Ind. Eng. Chem. Res. 1990,29, 2129-2133
2129
A Mathematical Model for Leaching in Insoluble Matrix Films J. J. Caprari,* J. F. Meda, M. P. Damia, and 0. Slutzky CIDEPINT, Research and Development Center for Paint Technology, CIC-CONICET, Calle 52 Entre 121 y 122, 1900 La Plata, Argentina
Marson’s model, which explains leaching in antifouling paint films of the insoluble matrix type, is a semiempirical simplification of the Stokes-Einstein equation. It explains the leaching rate as a function of film thickness; for this reason, Marson’s model is not predictive. In this paper is presented a modification of Marson’s model where the independent variable is time; in this way, the model changes to a predictive one. The experimental data obtained by different authors in different media are reported and interpreted by this model.
Introduction Antifouling paints are used for the protection of ship hulls and submerged structures to avoid the damaging action of sea organisms and base their effect on keeping an adequate toxic concentration in the paint film-seawater interphase. This implies that the pigment leaching rate should keep a value as near as possible to the limit that assures protection during all its useful life. On the other hand, if the leaching rate were higher than that needed to keep up protection, two unwanted effects would take place: a shortening in the useful life of the film and unjustified pollution in the sea media. Provided that the necessary trials to evaluate the efficiency of new formulations are expensive and take too much time, it is necessary to have a model that allows us to foretell quickly the service behavior of a determined formulation, thus avoiding the trial-and-error methodology. Although this topic has been studied in a considerable number of papers, the proposed models do not correlate efficiently the service behavior with the formulation parameters. Marson’s model is a semiempirical simplification of the Stokes-Einstein diffusion equation, and it has interesting characteristics since it is conceptually acceptable, it explains the leaching rate as a function of film parameters, and it has a good fit to the experimental data. But it has a disadvantage: the pigment leaching rate is expressed as a function of the leached film thickness, which makes it nonpredictive. In this paper is presented a modification of the mentioned model, where we use the leaching time as the independent variable, checking the fit by means of experimental data obtained from several leaching media by different authors; this has allowed us to show that this model has two main advantages over Marson’s one: it is predictive and applicable to any process in which the dissolution-diffusion of a solution through a membrane is involved, and it is independent from the media in which the phenomena take place, which makes it applicable to natural media as well as laboratory or plant production media, such as leaching of salts and metals from sintered powders, pellets, or ores, production by leaching of chemicals from clays or ores, pollutants from cement inclusions, deterioration of cements by leaching, leaching of wood preservatives, etc. Theoretical Development Pigment leaching from an insoluble paint film binder is a complex phenomenon, whose analysis has originated a series of statements. Ferry and Ketchum (1946) point out that the leaching rate decreases according to the time for this kind of paint, showing the exhaustion of chains of pigment particles in close contact, raising the fact that the probability of two
particles touching each other is less than one. Accordingly, an exhausted film should display pigment particles and channels where the leaching has been produced. On the other hand, since it is not possible to produce close contact of the particles, at a certain film thickness, the paint would stop the leaching of toxic compounds. Nevertheless, it is observed in microscopical transversal sections of leached paint films in several media that leaching is produced in a perfectly defined front, without finding evidence of cuprous oxide particles in the spongy matrix produced in the process (Marson, 1964a,b). Figure 1 shows a microscopical (320X) section of partially leached film (Caprari et al., 1986). The model proposed by Marson is limited to insoluble matrix films that have enough soluble pigment to make the particles meet at a minimum distance. This begins when the particles contact each other through the adsorption layers and the binder just fills the voids between particles. This point is called the CPVC (critical pigment volume concentration). This situation also takes place for pigment volume concentration (PVC), which is higher than the CPVC (Castells et al., 1983). Also the pigment dissolution rate needs to be greater than the diffusion rate in the system, so that the leached solution in contact with the pigment may be considered to be a saturated one. In these conditions, the leaching rate is determined by the diffusion of solvable pigment through the exhausted matrix; within this mechanism, which extends up to the boundary layer (film-leaching media interphase), the concentration tends gradually to zero, since this layer becomes negligible within the total volume of the leaching media that surround it (Figure 2). However, what is the probable mechanism of dissolution of each particle in an insoluble binder film, taking into account that chains of particles formed are not in direct contact with themselves but with the binder adsorbed layers that encircle each of them? According to Marson (1969), “When the film is immersed, the pigment particles at the surface of the film dissolve forming a saturated solution of the pigment at the pigment/leachate interface. The solvated ions or molecules diffuse outwards through the diffusion layer in contact with the paint surface. The overall rate of solution of the pigment in the early stage of leaching would, therefore, be governed by the rate of diffusion across this layer. When a particle has dissolved to uncover the thin film of vehicle separating it from an as yet unleached particle the solvent will diffuse through the thin membrane and dissolve some of the pigment. The resulting osmotic pressure will then rupture the membrane. As the solvated pigment will have to diffuse through the interconnecting holes so formed, the leaching rate will fall and as solution occurs further within the film, the exhausted matrix will play an increasing part in determining the leaching rate.”
0888-5885/90/2629-2129$02.50~0 0 1990 American Chemical Society
2130 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
Unleached
layer
kaching
front
Diffusional f r o n t
Leached
matrix
Figure 1. Microscopic cut (320X) of an antifouling paint layer (insoluble matrix), partially leached.
An experience that confirms this was made on acrylic test plates painted with antifouling paints submerged in seawater for 15 months (Caprari et al., 1986). By means of mathematical treatment that starts with the Stokes-Einstein equation, Marson obtained the equation that relates the leaching rate to parameters of the film and of the leaching media: B(PVC) F=(1) CD 1+P where F is the leaching rate (pg cm-2 min-l), B is a constant that depends on the leaching media, soluble pigment, and temperature (pg cm-2 min-l), PVC is the pigment volume concentration, D is the thickness of the exhausted matrix (pm), and P is the fraction of interconnected holes that is a function of the PVC and the particle size distribution. An analysis of Marson's model indicates that the constant C (pm-') should be related to some binder characteristics such as mechanical resistance and others inherent to the pigment and to the leaching media, like the overpressure produced within the membrane due to pigment dissolution. Since in Marson's paper only the C/P data are given, an identification of the individual values was made by means of the experimental results (Marson, 1964a,b),and an adjustment of the constant by an iterative method developed by Powell (1964) and Zangwill(l977) was made. The results of such an adjustment are indicated in Table I, and these data show that B will be greatly influenced by the nature of the leaching media if the osmotic pressure is sufficient to overcome the resistance of the polymer that enclosed the particles. The iterative model used suggests that C is constant, a fact that can be verified in the studied cases, and the fraction of holes P should by greatly influenced by PVC. The interconnected fraction voids are a function of the PVC, so when PVC tends to 0, P tends to 0 too, and when PVC tends to 1, P tends to 1 too. A characteristic point of this function of PVC = CPVC, where particles are at a minimum distance (as was explained before) only separated by their adsorption layers since the interparticulate volume is filled only by binder. Starting at this point, a PVC increase is traduced in the replacement of binder by air, which strongly increases the
Table I. Adjusted Leaching Variables leaching author sample media" PVC Marson 1 I 0.523 I 0.680 2 I 0.740 3 Caprari et al. 4 I1 0.485 5 I11 0.485 IV 0.485 6 7 V 0.485
adjusted variables B C
I#I
0.227 3.60 0.630 3.60 0.739 3.60 0.219 0.041 0.219 488.0 0.219 316.0 0.219 128.0
0.017 0.017 0.017 0.017 0.017 0.017 0.017
"I, sodium glycinate, 0.025 M; sodium chloride, 0.48 M. 11, seawater. 111, hydrochloric acid, 30 g/L. IVYhydrochloric acid, 20 g/L. V, hydrochloric acid, 10 g/L.
void interconnection probability. On the other hand, if PVC diminishes, increases the interparticulate distance drastically lower the interconnection probability. It was not possible to obtain an expression that gives a porosity value (P)as a function of the film parameters, since only some such parameters have been given in the references that contain the experimentaldata used. These statements justify the use of an empirical adjustment of such an expression, which could be represented by the equation -(1 - PVC)2 = A(PVC)
)
where 4 indicates an adjusted value of the interconnected holes fraction (P)and A is an adimensionalconstant. The expression indicates that when the pigment fraction tends to 1, 4 tends to the same value, while PVC decreases. The adjusted value of P diminishes quickly, down to values that make the leached diffusion through the exhausted matrix difficult. In Figure 3, a good relationship is observed between 4 and the experimental values, despite the mentioned differences in the PVC. This good fit is enough to justify the use of these empirical expressions until the right functional relationship is obtained. The dotted lines indicate values out of the studied PVC range, 0.48-0.74.
Relationship between the Exhausted Matrix and the Immersion Time We considered a coating divided into layers thin enough so that the thickness of the exhausted matrix in each layer
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2131
Unleached m a t r i x Leached matrix
7
Pigment
volume concentration I PVC)
Figure 3. Adjustment of the interconnected hole fraction as a function of pigment volume concentration.
,
\,
K is a gravimetric constant that relates the pigment molecular weight (MW,) and the leached compound molecular weight (MWb): lO*MW, K= (7)
L e a c h i n g holes
MWb
Since the function 4 has already been defined, we can arrange Marson’s equation as follows: B(PVC) F= Cne 1+4 The leaching time t, of a layer with an exhausted matrix of n layers of thickness e and unit area a will relate (3) and (8),and taking into account the gravimetric factor K , d,e d,Ce2n
M i n i m u m letal concentration
Depth ( c m )
Figure 2. Toxic concentration variation scheme, as a function of the film depth.
will not affect the leaching rate. This situation takes place if the layer thickness e (em) is smaller than the pigment particle size. Since the commercial cuprous oxide particle size is between 0.5 and 5 pm, for our calculations, we used a layer thickness e = 0.1 pm, and this value must be taken for other particle size ranges. The pigment mass W (g contained in an unit area a and thickness e will be W , = d,ePVC (3) where d, is the pigment density (g At the initial immersion time, there is no exhausted matrix above the layer to be considered. The leaching rate F will be directly proportional to the PVC. Therefore, F = B(PVC) (4) Thus, the value of to,which is the needed time to dissolve the first layer, will be to = W,/F (5) so to =
dPe KB
(6)
t,=-+-
(9)
BK BK4 The total leaching time from the beginning of the leaching up the film to the nth layer will be
T,= Cti i=O
and
dPe Cdpe2n2 + n + 1)+ (11) KB KB4 2 The number of leached layers of thickness e is obtained for a given total leaching time T,:
T, = -(n
KB Thus, for a given leaching time, the number (n,) of leached layers of thickness e will be given by the solution of the quadratic expression (12), and the leaching rate for that time will be B(PVC) F, = Cent 1+4
2132 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
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Figure 4. Curves developed by the model for the experimental Marson values.
The main difference between this and Marson's equation is the introduction of data corresponding to the adjusted fraction value of interconnected holes (4) and the total number of leached layers of thickness e, for a total leaching time of T,. So n, turns the model into a predictive one, since it allows us to calculate the leaching rate (F,) as a function of time. Analysis of t h e Model Leaching in Sodium Glycinate-Sodium Chloride Solution. In Table I are included samples 1-3 corresponding to the trials made by Marson (1964a,b) in sodium glycinate (0.025 M)-sodium chloride (0.48 M) solution. Applying such values to the model, we obtain values that are represented in Figure 4 for different PVC levels. Leaching in Natural Sea Media In the tests made, insoluble matrix paint films (three types, prepared with 86% vinyl chloride, 14% vinyl acetate copolymer and red cuprous oxide) were used, submerging them in Mar del Plata's harbor in an experimental raft 1.20 m deep for a period of 15 months, including two seasons (spring and summer) of intense fouling fixation. Test plates where removed at periods of 3 months. The leached layer thickness for each period was carefully measured by optical and electron microscopy and the leaching rate calculated (Caprari et al., 1986). Comparison of raft trials and the prediction obtained with the model showed good correlation (Figure 5). The leached layer thickness was obtained by resolution of eq 12 for each month. It is important to see as well that the equation is a parabolic function; the coefficient of n2 is small enough to have a curvature in the used interval that is not noticeable. As shown by the diagram, the results obtained in sea media are less than those obtained by the model. This disagreement is logical since some factors such as temperature, pH, and biological and shaking factors where not considered by the model. Temperature varies greatly, as can be seen in Figure 6. Comparing Figure 5 with Figure 6, the influence of this parameter on the red cuprous oxide leaching is shown. These same experiences are represented in Figure 7, where a horizontal line indicates the protection threshold gen-
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Immersion
Time [days)
Figure 5. Model response to values obtained in raft and sea media; the function is a parabol of w r y low curvature. T
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Figure 6. Water temperature fluctuation during a year at Mar del Plata's harbor.
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Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2133
3. With regard to the large bibliography existing on this subject, the lack of numerical values of constants such as CPVC and CPVC* and the particle size distribution also prevents the attainment of more exact equations that would lead to the determination of the probability of interconnection within the exhausted matrix. 4. After making laboratory and seawater trials with coatings with some known parameters, data were obtained that should be applied to feed the system to improve the model. 5. In this way, the long-duration trial-and-error experiences can be replaced by a system where successive simulations are made by means of a model to establish the formulation parameters that allow us to obtain potentially more adequate paints. 6. This work scheme takes time, but it is justified taking into account the economic and ecological impact of toxic compounds in sea media.
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10
20
30
50
60 70 80 90 immersion tlme (min)
40
100 110
120 130
Figure 8. Leached depth aa a function of time for a high aggressive media (hydrochloric acid in different concentrations).
indicates that it is under the threshold value, fouling fixation is not produced, since it was winter time in the southern hemisphere (period of minimum biological activity in the sea media). Leaching i n Acid Media It is interesting to observe the model response to a much more aggressive media, such as the acid one, with hydrochloric acid being chosen in concentrations of 10,20, and 30 g/L, using the same films as in the previous experiment and measuring by microscopy the leached matrix thickness for 30- and 120-min exposure. The results are indicated in Figure 8. A very good fit of the model to the experimental data in such an aggressive media is evident. Final Considerations 1. Starting with Marson's equation, an ideal film model is proposed where the leaching rate is related to the immersion time. 2. This model is a simplification of the actual phenomenon, since in leached products in a certain media there are effects that are not considered, such as pH, temperature, shaking, etc., and that prevent a more precise adjustment to experimental conditions. Nevertheless, these deviations can be considered to be acceptable, and the model might be improved mathematically by introducing in it factors that take into account these variations.
We thank the CIC (Comisibn de Investigaciones Cientificas de la Provincia de Buenos Aires) and the CONICET (Concejo Nacional de Investigaciones Cientificas y TBcnicas) for their economical support and Mr. Pedro L. Pessi (CIDEPINT) for the laboratory experiments. Registry No. (Vinyl chloride)(vinyl acetate) (copolymer), 9003-22-9;red cuprous oxide, 1317-39-1.
Literature Cited Caprari, J. J.; Slutzky, 0.; Pessi, P.; Raacio, V. A Study of the leaching of Cuprous Oxide from Vinyl Antifouling Paints. Prog. Org. Coat. 1986,13, 431-444. Castells, R. C.;Meda, J. F.; Caprari, J. J.; Damia, M. P. Particle Packing Analysis of Coating Above the Critical Pigment Volume Concentration. J. Coat. Technol. 1983,55,12,53. Ferry, J. D.;Ketchum, B. H. Maintenance of the Leaching Rate of Antifouling Paints Formulated with Insoluble Impermeable Matrices. Znd. Eng. Chem. 1946,38, 806. Marson, F. J. An Accelerated Leaching Rate Technique for Cuprous Oxide Based Antifouling Paints. J. Oil Colour Chem. Assoc. 1964a,47,323. Marson, F. J. Leaching of Toxic Pigments from Contact Leaching Antifouling Paints. J. Oil Colour Chem. Assoc. 1964b,47,831. Marson, F. J. I: Theoretical Approach to Leaching of Soluble Pigments from Insoluble Paint Vehicles. J. Appl. Chem. 1969,19, 93. Powell, M. J. An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives. Comput. J. 1964,7, 155-162. Zangwill, W.Minimizing a Function without Calculating Derivatives. Comput. J. 1977,10,213-276. Receiued for reuiew October 30, 1989 Revised manuscript received May 18, 1990 Accepted May 30, 1990
