [-1 - (2[[alpha].sub.1])] + [C.sub.3L1] [[[alpha].sub.1]] = [-C.sub.2L1.sup.0] (11)

[C.sub.(P-1)L1][[[alpha].sub.1]] + [C.sub.PL1] [-1 -3[[alpha].sub.1] + (2[[alpha].sub.1][m.sub.1])] + [C.sub.(P+1)L1][2[[alpha].sub.1][m.sub.2]] = [-C.sub.PL1.sup.0] (12)

[C.sub.PL1] [2[[alpha].sub.2][m.sub.1]] + [C.sub.1L2][-1 -3[[alpha].sub.2] + (2[[alpha].sub.2][m.sub.2])] + [C.sub.2L2][[alpha].sub.2] = [-C.sub.1L2.sup.0] (13)

[C.sub.1L2][[[alpha].sub.2]] + [C.sub.2L2][-1 -2[[alpha].sub.2]] + [C.sub.3L2][[[alpha].sub.2]] = [-C.sub.2L2.sup.0] (14)

[C.sub.(M-1)L2][[[alpha].sub.2]] + [C.sub.ML2][-1 - [[alpha].sub.2]] = [-C.sub.ML2.sup.0] (15)

Where,

[[alpha].sub.i] = [[[DELTA]t]/[[([DELTA][z.sub.i]).sup.2]]] [[D.sub.ALi]/[R.sub.i]] (16)

[k.sub.cs] = [[2[D.sub.AL1]/[R.sub.1]]/[(2[D.sub.AL1]/[R.sub.1]) + [k.sub.a] * [DELTA][z.sub.1] (17)

[m.sub.1] = [[([D.sub.AL1]/[R.sub.1]) * [DELTA][z.sub.2]]/[([D.sub.AL1]/[R.sub.1]) * [DELTA][z.sub.2] + ([D.sub.AL2]/[R.sub.2]) * [DELTA][z.sub.1]]] (18)

[m.sub.2] = [[([D.sub.AL2]/[R.sub.2]) * [DELTA][z.sub.1]]/[([D.sub.AL1]/[R.sub.1]) * [DELTA][z.sub.2] + ([D.sub.AL2]/[R.sub.2]) * [DELTA][z.sub.1]]] (19)

Estimation of parameters

Results from two cases for the paint film mechanisms will be presented here. A constant diffusion coefficient for TMPD-MIB transport through a paint layer of constant thickness was assumed for Case I. For Case II a variable diffusion coefficient that changes as the water evaporates from the film was assumed. The diffusion coefficient for Case II was assumed to vary linearly depending on the water content, which dictates the shrinkage of the paint layer and the time required for the water to completely evaporate from the film. The rate of water depletion from the liquid paint is used to develop an estimate of the water evaporation time [t.sub.evap], in conjunction with a linear function that represents the dynamic behavior of the diffusion coefficient as shown in equations (20) and (21).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[t.sub.evap] = [[[DELTA][delta]]/[[[k.sub.c]/[rho]] * [C.sub.w] (1 - [RH/100])] (21)

where [DELTA][delta] is the change in thickness of the paint film due to water evaporation (m), [k.sub.c] is the water mass transfer coefficient (m [s.sup.-1]), [rho] is the water density (kg [m.sup.-3]), [C.sub.w] is the saturation vapor density of water (kg [m.sup.-3]), and RH is the relative humidity.

Model parameters were estimated based on previous small chamber experiments completed at the University of Texas at Austin. (24), (25) In those experiments, painted materials were placed in small electro-polished stainless steel flow-through source chambers (3.33 L, air flow rate = 5 L [h.sup.-1]) to characterize TMPD-MIB emissions during the wet paint stage for 48 h, period after which the specimens were rotated in and out from the source chambers to quantify emissions for periods of up to 15 months.

The mass transfer coefficient ([k.sub.c]) 0.72 m [h.sup.-1] for water was obtained based on experimental conditions from [Lin.sup.24] where a series of temperature and RH values were recorded in the chamber inlet and outlet to characterize mass transfer in experimental chambers. A TMPD-MIB mass transfer coefficient ([k.sub.a]) of 0.21 m [h.sup.-1] was estimated based on comparison of the molecular weight between water and TMPD-MIB. Other experimental conditions used by [Lin.sup.24] are summarized in Table 1.
Table 1: Experimental data for model validation

Paint parameter (a)                    LPVC               HPVC
                                       paint              paint

Water content (%)                      ~16                ~43

Polymer content (%)                    ~57                ~16
                                       (b)                (c)

Pigment content                        ~25                ~40
[TiO.sub.2] (%)                        (d)                (e)

PVC (%)                                ~18                ~42

Density (g                             1.254              1.329
[mL.sup.-1])

TMPD-MIB mass fraction                 0.0154             0.0067

Layer thickness (m x                   134                273
[10.sup.-6])

Mass (g)                    1.07               2.18  139          277

Test protocols          Small chamber experiments

Chamber volume          3.33 L, air flow rate = 5 L [h.sup.-1]

Material/dimensions     Gypsum board: 8 cm x 8 cm 1.5 cm,
                        Concrete: 8 cm x 8 cm 1.5 cm

Paint application       Small roller

Sample extraction       Tenax[TM] tubes at 85 mL [min.sup.-1] for
                        5-25 min

Analysis                GC/FID (Agilent 6890)

(a) These and other experimental conditions as reported by [Lin.sup.24])
(b) Acronal 296d with 50% solids content ([rho] = 1.04 g [mL.sup.-1])
(c) Flexbond 325 with 55% solids content ([rho] = 1.09 g [mL.sup.-1])
(d) Kronos 2090 ([rho] = 4.1 g [mL.sup.-1])
(e) Tronox CR813 ([rho] = 3.7 g [mL.sup.-1])


Effective diffusion coefficients for sorptive materials can be estimated through the use of theoretical relationships between a chemical of interest and diffusion experiments with inert surrogate chemicals such as sulfur hexafluoride ([SF.sub.6]). (26) A mathematical expression that allows estimation of the diffusion coefficient for one compound based on that of another is:

[[[D.sub.e], VOC]/[[D.sub.e],[SF.sub.6]]] = [[[D.sub.air],VOC]/[[D.sub.air],[SF.sub.6]]] = [PSI] (22)

where [D.sub.eVOC] is the effective diffusion coefficient of a VOC in material pores ([m.sup.2] [h.sup.-1]), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the effective diffusion coefficient of [SF.sub.6] in material pores ([m.sup.2][h.sup.-1]), [D.sub.air,VOC] is the diffusion coefficient of a VOC in air ([m.sup.2][h.sup.-1]), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the diffusion coefficient of [SF.sub.6] in air ([m.sup.2][h.sup.-1]). The Fuller-Schettler-Giddings (FSG) (27) equation was used to calculate values of [D.sub.air,VOC] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] obtaining 0.0183 and 0.032 [m.sup.2] [h.sup.-1] for TMPD-MIB and [SF.sub.6], respectively, and a corresponding value of [psi] = 0.56. Using the [SF.sub.6] effective diffusion coefficient (5.2 x [10.sup.-3] [m.sup.2] [h.sup.-1]) for gypsum board obtained by Corsi et al., (26) and [psi] = 0.56, a value of 2.9 x [10.sup.-3] [m.sup.2] [h.sup.-1] was estimated for the TMPD-MIB effective diffusion coefficient for gypsum board ([D.sub.eff2]).

Other parameters needed for the 2LM are summarized in Table 1. The two paint formulations shown in Table 1 contain different amounts of TMPD-MIB, water, polymer, and pigment (resulting in low and high pigment volume concentrations, LPVC and HPVC, respectively). The PVC of the paint can be estimated using equation (23).

PVC = [[V.sub.p]/[[V.sub.p] + [V.sub.b]]] x 100 (23)

where [V.sub.p] is the pigment volume and [V.sub.b] is the nonvolatile binder (polymer) volume.

HPVC paints are generally characterized as being more "flat" than LPVC paints. They contain more fillers and the film formed is not as homogeneous as that for LPVC paint. Experimental values for the diffusion coefficient and retardation factor of TMPD-MIB in latex paint are nonexistent in the published literature; hence, a back-calculation of both parameters was obtained using a best fit of the model described herein to experimental data for latex paint applied on gypsum board. Details of the chamber experimental setup and parameters appear elsewhere. (25) The 2LM was fitted to minimize the root mean square error (RMSE) of the emission factor for the initial 150 h of experimental data, since the experimental error increased after t > 150 h.

Results and discussion

Typical single coating wet paint film thicknesses are in the range of 100 to 150 [micro]m. For this reason, previous experimental samples with a film thickness of 134 [micro]m. for LPVC paint and 139 [micro]m for HPVC paint were selected for estimation of the diffusion coefficient for both cases by minimizing the RMSE. The emission factors were in all cases normalized by the initial amount of TMPD-MIB applied to better account for differences in the TMPD-MIB content of the paint mixtures. A summary of these results for both cases is provided in Table 2.
Table 2: Parameters for 2LM best fit to experimental data

Film size ([mu]m)  Case     [D.sub.AL1]([m.sup.2]             R
                   type        [h.sup.-1])

134 (LPVC)           I     1.2 x [10.sup.-7]           4.5 x [10.sup.5]

139 (HPVC)           I     1.9 x [10.sup.-7]           9.1 x [10.sup.4]

134 (LPVC)           II    1.5-0.03 x [10.sup.-5] (a)  1.3 x [10.sup.6]

139 (HPVC)           II    1.6-0.16 x [10.sup.-6] (a)  2.1 x [10.sup.5]

Film size ([mu]m)  Case  [t.sub.evap] (h)  RMSE (%)  [R.sup.2]
                   type

134 (LPVC)           I         -              8.6       0.95

139 (HPVC)           I         -             51.8       0.90

134 (LPVC)           II      7.3 (b)          4.2       0.99

139 (HPVC)           II     21.6 (b)         15.8       0.99

(a) Range corresponds to initial and final diffusion coefficient
(b) Water evaporation time not needed for constant diffusion
coefficient case


Case I (constant diffusion coefficient)

For this case, differences in TMPD-MIB emissions of the two types of paint are apparent in Figs. 2a and 2b. The TMPD-MIB emission rate from HPVC paint is more than three times the rate from LPVC paint. For this reason, the HPVC paint emissions decay faster than emissions from LPVC paint. The diffusion coefficient ([D.sub.AL1]) was on the same order of magnitude for both paints, while the retardation factor was nearly an order of magnitude greater for the LPVC paint. This might be explained by the larger amount of polymer present in LPVC paints. Although the coefficient of determination ([R.sup.2]) was above 0.90 for both types of paint, RMSE was lower for the LPVC than for the HPVC paint. This might be due to the fact that the LPVC paint behaves more uniformly as the paint film dries, while the HPVC paint is more prone to cracks and irregularities due to its physical composition.

[FIGURE 2 OMITTED]

Case II (variable diffusion coefficient)

For Case II, the complete water evaporation time was calculated to be 7.3 and 21.6 h for LPVC and HPVC paints, respectively. The RMSE minimization for the LPVC paint resulted in a diffusion coefficient that decreases linearly from 1.5 x [10.sup.-5] to 0.03 x [10.sup.-5] [m.sup.2] [h.sup.-1]. The obtained RMSE for LPVC in Case II (4.2) was lower than that obtained for Case I (8.6). Results for HPVC paint (Fig. 3) indicate a slower linear decay for the diffusion coefficient, from 1.6 x [10.sup.-6] to 0.16 x [10.sup.-6] [m.sub.2] [h.sup.-1]. The obtained RMSE for HPVC paint in Case II was reduced 70% from the value for Case I, significantly increasing the [R.sub.2] to 0.99.

[FIGURE 3 OMITTED]

For Case II, the retardation factor for the LPVC paint was nearly one order of magnitude greater than the value for HPVC, just as in Case I, but with predicted emissions closer to the experimental data. This result suggests that even though average internal sorption effects are captured by using a constant diffusion coefficient as in Case I, the overall mechanisms of latex paint drying are better explained by a model that accounts for paint layer shrinkage and initial water content of the paint.

Due to the comparable size of TMPD-MIB to that of the copolymer, it can be argued that the effects of internal surface resistance prevent absorption of TMPD-MIB into the polymer matrix. Resistance should thus become highly influenced by surface resistance phenomena while water is still present in the film and polymer particles are not in close contact to each other. Since the HPVC paint contains less polymer particles than the LPVC paint, there are fewer available spaces for adsorption; thus, the amount of TMPD-MIB used in each type of paint correlates with the amount of polymer in the paint.

For Case II, the diffusion coefficient is proportional to water content of the film and leads to improved estimates of TMPD-MIB emissions for both types of paint during the initial emissions period. Predicted emissions of TMPD-MIB for the HPVC paint (Fig. 3) are reduced and result in a better fit to experimental data than those from Case I (Fig. 2b) due to the decrease in the diffusion coefficient of the paint while the water is evaporating (t < [t.sub.evap]). The higher water content of the HPVC paint leads to more rapid changes in film properties than for LPVC paint, and this might be the reason why Case II improves the fit further for HPVC than for LPVC.

The diffusion coefficients obtained from these initial analyses can be used to predict long-term emissions of TMPD-MIB since the emission curve lends to flatten after 150 h. After 150 h, TMPD-MIB slowly diffuses out of the material and through the paint layer. The diffusion coefficient is sufficiently small that further changes in the paint film will not appreciably affect emissions.

Model validation and paint thickness effects

The effects of paint thickness were explored using an independent set of painted samples analyzed by Lin and Corsi. (25) A thicker set of painted samples was used to establish differences in emissions compared to the base case used for parameter estimation. The thickness for the new set of painted samples was 273 and 277 [micro]m for LPVC and HPVC paints, respectively. Figure 4 shows the results of model predictions using the previously back-calculated diffusion coefficients, and a summary of the results obtained is given in Table 3. Up to 10,000 h for the LPVC paint and 4000 h for the HPVC paint are predicted by the 2LM for validation of the model with experimental data available.

[FIGURE 4 OMITTED]
Table 3: Prediction results for thick paint layers

Film size ([mu]m)  Case type  RMSE (%)  [R.sup.2]

273 (LPVC)             I        3.4       0.98
277 (HPVC)             I       30.7       0.88
273 (LPVC)             II       6.0       0.93
277 (HPVC)             II      16.2       0.97


The model compares favorably with experimental data from the independent dataset. The RMSE were 3.4 and 30.7 for LPVC and HPVC, respectively. Both the experimental data and model predictions indicate two important trends in TMPD-MIB emissions. First, although the mass of TMPD-MIB applied with the HPVC paint was only about one-half that of LPVC paint, its initial mass normalized emission rate was nearly four times that of LPVC paint (actual emissions from HPVC paint are roughly twice that from LPVC paint). After approximately 150 h this trend inverts. This result suggests the importance of pigment volume concentration on TMPD-MIB emission dynamics, particularly during the transition from water to polymer diffusion-dominated mechanisms as the film cures. Second, model predictions are consistent with experimental results in suggesting long-term emissions of TMPD-MIB from dry paint films. In fact, integration of the modeled emission curve for a typical LPVC paint thickness suggests that only 54% of the initial TMPD-MIB mass applied is emitted after 10,000 h (1.14 years). This prediction is consistent with previously reported experimental findings (25) for an LPVC paint with a 134 [micro]m paint film thickness.

TMPD-MIB concentration profile

The proposed model can be used to predict the concentration profile of TMPD-MIB inside both the paint film and the material, although no experimental data were found in the current published literature to validate the results. Simulations of the concentration profile were obtained for 1, 10, 100, and 1000 h after the paint event. No significant differences were found in the TMPD-MIB concentration profile for the LPVC paint when using either Case I or Case II. On the contrary, the peak concentration profile in the paint layer for the HPVC paint using Case I resulted in a higher value as time increased when compared to the results for Case II. The concentration profile for a 139 [micro]m wet film thickness and HPVC paint, and material layers for Cases I and II can be seen in Figs. 5 and 6, respectively.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

The initial uniform concentration for the paint layer remains almost unchanged for the 1- and 10-h profiles, as shown in Fig. 5a. As time increases, the concentration profile for TMPD-MIB results in a less pronounced peak at the middle of the paint layer. The large difference between the magnitude of the diffusion coefficient for paint and material layers results in a uniform concentration profile for the material as shown in Fig. 5b. For Case II, the initial uniform concentration for the paint layer was slower to disperse, resulting in a TMPD-MIB concentration peak of more than 90% higher value than the peak value obtained for Case I after 1000 h, as depicted in Fig. 6a. The TMPD-MIB concentration profile in the material layer was also uniformly constant as shown in Fig. 6b, but results after 1000 h show a 30% decrease of the uniform concentration value in the material layer compared to the results for Case I. This decrease is reasonable, since the higher mass of TMPD-MIB remaining in the paint layer will increase the uniform TMPD-MIB concentration value in the material layer, while decreasing the peak TMPD-MIB concentration in the paint layer as time continues to increase past 1000 h.

Sensitivity of predicted TMPD-MIB emissions to air velocity

The air velocity over the material surface affects the convective mass transfer coefficient; the higher the velocity, the greater the mass transfer coefficient. The impact of air velocity on the emission rate was analyzed through a parametric study. The short-term (t < 150) emission rates of TMPD-MIB at four different air velocities are shown in Fig. 7. The value for the TMPD-MIB mass transfer coefficient was changed over three orders of magnitude, to 100 [k.sub.a]. Predicted emissions were compared with the experimental base case for the 134 [micro]m thickness LPVC paint experiment reported by Lin and Corsi. (25)

[FIGURE 7 OMITTED]

As the mass transfer coefficient increases, the emission rate increases up to a point of 10-fold increase in [k.sub.a] for t < 150 h. Results at 100-fold increase are identical to those at a 10-fold increase, indicating that above 10 times [k.sub.a] the mass transfer coefficient is so large that emissions no longer have any dependence on gas-side mass transfer processes. In fact, four times [k.sub.a] results in a convergence to 95% of the 100-fold [k.sub.a] emissions curve. Therefore, for [k.sub.a] greater than 0.84 m [h.sup.-1] the rate limiting factor for TMPD-MIB emissions is diffusion rather than gas-side mass transfer. At times greater than 150 h, the dependency of emissions on the mass transfer coefficient is diminished dramatically, suggesting that emissions from the dry film are dominated by diffusion processes within the film and substrate.

Conclusions

The water content of latex paint is an important parameter with respect to short-term emissions of TMPD-MIB; the higher the water content of the paint, the faster the initial release of TMPD-MIB. The model described herein was successfully used to predict emissions of TMPD-MIB from paints with different PVC and thicknessess. Results indicate that the diffusion coefficient does not depend on the thickness of the wet paint film. A constant diffusion coefficient proved successful for modeling emissions of TMPD-MIB from LPVC paints. For HPVC paints, which contain higher water content and a less homogeneous film, a single diffusion coefficient failed to capture the physical mechanisms of the drying film. The inclusion of a linear decrease in the diffusion coefficient to account for water evaporation from the film resulted in a better prediction of TMPD-MIB emissions, and makes physical sense given the rapidly changing nature of the paint film as it dries. For times greater than 150 h after a paint event, mass transfer processes in air above the paint film did not appear to significantly affect TMPD-MIB emissions. The TMPD-MIB concentration profile was estimated for both the paint and material layers, but it could not be validated with currently published experimental data. Additional verification of the model is necessary, but initial results indicate the possibility of accurately predicting TMPD-MIB emissions using the model described herein.

Acknowledgments This research was supported in part by the National Science Foundation through its IGERT program at the University of Texas at Austin "Indoor Environmental Science and Engineering - An Emerging Frontier" (NSF grant # DGE-0549428). The authors wish to thank Dr. Neil Crain for discussions related to small-scale chamber results.

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L. A. Ramirez, H. M. Liljestrand (*),

R. L. Corsi

Department of Civil, Architectural and Environmental Engineering, The University of Texas at Austin, 1 University Station C1786, Austin, TX 78712-0273, USA

e-mail: liljestrand@mail.utexas.edu

J. Coat. Technol. Res., 7 (3) 291-300, 2010

DOI 10.1007/s11998-009-9194-0
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Author:Ramirez, Leonardo A.; Liljestrand, Howard M.; Corsi, Richard L.
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Date:May 1, 2010
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