Analysis of Self-Polishing Antifouling Paints Using ... - ACS Publications

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Ind. Eng. Chem. Res. 2001, 40, 3906-3920

Analysis of Self-Polishing Antifouling Paints Using Rotary Experiments and Mathematical Modeling Søren Kiil,*,† Claus E. Weinell,‡ Michael Stanley Pedersen,‡ and Kim Dam-Johansen† Department of Chemical Engineering, Technical University of Denmark, Building 229, DK-2800 Kgs. Lyngby, Denmark, and Hempel’s Marine Paints A/S, Lundtoftevej 150, DK-2800 Kgs. Lyngby, Denmark

A detailed mathematical model for a self-polishing antifouling paint has been developed. The important rate-influencing steps, dissolution of pigment particles, hydrolysis and erosion of the active polymer binder, effective diffusion in the leached layer, and external mass transport of relevant species, were all included. The aims have been to produce a tool for estimating paint lifetimes at various seawater conditions and paint compositions, for possible product optimizations, and for supporting the development of novel and environmentally friendly antifouling paints. Experimental data for model verification, such as polishing rates and extent of pigment leaching, were obtained using a laboratory rotor. Simulations performed for different rotary speeds and temperatures matched experimental data for two of the three paints investigated. In the last case, the disagreement between model and experiment was explained by significant water swelling of the hydrolyzed polymer. The modeling tools developed are applicable to other types of self-polishing antifouling paints than the ones investigated here. Introduction Fouling is the term employed to describe the undesired assemblages of animals and plants which grow on artificial structures immersed in marine waters. With time, this settlement can result in the formation of a thick, uneven, and hard crust. On ocean-going ships, some of the problems from fouling are:1,2 (1) the increased drag on the ship raises fuel and machine requirements for maintaining speed, (2) the dry-dock cleaning and painting costs, which, taking into account the loss of earning while the ship is immobile, can be significant, (3) maneuverability of the ship being reduced when heavily fouled. The US Navy has estimated that biofouling costs over $150 million annually in excess fuel consumption and cleaning costs for naval vessels.3 Worldwide, the maritime industry spends an estimated one billion dollars annually to address problems caused by biofouling.3 Many versatile ways of avoiding the fouling of ships have been tried out, as demonstrated in the detailed review of Swain.4 The methods count various chemical, electrical, radiation, surface, and thermal techniques. Presently, however, fouling is fought by the use of socalled antifouling paints, which slowly release biocides into the seawater. These paints are usually classified into three types, known as soluble and insoluble matrix paints and self-polishing paints.5,6 In the soluble matrix paints, the biocide is dispersed in a polymer binder that is physically soluble in seawater. Upon dissolution of the binder, the biocide is released. This paint type may be reinforced by adding an insoluble polymer, in which case it is referred to as a polishing or an ablative paint.7 Insoluble matrix paints, on the other hand, are based on the use of hard insoluble polymer matrixes that do not erode at all over time. Here, the biocide is released * Corresponding author. Telephone: 45 45 25 28 27. Fax: 45 45 88 22 58. Email: [email protected]. † Technical University of Denmark. ‡ Hempel’s Marine Paints A/S.

only through the dissolution of the biocide particles. The self-polishing paint contains as main binder an acrylic backbone polymer with an ester linkage to some kind of organometallic sidegroup. The latter (as well as biocides such as Cu2O) are released by chemical reaction (hydrolysis) of the polymer with seawater. Upon continued reaction, the remaining polymeric backbone becomes water-soluble and is also released. Since the early 1970s, the most effective systems have been the tributyltin self-polishing copolymer (TBT-SPC) based paints, which are presently used on approximately 70% of the world fleet of ocean-going ships.2 These systems are able to provide in excess of five years protection, a roughness not exceeding 100 µm average hull roughness (AHR), and very good protection against biofouling.4 The wide spectrum protection is due to the dissolved copper, being particulary effective against animal fouling, whereas the TBT, which is released through hydrolysis, is effective against copper-resistant slime and weed fouling.8 However, concern about the environmental effects of TBT-based antifouling paints has grown in areas where molluscs, such as oysters and marine snails, are present in the neighborhood of marinas and harbors.9 According to Ryle,10 this has motivated the Marine Environment Protection Committee (MEPC) to work toward a global legal binding instrument to ensure prohibition on the application of organotin compounds, which act as biocides in antifouling systems on ships by January 1, 2003, and a complete prohibition of these biocides in antifouling paints on ships by January 1, 2008. Some countries have already banned TBT in antifouling paints, e.g., Japan, which in 1991 was the first country to introduce an almost total ban.8 Thus, the future challenge is to develop a polymer system, which may act as a controlled release vehicle for suitable antifouling biocides but having no side effects to nontarget organisms and no accumulation potential in the environment. Alternatively, novel ways of avoiding the fouling must be introduced.

10.1021/ie010242n CCC: $20.00 © 2001 American Chemical Society Published on Web 08/11/2001

Ind. Eng. Chem. Res., Vol. 40, No. 18, 2001 3907

The present tin-free SPC paint systems, which all use some kind of acrylic backbone polymer similar to that used in the TBT-SPC systems, have been summarized elsewhere.2,8 The paint systems are based on copper, zinc, or silyl acrylats. Other self-polishing paints, which use gum rosin as their base, are available.10 Fiber composites are also described to reinforce mechanical properties of self-polishing paints.10 Recently, a selfpolishing paint employing a cross-linked copolymer with a hydrolyzable group built into the backbone was presented.11 However, these systems have to demonstrate their long-term performance under different fouling conditions and prove that the increased release of new biocides and their metabolites do not bear unknown environmental risks. Another alternative to TBT-based paints under development is the foul release paint.10 This is a form of nonstick coating, based on silicone, which provides a very smooth, slippery surface. No biocides are needed because the action of the vessel moving through the water simply washes fouling off. At present, its use is restricted to fast moving vessels (around 30 knots), and the price is about 5 times the cost of TBT-based paints. A further drawback is that the coatings are soft and therefore prone to damage. In the development of suitable antifouling paints, the approach has mainly been that of experiments.11-14 Any subsequent data analysis often lead to an empirically based correlation relating the biocide release rate to paint composition and/or seawater conditions. Only a small number of theoretical studies have been reported,15-20 and all of these were done on insoluble matrix paints. To our knowledge, there has been no attempt in the literature to model self-polishing antifouling paints. The purpose of this work is to illustrate how mathematical modeling, combined with rotary experiments, can be a strong tool in the analysis and development of self-polishing antifouling paints. A fundamental model of the paint behavior can be used to give reliable estimates of the paint lifetime at given seawater conditions and paint composition, as well as to suggest ways for optimization of the paint with respect to biocide release rates. Furthermore, insight into the release mechanism of existing paints may help to identify the properties that new efficient and environmentally friendly alternatives should possess. As a case study, model paints based on the well-known TBT-technology have been chosen. One of the reasons for this choice is that several kinetic studies of the hydrolysis of TBT-based polymers, which are needed in order to develop a mathematical model, are available in the open literature. No kinetic data for the active polymers of the novel self-polishing antifouling paints mentioned earlier have yet been released. Additionally, as stated elsewhere,2 a detailed understanding of the unique mechanism of TBT-SPC systems may provide a basis for the development of alternative environmentally safe systems. Finally, the principles used in this work to model TBTbased paints are of a general nature and therefore applicable to other antifouling paints based on similar controlled release mechanisms. Outline of Paint-Seawater Chemistry The copolymer used in tin-based antifouling paints, which can undergo hydrolysis in seawater, is that of tributyltin methacrylate (TBTM) and methyl methacry-

Figure 1. Chemical formula of a repeating unit of a copolymer of tributyltin methacrylate (TBTM) and methyl methacrylate (MMA). For the polymer used in this work, the molar ratio m:n is close to 1:2.

late (MMA), shown schematically in Figure 1. The pigment phase usually consists of seawater-soluble pigments such as Cu2O and ZnO, as well as insoluble TiO2. In the remainder of this section, the chemistry and kinetics of the TBT-hydrolysis and the Cu2O dissolution in seawater are considered in detail. No data have been found for the rate of dissolution of ZnO at seawater conditions. Chemistry of the Active Polymer (TBTM/MMA). The self-polishing mechanism of the polymer begins with the following reversible reaction:21

Polymer-COO-TBT(s) + Na + + Cl - a TBTpolymer(insoluble) Polymer-COO- Na + (s) + TBTCl(aq) (1) Acidpolymer(soluble) Attack by seawater (Na+ and Cl- ions) on the film surface causes hydrolysis of the organotin-ester linkage. The organotin moiety is released into the seawater mainly as TBTCl.22 If the TBTCl formed is allowed to react to equilibrium in seawater, other species, such as TBTOH, bis-TBTO, TBTOH2+, and TBTCO3-, may form, but the dominant species is still TBTCl.23 Whether a hydrolyzed TBT-polymer site actually combines with an Na+-ion, as suggested in eq 1, or perhaps an H+ ion has not been confirmed in the literature. However, at seawater conditions (i.e., a pH of about 8.2 and a concentration of NaCl of 0.5-0.6 M), reaction 1 seems the most likely. As will be shown later, this piece of information is not critical for the modeling of TBT-based antifouling paints. The partially reacted outer layer of the polymer, now containing hydrophilic free carboxylate groups, has little strength and is easily eroded by moving seawater (self-polishing effect), exposing a fresh layer of organotin acrylate polymer Xmax

Polymer-COO- Na+ (s) 98 Polymer-COO- Na+ (aq) (2) The TBT conversion of the polymer, at which it is released, is denoted here as Xmax. The hydrolysis and erosion mechanism is continually repeated until no polymer is left. Upon conversion of the TBT polymer, the glass transition temperature increases from about 25 to about 100 °C, making it brittle.24 Several researchers25-28 have measured the rate of hydrolysis of TBTM/MMA copolymers (in the following referred to as TBTCP) in seawater using different experimental techniques. Unfortunately, the reported rates, under

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Ind. Eng. Chem. Res., Vol. 40, No. 18, 2001

what appears to be similar conditions, do not correspond too well. However, those results of Takahashi and Ohyagi26 and van der Walle,28 which allow a comparison, agree very well. Considering that the data of Takahashi and Ohyagi were obtained using gas chromatography and those of van der Walle using atomic force microscopy, the reported data seem reliable. For reaction 1, the kinetics of the forward reaction are given by

kSLxTBTM[OH-]a[Cl-] (-rTBTCP) ) , (1 + k2[OH-]b)

xTBTM e 0.28 (3)

and

kSH(xTBTM - σ)[OH-]a[Cl-] , (-rTBTCP) ) (1 + k2[OH-]b) 0.28 < xTBTM < 0.52 (4) where a ) 0.32 ( 0.05, b ) 0.43 ( 0.04, σ ) 0.270 ( 0.001, k2 ) 8 ( 4 (m3/mol)b (independent of temperature), kSL(25 °C) ) (3.2 ( 0.2) × 10-11, and kSH(25 °C) ) (9.3 ( 0.5) × 10-10 (m3/mol)1+a mol TBTCP m-2 s-1. In the above expressions, the dependency of the rate on the TBTCP content of TBTM, xTBTM, and the value of the rate constant were found by correlating the data of Takahashi and Ohyagi.26 The discontinuity in reaction rate was explained by Takahashi and Ohyagi as being due to steric hindrance of TBT, but no further comments were given. The relatively weak pH dependency (expressed as [OH-]) of the hydrolysis rate was found by correlating the data of Hong-xi et al.,27 which are valid for 6 < pH < 9. The large uncertainty of k2 is not so important because of the weak dependency of the rate on pH. The first-order dependency of the rate on the concentration of Cl- was assumed, due to lack of data, and this should be kept in mind when performing simulations at different values of seawater salinity. Under the assumption that the rate of hydrolysis is dependent on the number of TBT-groups only and not the number of acid groups formed during hydrolysis, the current polymer content of TBTM, needed in the above kinetic expressions, can be calculated from

xTBTM )

xoTBTM(1

- XTBTCP)

(5)

where XTBTCP is the conversion of TBT-groups and xoTBTM is the initial molar content of TBTM in the TBTM/MMA copolymer. Here, it should be mentioned that the solubility of the TBT-polymer, contrary to the reaction rate, is strongly dependent on the occurrence of acid groups. Assuming, that the rate of the reverse reaction 1 is first order in the concentration of TBTCl and the number of hydrolyzed TBT-groups, the rate of the reverse reaction can be written as

(-rTBTCl) ) k-S[TBTCl]XTBTCPxTBTMo

(6)

where

k-S )

kSL , RL,eq

xTBTM e 0.28

(7)

At equilibrium

(-rTBTCP) ) (-rTBTCl)

(8)

which leads to an expression for RL,eq

RL,eq )

XTBTCP,eq(1 + k2[OH-]beq)[TBTCl]eq (1 - XTBTCP,eq)[OH-]aeq[Cl-]eq

(9)

For 0.28 < xTBTM < 0.52, an expression similar to eq 9 can be derived. Pigment Chemistry. Ferry and Caritt29 conducted a detailed experimental study on the rate of dissolution of Cu2O particles in seawater. They found the relevant reactions to involve the formation of the copper chloride complexes, CuCl2- and CuCl32-

/2Cu2O(s) + H+ + 2Cl- a CuCl2- + 1/2H2O(l) (10)

1

CuCl2- + Cl- a CuCl32-

(11)

Reaction 10 is reversible, though this was left unmentioned by Ferry and Carritt. Reaction 11, which is reversible and instantaneous, can be considered at equilibrium at all times. If O2(aq) is present in seawater, the copper complexes may be oxidized to Cu2+.30 The kinetics of the forward reaction 10 was determined to be29

(-rCu2O) ) k1[H+][Cl-]2

(12)

where k1(25 °C) ) 2.35 × 10-7 mol Cu2O (mol/m3)-3 m-2 s-1. Assuming the reverse reaction 10 to be first-order in the concentration of CuCl2-, the kinetics for this reaction can be written as

(-rCuCl2-) ) k-1[CuCl2-]

(13)

2(-rCu2O) ) (-rCuCl2-)

(14)

At equilibrium

which leads to an expression for k-1

2KW k-1 ) k1 KCuCl2- LCuOHγ(2

(15)

Seawater Equilibria. To complete the description of the seawater-paint chemistry, the two equilibria of seawater must be included:

HCO3- a H+ + CO32-

(16)

H+ + OH- a H2O(l)

(17)

Mechanism of an Antifouling Paint System. The physical process that arises when a TBT, retarder (defined later), and Cu2O-based antifouling paint is exposed to seawater is shown schematically in Figure 2. Moving seawater slowly erodes the copolymer through hydrolysis, thereby releasing the TBT-groups as TBTCl. Seawater diffuses into the porous polymer matrix, ahead of the eroding polymer front, and dissolves the pigment particles, at the dissolving pigment front, resulting in the formation of a leached (i.e., Cu2O

Ind. Eng. Chem. Res., Vol. 40, No. 18, 2001 3909 Table 1. Composition (percentage by solids volume) of the Various Antifouling Paints Used in the Rotary Experiments paint components

P1

P2

P3

P4

P5

Cu2O 39.91 39.70 39.94 39.96 0.0 TBTM/MMA (molar ratio 1:2) 60.09 56.70 54.07 48.05 89.80 BMA/MMA (molar ratio 1:1) 0.0 3.61 5.99 11.99 10.21

Figure 2. Schematic illustration of the behavior of a self-polishing antifouling paint comprised of an active polymer, a retarder, and Cu2O particles.

depleted) layer. The dissolved copper forms a complex with Cl- (CuCl2- or CuCl32-) diffuses out through the leached layer and across the external solid-liquid boundary layer into the bulk seawater. Consequently, two moving fronts, the dissolving pigment front and the eroding polymer front, develop. After some time, the thickness of the leached layer becomes constant. In Figure 3, a scanning electron microscopy (SEM) photograph of a real antifouling paint clearly shows the two fronts. Experimental Section Commercial antifouling paint systems may contain more than 15 different components. The bulk products are binder materials (polymers), pigment particles, and solvents. The remaining (typically present in up to a few weight percent each) are additives of various kinds, present to give specific properties to the paint, and extenders, which are chemically inert materials that reduce the price and/or increase the wearing qualities of the paint. To investigate some well-defined paint systems, for which a model verification can be reliably done, model paints consisting of a two-component binder phase and one type of pigment particles were produced. The main polymer making up the binder phase of the model paints is the copolymer shown in Figure 1. The copolymer present in smaller amounts and known as a retarder is a butyl methacrylate methyl methacrylate (BMA/MMA). This component is chemically inactive and present to reduce the solubility of the hydrolyzed TBTpolymer and to give strength to the polymer matrix, thereby reducing the polishing rate of the paint system. The pigment phase consists of Cu2O particles (impurity content of 4 wt %), which, together with the TBTpolymer, is the main biocide of self-polishing antifouling

paints. These three-component model paint systems will illustrate the behavior of self-polishing antifouling paints without the influences of insoluble pigments, extenders, and additives. Formulation of Model Paints. Five different model paints, denoted P1-P5 and with compositions as shown in Table 1, were produced in batches of about 100 mL using xylene as solvent. The binder phase of P2-P5, was prepared by mixing a 60 wt % solution of TBTM/ MMA in xylene with a 40 wt % solution of BMA/MMA in xylene. The two copolymers are completely miscible and form a one-phase binder system. For each paint, the binder phase, pigment particles, and about 100 mL of glass beads (diameter of 3 mm) were mixed in a jar, supplied with a lid, and placed for 30 min on a mechanical shaker. After removal of the glass beads, the fineness of grind (i.e., the maximum particle size) was measured using a grindometer and found to be about 40 µm for the paints containing Cu2O. The critical pigment volume concentration (CPVC) of the paints was estimated to about 57 vol. %, using density and oil absorption data for Cu2O.31 The paints were applied to foils within a few days following preparation. Due to the absence of additives, phase separation or sedimentation problems could occur upon storage. The dry film thickness was about 300 µm. Finally, an inert acrylic paint was applied to part of each dry paint sample, thereby establishing a reference point where no polishing can occur. Paint Rotor. Schatzberg14 has reviewed some earlier devices for measurement of biocide release rates, and others have used traditional paint rotors.11,13 In this work, paints were tested on the rotary rig shown schematically in Figure 4, which consists of two concentric cylinders with the innermost in rotation. In this configuration, the intention is to create a close approximation to couette flow (flow between two parallel walls, where one wall moves at a constant velocity). For both laminar and turbulent couette flow, the shear stress will be constant across the channel width.32 Centrifugal forces in the system require that there is a pressure gradient in the radial direction and this is a deviation form “true” couette flow in which there are no pressure gradients. The effects hereof are expected to be small, however, as the ratio of the spacing between

Figure 3. SEM photograph showing part of the cross section of a self-polishing antifouling paint (magnification 5000×). The leached layer and the two moving fronts can be clearly seen.

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Figure 4. Schematic illustration (seen from above) of paint rotor set up.

Figure 5. Microscopical examination of the cross section of a paint sample (P2, magnification 200×). One unit on the ruler corresponds to 5 µm. The paint has polished 15-17.5 µm ,and the thickness of the leached layer is 7.5-10 µm. Table 2. Composition of the Artificial Seawater Used in the Rotary Experiments parameter pH NaCl (g/L) MgSO4, 7H2O (g/L) NaHCO3 (g/L)

8.2 32 14 0.2

the cylinders and the radius of the inner cylinder is reduced. The wall shear stress at rotary speeds of 20 and 30 knots was calculated to 121 and 242 N/m2, respectively, comparable to what a ship travelling at 20-30 knots experience. The rotor is placed in about 500 liters of artificial seawater with a composition (see Table 2) according to the recommendation of Grasshoff.33 Initially, the pH is adjusted using NaOH and HCl. The temperature can be controlled using a heat exchanger and Cu2+ ions, originating from the paints, are continuously removed using an ion exchanger. Paint samples are attached to the rotor by the use of foils. At selected time intervals (typically 3-4 weeks), the extent of polishing (reduction of paint thickness) and leaching (i.e., the thickness of the leached layer) of the paints can be measured using sampling techniques described below. The total run time is at least 2 months, but typically 3-4 months. Paint Analyses. During a rotary run, the model paints were analyzed at regular time intervals using a microscopical examination of the paint samples similar to the technique used by Urban et al.34 and Anderson.8 This is a destructive technique, which requires several samples on the rotor of each paint type. The paint sample is mounted in paraffin, which is subsequently sliced on a fine plane (microtome). The cross section of the paint sample can now be viewed under a microscope and the extent of polishing and leaching be determined. An example (model paint P2 after 47 days at 20 knots and 25 °C) is shown in Figure 5. The leached layer is transparent because very few pigment particles are present here.

The uncertainty of a measurement obviously depends on the quality of the paint sample under inspection. If the positions of the moving boundaries are clear (as in Figure 5), then the reading can be done with an accuracy of about 3 µm. However, if visual inspections are done by different individuals the uncertainty may be larger during a rotary run. Additionally, there may be small variations through the paint sample so that different cross section cuts may result in somewhat different measurements. At worst, the absolute error may be 5-10 µm or more. The higher polishing thickness evaluated, the lower the relative error of the measurement will be. In the experiments of this work, the absolute error was estimated to 3-5 µm for most paint samples. Presently, the microscopical examination of the paints appears to be the best method available for obtaining both polishing rate and leaching. It should be mentioned that if paints are produced from different batches of raw materials, contrary to the ones of this work, then a larger variation in the results could be found. The particle size distribution (PSD) of Cu2O in the paints was measured by a Malvern Analyser employing laser diffraction.35 To establish Cu and Sn conversion profiles in the leached layer of some of the model paints, SEM combined with energy-dispersive X-ray analysis (EDX) was employed. Prior to examination of the paints, the samples were sputted with gold. The element maps obtained were converted to profiles using the software package ImagePro. Subsequently, the data were smoothed using an exponential smoothing tool. Mathematical Modeling A model capable of describing the seawater behavior of model paints P2-P4 is described here. The model takes into account the hydrolysis and erosion of the TBT polymer, the dissolution of Cu2O particles, as well as external or internal mass transport resistances. The physical process is schematically shown in Figure 2. The overall purpose of the model is to provide a tool that can estimate the position of the moving polymer and pigment fronts at all values of time. The following species are considered in the model: H+, OH-, Na+, Cl-, TBTCl(aq), CuCl2-, CuCl32-, Cu2O(s), HCO3-, CO32-, TBTCP(s), and CP-Na+(s). The assumptions underlying the model development are: (i) The rate of dissolution of Cu2O particles is sufficiently fast, compared to diffusion of ions in the leached layer, that it takes place at the dissolving pigment front only (i.e., the leached layer is particle-free). (ii) There is no penetration of any species beyond the dissolving pigment front (i.e., into the unreacted paint film). (iii) The content of Cu2O particles in the solid paint film is high enough to ensure that most particles are in contact with other particles, thereby creating a pore network in the leached layer when the particles dissolve. On the other hand, the content of Cu2O does not exceed the estimated CPVC of about 57 vol. %.31 (iv) The hydrolysis of the TBT polymer takes place throughout the leached layer, except at the dissolving pigment front, and the only TBT product of the reaction is TBTCl. (v) Any formation of CuCl43- is neglected. The existence of this species has been mentioned.36 (vi) The oxidation of CuCl2- and CuCl32- to Cu2+ takes place in the bulk seawater only.


(vii) Pseudo-steady-state concentration profiles of all species are established in the leached layer. (viii) The walls of the polymer matrix are stable over time, and no swelling occurs due to the presence of water in the leached layer. (ix) The rate of diffusion of water into the leached layer is so rapid, due to the large pores created by dissolving Cu2O particles, that it does not influence the overall rate of polishing and extent of leaching. (x) The Cu2O particles present in the paint are assumed to be nonporous and spherical. (xi) The seawater-polymer interfacial area per unit volume of leached layer is equal to the surface area of Cu2O particles per unit volume of unreacted paint film (i.e., the empty polymer matrix is a template of the Cu2O particles which have dissolved). (xii) Due to the high ionic strength of seawater, electric potential gradients in the leached layer are neglected. (xiii) Diffusion of residual solvent from the interior of the paint film and into the bulk seawater is neglected. (xiv) The partially hydrolyzed TBT polymers are released into the seawater at a certain value of conversion termed Xmax. (xv) The ions Mg2+ and SO42-, both present in seawater, are considered inert. (xvi) The equilibrium of HSO4- a H+ + SO42- is neglected at seawater conditions due to a pKs value of 1.97 at 25 °C.37 (xvii) The paint geometry is that of a slab. The validity of the above assumptions is addressed in the Results and Discussion section. On the basis of the kinetics and assumptions stated above, the mathematical model can now be developed. Moving Pigment Front. A mass balance over the dissolving pigment front, stating that the rate of dissolution of Cu2O must equal the rate of diffusion of CuCl2- and CuCl32-, provides an equation for the rate of movement of this front dls1

) dt MCu2O(De,CuCl2-∇l[CuCl2-]|l)ls1 + De,CuCl32-∇l[CuCl32-]|l )ls1) 2(1 - o)VCu2OFCu2O(1 - VI)

(18) with the initial condition

ls1(t)0) ) 0

(19)

Moving Polymer Front. The assumption of a constant conversion, Xmax, at the polymer front (i.e., the paint surface) leads to an equation for the rate of movement of this front38

(

)| )| )

∂XTBTCP dls2 ∂t l)ls2 ) dt ∂XTBTCP ∂l l)ls2 t

((

MunitSo fR ∂XTBTCP ) ((-rTBTCP) ∂t (1 - o)FTBTCPVTBTCP (-rTBTCl)) (22) with the initial condition

XTBTCP(l,t)0) ) 0

(23)

The concentration profiles of the species present in the leached layer, needed in the solution of the above equations, can be determined by the mass transfer model derived below. Mass Balance for TBTCl. A differential mass balance for TBTCl in the leached layer yields

De,TBTCl∇l2[TBTCl] + ((-rTBTCP) - (-rTBTCl))So fR ) 0 (24) The boundary condition at ls2 is given by the condition that the rate of transport of TBTCl to the surface of the paint must match the rate of transport of this species across the external solid-liquid film

De,TBTCl∇l[TBTCl]|l)ls2 ) kL,TBTCl([TBTCl]l)ls2 [TBTCl]o) (25) At ls1, the boundary condition is given by the assumption of no penetration of TBTCl beyond the dissolving pigment front

∇l[TBTCl] ) 0

(20)

(26)

Total Component Mass Balances. Now follows a number of balances derived from the principles of Olander.39 Subsequently, all the necessary boundary conditions for these equations are given. CuCl2- and CuCl32-.

De,CuCl2-∇l2[CuCl2-] + De,CuCl32-∇l2[CuCl32-] ) 0 (27) This equation is valid because reaction 10 only takes place at ls1. Cl- Species.

2De,CuCl2-∇l2[CuCl2-] + De,Cl-∇l2[Cl-] + 3De,CuCl32-∇l2[CuCl32-] + De,TBTCl∇l2[TBTCl] ) 0 (28) HCO3- and CO32-.

De,HCO3-∇l2[HCO3-] + De,CO32-∇l2[CO32-] ) 0

(29)

Various Species. A combination of mass balances for relevant species gives

De,H+∇l2[H +] - De,OH-∇l2[OH-] + De,CuCl2-∇l2[CuCl2-] + De,CuCl32-∇l2[CuCl32-] +

The initial condition is given by

ls2(t)0) ) 0

Local Conversion. The local conversion of TBTCP in the leached layer is given by a mass balance

(21)

De,HCO3-∇l2[HCO3-] ) 0 (30)

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Charge Balance.

Water.

[H +][OH-]γ (2 - Kw ) 0

De,H+∇l2[H+] + De,Na+∇l2[Na+] - De,OH-∇l2[OH-] -

2

2

2-

De,CuCl2-∇l [CuCl2 ] - 2De,CuCl32-∇l [CuCl3 ] -

2

-

2

De,Cl-∇l [Cl ] - De,HCO3-∇l [HCO3 ] 2

2-

2De,CO32-∇l [CO3 ] ) 0 (31)

(39)

HCO3-/CO32- Buffer.

KHCO3- [HCO3-] - [CO32-][H+]γ ( ) 0

(40)

Cl- Complexes.

( )

Boundary conditions at ls1 now follow. Flux balance for Cl- species

KCuCl32KCuCl2-

-

-

2De,CuCl2-∇l[CuCl2 ] + De,Cl-∇l[Cl ] + 3De,CuCl32-∇l[CuCl32-] + De,TBTCl∇l[TBTCl] ) 0 (32) and for HCO3- and CO32-

De,HCO3-∇l[HCO3-] + De,CO32-∇l[CO32-] ) 0 (33) Flux balance for selected species

De,H+∇l[H +] - De,OH-∇l[OH-] + De,CuCl2-∇l[CuCl2-] + De,CuCl32-∇l[CuCl32-] + De,HCO3-∇l[HCO3-] ) 0 (34) and no charge transport across the pigment front

De,H+∇l[H+] + De,Na+∇l[Na+] - De,OH-∇l[OH-] De,CuCl2-∇l[CuCl2-] - De,Cl-∇l[Cl-] 2De,CuCl32-∇l[CuCl32-] - De,HCO3-∇l[HCO3-] 2De,CO32-∇l[CO32-] ) 0 (35) Also, at the dissolving pigment front, the rate of dissolution must equal the rate of diffusion of Clcomplexes -

2-

De,CuCl2- ∇l[CuCl2 ]|ls1 + De,CuCl32- ∇l[CuCl3 ]|ls1 ) (2(-rCu2O)|ls1 - (-rCuCl2-)|ls1)fA (36) Due to the assumption of all Cu2O particles dissolving at the pigment front, the fraction of the total cross section area that is Cu2O at this front, fA, is equal to the volume fraction of Cu2O (corrected for impurities) in the unreacted paint film. Thus, the rate of conversion of Cu2O particles at the pigment front is independent of particle size. However, the rate of conversion of TBTCP in the leached layer is dependent on Cu2O particle size because the initial active surface area per unit volume of the leached layer is a function of the PSD of Cu2O in the unreacted paint film

φi

N ∑i)1 dpi

So ) 6VCu2O(1 - o)

(37)

Five boundary conditions are supplied at ls2

De,i∇l[i]|l)ls2 ) kL,i([i]l)ls2 - [i]o)

(38)

where i ) H+, Na+, Cl-, HCO3-, or CuCl2-. Equilibria. The remaining equations, which close the system of equations, are supplied by three equilibria, which are valid throughout the leached layer.

[CuCl2-][Cl-]γ ( - [CuCl32-] ) 0 (41)

In eq 41, the two equilibrium constants represent the formation of CuCl2- and CuCl32- from Cl- and Cu+. The ratio of the two constants is equal to the equilibrium constant of eq 11. Bulk phase concentrations are calculated by assuming that equilibrium has been established among the various salts (see Table 2) added to the rotor tank. In the case of TBTCl(aq), CuCl2-, and CuCl32-, the bulk phase concentrations were all taken to be 0 mol/m3. Numerical Solution Strategy. The model is rendered dimensionless by introduction of dimensionless variables and solved by the method of orthogonal collocation.40 Since the reaction zone is confined to a very thin layer at the surface of the paint film, a very steep solid-phase conversion profile develops. This requires an advanced numerical solution technique, whereby the reaction zone is immobilized by introduction of suitable coordinate transformations.38 Using this method, only four interior collocation points were needed in the active zone (i.e., in the leached layer) to obtain convergence. The computation time was less than 1 min on a Pentium 450-MHz PC. Estimation of Model Parameters The paint model requires a number of physical and chemical constants. These were taken from various literature sources and are available as Supporting Information. Equilibrium constants for reactions 16 and 17 as functions of temperature were taken from Brewer.37 The equilibrium constant of reaction 11 at 25 °C was calculated from data in Iselin,30 but no temperature correction was possible. Molecular diffusion coefficients at 25 °C and infinite dilution for H+, OH-, Cl-, and Na+ were taken from Lide.41 In the cases of HCO3- and CO32-, the diffusion coefficients were found in Gage,42 and that of CuCl32was available in White et al.43 No diffusion coefficient has been found for CuCl2-, but Dudek and Fedkiw44 have reported a value for Cu+, and since the value for CuCl2- is expected to lie between that of Cu+ and that of CuCl32-, an average value of the latter two was used. The diffusion coefficient for TBTCl(aq) was calculated by the Wilke-Chang correlation45 using the molar volume of TBTCl(aq) at 4 °C.46 All molecular diffusion coefficients were extrapolated to other temperatures using the Stokes-Einstein relation42 and corrected for ionic strength by Gordon’s method.45 Effective diffusion coefficients of the leached layer were calculated from47

De,i )

LL D τ m,i

(42)

The tortuosity factor, τ, was calculated by the method

Ind. Eng. Chem. Res., Vol. 40, No. 18, 2001 3913 Table 3. Overview of Rotary Experimentsa run

rotary speed (knots)

temperature (°C)

1 (base case) 2 3

20 30 20

25 25 35

a

1.0 knot ) 0.514 m/s ) 1.850 km/hour.

of Wakao and Smith,48 and the porosity, LL, is equal to the volume fraction of Cu2O in the unreacted solid paint film. The solubility product of CuOH at 25 °C was found in Iselin,30 but no temperature correction was possible. Liquid viscosities and densities were taken from DIPPR Tables.49 The activation energy for the rate constant of reaction 10, between temperatures of 0 and 37 °C, was determined by Ferry and Caritt,29 and that of reactions 3 and 4, at temperatures between 15 and 35 °C, could be estimated by extrapolation of the data of Hong-xi et al.27 External liquid-solid mass transfer coefficients were calculated from the correlation of Sherwood et al.50 However, to perform simulations at conditions of our test rig, it was necessary to extrapolate the data to higher Reynold’s numbers than the authors recommend. The model was, fortunately, practically insensitive to the value of the mass transfer coefficient at the rotation speeds used (20-30 knots). At seawater conditions corresponding to Table 2 and a temperature of 25 °C, the equilibrium conversion of TBTCP can be estimated from literature data22 and similarly with the solubility of TBTCl by assuming that the solubility of TBTCl is similar to that of TBTF.51 With these data in hand and the concentrations of Cl- and OH- determined from the data in Table 2, the equilibrium parameter of eq 9, RL,eq, can be estimated to 0.006 (m3/mol)a. The uncertainty of this number may be quite high, but the model is insensitive to the value of RL,eq for values larger than 0.0002 (m3/mol)a. The mean ion activity coefficient of seawater was calculated according to Meissners method,52 which is valid at moderate ionic strengths. It was assumed that NaCl is the dominating electrolyte, and MgSO4 was ignored in the calculation of the activity coefficient. In this way, the Meissner parameter for NaCl could be used, and the mean ion activity coefficient was assumed to be the same for all species. To estimate if any precipitation occurs in the leached layer, the solubility product of CuCO3 and Cu(OH)2 was taken from Mortimer53 and Skoog et al.,54 respectively. The porosity of the unreacted paint film was assumed to be negligible. Results and Discussion To verify the paint model and to investigate certain aspects of paint behavior, three experimental series (see Table 3) were conducted with the paint rotor using the model paints of Table 1. The first run, in the following referred to as the base case, involved using a rotary speed of 20 knots and a temperature of 25 °C. The remaining two experiments involved pertubations from the base case with respect to two important parameters: rotary speed (increased from 20 to 30 knots) and temperature (increased from 25 to 35 °C). In subsequent sections, data obtained under the different conditions are compared and discussed. Experiments with a Paint of TBTM/MMA Copolymer and Cu2O Particles. In Figure 6, data of rotary experiments performed at 20 and 30 knots with a paint of TBTM/MMA copolymer and Cu2O particles

Figure 6. Measured polishing data of model paint P1 at two values of rotary speed. Error bars are shown on the figure.

(paint number P1) are shown. In the figure, 1 day equals 24 h. It can be seen that the paint polishes very fast (about 150 µm/month). In comparison, a commercial TBT-based antifouling paint typically polishes 5-20 µm/ month.24 The reason for the high polishing rate is the absence of a retarder (such as BMA/MMA). Upon reaction of the TBTM/MMA copolymer with seawater, it becomes hydrophilic, and since there is no retarder present in the binder phase to keep the solubility low, water can probably penetrate the polymer itself and not just the porous leached layer remaining from dissolution of Cu2O particles. This may cause the polymer to swell and the surface area available for the hydrolysis reaction to increase by several orders of magnitude. The result is that the binder phase reacts and dissolves very rapidly. In fact, the polishing happens so fast that there was not detected any formation of a leached layer. It can also be seen in Figure 6 that the paint polishes at equal rates at 20 and 30 knots. This observation suggests that the rate of polishing is controlled by the solubility of the hydrolyzed polymer alone and not the shear stress experienced by the paint. At lower rotary speeds, however, this may not necessarily be the case. Since commercial antifouling paints contain retarders, there has been no attempt to model the behavior of P1. Comparison of Model Simulations with Experimental Data. In this section, model simulations are compared to experimental rotary data for the three paints P2, P3, and P4. Simulations were done using just one adjustable parameter, Xmax. All other parameters were determined from literature data, as described in an earlier section. In the next paragraph, the values of Xmax used in the simulations are justified. In Figure 7, simulated positions of the moving pigment and polymer fronts, at values of time between 0 and 80 days, are compared to experimental data for the paint P2. For visual reasons, error bars have only been added to the data obtained for the moving polymer front. A value of 0.65 was used for Xmax at both 20 and 30 knots. The gap between the positions of the two fronts represents the thickness of the leached layer. It is evident that after about 25 days of seawater exposure, a stable thickness of approximately 10 µm is reached. Good agreement between simulations and experimental data can be seen, keeping the uncertainty of the measurements (about 3-5 µm) in mind. Notice, by comparison with the data of P1 in Figure 6, that the rate of polishing is drastically reduced (from about 150 to about 15 µm/month) by the introduction of a small amount (about 3 vol. %) of retarder in the paint. It can also be seen that there appears to be no significant difference between the

3914


Figure 7. Polishing rates and extent of leaching for paint P2. Values estimated by the model (Xmax ) 0.65) are represented by lines and measured data by symbols. The temperature was 25 °C.

experimental results at 20 and 30 knots, suggesting that the rate of polishing in this interval of rotary speed is controlled by the solubility of the hydrolyzed polymer alone and not the shear stress experienced by the paint. One may also notice the lag time (about 15 days in Figure 7) prior to the initiation of polishing. The length of this lag time is determined by the rate of hydrolysis at the paint surface, which must produce a certain conversion, Xmax, of the TBTCP before any binder is released to the seawater. Once the outer surface has reached Xmax, the polishing is initiated because the polymer layer next to the surface will have a conversion close to Xmax. The latter is due to the hydrolysis taking place throughout the leached layer. In Figure 8, the comparison is shown for P3, which contains about twice the amount of retarder of P2. A value of 0.75 was used here for Xmax in order for simulations to match experimental data. Also for this paint, there appears to be only a small difference between the data recorded at 20 and 30 knots. According to simulations, the lag time is about 20 days, as opposed to 15 days for P2, but an experimental confirmation of this difference between the behavior of the two paints was not possible due to the uncertainty of the experimental data. In fact, the two paints P2 and P3 appear to polish at very similar rates. The paint P5, which is identical to P3 except that the paint is particle-free, showed no polishing at all during the test period. This suggests that Cu2O particles play an important part in the maintenance of the polishing behavior of selfpolishing antifouling paints. In Figure 9, the comparison is shown for paint P4, which contains a very large amount of retarder (twice the amont of P3). Here, it can be seen that the model does a poor job of predicting the experimental data. The rate of movement of the pigment front is overestimated, whereas the polishing data, due to a long lag time of at least 45 days, are too few to confirm the simulations. Thus, it appears that some kind of mechanism, which is not included in the model, slows down the rate of dissolution of the pigment particles. Probably, due to the large amount of retarder, the binder phase is so



hydrophobic that more TBT-groups than was the case for P2 and P3 will have to be hydrolyzed to bring the polymer chains in solution. In terms of the model, this means that Xmax will be higher. The latter is confirmed by the simulations of Figure 9, where a value of 0.95 was required for Xmax to match the polishing data. Now, the almost completely hydrolyzed polymer chains, trapped in the retarder at the surface of the paint film, are so water soluble that they may swell, due to water uptake, and partly block the pores of the leached matrix. This was also seen as a very light color of the paint surface after exposure to seawater, which is likely due to a higher difference in refractive index of the leached layer with air or water. The swelling phenomenon will slow the rate of effective diffusion in the leached layer


and thereby the rate of movement of the pigment front. A simulation performed with a tortuosity factor of 10 (as opposed to a value of 2.5 estimated in an earlier section) suggests that a hindered diffusion could indeed be the explanation, as shown in Figure 9. For P4 also, there appears to be no influence of rotary speed on the rate of polishing between 20 and 30 knots. The simulated rate of polishing (i.e., the slope of the polishing curve produced by the model), once it has begun, looks quite high for paint P4. However, the curve levels off at later times (cannot be seen on the figure) resulting in a steady polishing rate of 10.0 µm/month. In comparison, the simulated steady polishing rates of P2 and P3 are 15.1 and 13.7 µm/month, respectively. In summary, simulations and experimental data show that the introduction of a small amount of retarder in the paint (about 3 vol %) causes a drastic reduction in the rate of polishing, but further substitution of TBTM/ MMA by retarder produces only slight decreases on polishing. The lag time, however, is increased with increasing amounts of retarder in the paint, and this is important because longer run times with the paint rotor may be required to establish the stable rate of polishing. Effect of the Model Parameter Xmax. In the above simulations of P2, P3, and P4, a single adjustable model parameter, Xmax, was allowed to vary among the three paints. The parameter has a physical meaning because it represents the conversion of TBT groups at the surface of the paint film at which the active polymer is released into seawater. It may vary according to any changes in seawater conditions, paint composition, or rotary speed. If Xmax can be estimated as a function of all these relevant process parameters, then a complete description of the paint behavior would be possible. No such method has yet been identified. Presently, however, it is possible to measure the value of Xmax for a paint film that has been exposed to given conditions. This can be done using SEM combined with EDX as described in the Experimental section. A sample of each of the paints P2, P3, and P4, all taken after 47 days on the rotary experiment employing 20 knots, was analyzed by this technique. In Figure 3, the SEM recording for P2 is shown as an example, and in Figure 10, Cu and Sn conversion profiles are compared to model predictions. The SEM/EDX data of Figure 10 have been combined with rotary data for these three paints so that a direct comparison of the profiles with simulations can be done. The Cu conversion profile was obtained by comparing the Cu level in the unreacted paint film with the level found at various positions in the leached layer. In the case of Sn, it was not possible to compare the levels in the unreacted film with those in the leached layer because the content of Cu2O particles was lower in the leached layer. In the unreacted paint film, Sn will not be detected in the binder areas that are right below Cu2O particles. Thus, if the levels were compared, the Sn concentration in the leached layer would be overestimated because here Sn is also detected in the pores created where the Cu2O particles have dissolved. Thus, it was necessary to compare the Sn concentration in the leached layer with the Sn concentration detected close to and behind the pigment front. Additionally, determination of the exact position of the paint surface requires a smooth and completely horizontal surface, which is hard to obtain, as shown in Figure 3. However, a good indication of where the surface begins could be obtained from the Cu profiles. Due to these modifica-

Figure 10. Cu and Sn conversion profiles in the leached layer for paints P2-P4 after 47 days on the rotor (25 °C and 20 knots). Lines represent model estimations and symbols experimental data. For each of the experimentally determined Cu profiles, the position of the paint surface is indicated by an arrow.

tions, the Sn profiles are somewhat inaccurate and should be used with caution when compared to simulations. It can be seen in Figure 10 that the experimental Cu profiles for P2 and P3 are very steep. Most of the pigment front extends over a few micrometers, verifying the model assumption of dissolution of Cu2O taking place at the moving pigment front only. Somewhere between 10% and 20% of the Cu2O is present in the leached layer (defined here as the region covered by the Sn profile), and this is probably due to a combination of CuO or Cu, present as impurities in Cu2O, as well as some Cu2O being isolated in the binder without any contact to the seawater-filled pores. For P2 and P3, there is a reasonable agreement between model simulations and measured Cu profiles, again keeping the uncertainties of the experimental data in mind. For P4, the observed front is not so steep, and the simulation does not match for the reasons discussed earlier. The simulated and measured Sn profiles for P2 and P3 are very steep and in good agreement, and the values of Xmax used in the simulations (0.65 and 0.75, respectively) seem reasonable. A better match is probably not possible considering the accuracy of the measurements. The relative uncertainty of the measured values of Xmax was estimated to 10-15%, but the (unknown) error introduced from the varying content of Cu2O in the paint film near the dissolving pigment front may increase this number even further. Presently, however, this technique appears to be the best one available for estimation of Xmax. For P4, the value of Xmax used as model input (0.95) looks right, but the profile is all wrong for reasons discussed above. Thus, the approach of using Xmax as the adjustable parameter in the model corresponds reasonably well with what is observed for

3916


Figure 11. Simulations showing the effect of the model parameter Xmax (25 °C and 20 knots) on the polishing rate (at conditions of a stable leached layer thickness), the lag time, and the stable thickness of the leached layer.

real paints, where the surface conversion of the polymer does indeed vary with paint composition and seawater conditions. The sensitivity of the model with respect to Xmax is shown in Figure 11. Strong sensitivity for the thickness of the leached layer (taken here as the stable thickness that is reached after some time of seawater exposure) and the polishing lag time for values of Xmax larger than about 0.80 can be seen. Contrary to this, the rate of polishing (evaluated at a stable leached layer thickness) is not very sensitive here. In the region of interest, from about 0.50 to close to 1.0, the rate of polishing decreases 40%. Thus, a very accurate value of Xmax is not needed to perform a proper simulation. If no experimental value is available for Xmax, then one may still get a reasonable estimate of what the polishing rate could be under given conditions. The reason the polishing rate shows a low sensitivity to Xmax for values of this parameter close to unity can be ascribed to the stable thickness of the leached layer. The rate of hydrolysis decreases (for conversions close to Xmax) as Xmax is increased, but at the same time, the stable leached layer thickness increases. The latter means that the time of seawater exposure the active polymer in the leached layer experiences, prior to detachment at the surface, is longer the higher the value of Xmax. According to simulations, the effects of a lower hydrolysis rate and a longer reaction time more or less counterbalance one another. At values of Xmax larger than 0.97 and smaller than 0.02, simulations could not be performed. The value of the lag time is essential for estimation of the lifetime of a paint from rotary experiments, but it is not so important with respect to fouling control because the polymer anchored biocide is released from the very beginning even though the polymer front does not move. The stable thickness of the leached layer goes up as the polishing goes down and this means a reduced release rate of Cu-based biocides. When Xmax increases from 0.5 to close to 1.0, the rate of release of CuCl2and CuCl32- is reduced almost 50% (not shown). However, the higher the value of Xmax is, the more the TBTbiocide is exploited. If the TBT group is still attached to the polymer when the latter is released, then it probably does not contribute to fouling control, only to pollution of seawater. A thing to notice about the parameter Xmax is that it is used to explain rate phenomena related to the polymer hydrolysis and erosion. Other aspects, such as

Figure 12. Polishing data (at conditions of a stable leached layer thickness) for a commercial antifouling paint. Model estimations are represented by lines and measured data of Anderson24 by symbols.

the presence in the paint of extenders or insoluble pigments or the occurrence of cross linking reactions in the leached layer,5 may need their own mathematical treatment. If this is not done properly, then the errors imposed on the physical system represented by the model may be lumped into the value of Xmax, which may consequently be in error, thereby resulting in incorrect simulations. Additionally, certain formulations could result in paints that do not initiate polishing even when 100% conversion of the active polymer has been achieved. This could, for instance, happen to paints with very high retarder contents. In such cases, the paint will polish due to the shear stress applied only, and the model of this work would be incapable of describing the behavior. Detailed paint models can perhaps be used to accelerate rotary experiments. Initially, when a new model has been developed, experiments at some selected base case conditions are always needed to verify simulations (as shown earlier). At the subsequent application of the model at other conditions, a measurement of Xmax for the pertinent paint is required to obtain accurate results. This one measurement can be recorded using for example SEM/EDX as soon as the paint sample on the rotor has begun to polish because at this point Xmax no longer changes (assuming constant seawater and rotary conditions). Thus, it may only take 3-4 weeks (corresponding to the length of the lag time) to obtain a value of Xmax and the polishing rate and extent of leaching can then be estimated by the model. In comparison to several months or even years8 of rotary experiments or ship tests, this may in some cases be a useful and economical procedure. It should be stressed, however, that mathematical models are always to be used in combination with rotary experiments, especially when new paint formulations are tested. Finally, it should be mentioned that in order to use paint models to estimate paint lifetimes at varying conditions of seawater and ship speed, a correlation of Xmax as a function of the pertinent parameters would need to be established. Comparison of Model Simulations with Experimental Data of Anderson24. It has been shown above that the three paints P2-P4 were insensitive to rotary speeds in the interval 20 to 30 knots. No investigations were done at lower speeds, but Anderson24 has supplied data for a commercial TBT-based antifouling paint at rotary speeds between 0.0 and 30 knots and close to 25 °C. In Figure 12, the model of this work has been used to simulate the experimental data. In the figure is also shown how Xmax was varied in order to match the data. First of all, it can be seen that the experimental data


Figure 13. Simulated concentration and pH profiles through the leached layer of paint P3 (Xmax ) 0.75) after 47 days. The thickness of the stable leached layer is estimated by the model to about 9 µm. The temperature was 25 °C and the rotary speed 20 knots.

shows a very little (if any) influence of rotary speed between 20 and 30 knots, in good agreement with our observations. Below about 20 knots, however, the shear stress experienced by the paint becomes important. Second, by varying Xmax between 0.6 and 0.97, the model can match the data despite the fact that the composition of the paint probably is somewhat different to the model paints of this work. The latter suggests that the main features of the biocide release mechanisms have been captured in the model. However, it should be emphasized again that in order to conduct accurate simulations of commercial antifouling paints, all relevant rateinfluencing steps must be included. Notice in Figure 12 that the polishing rate decreases quite rapidly below about 10 knots, even though an almost constant value of Xmax is used in this region. The reason for this is that the external mass transport resistances of CuCl2- and CuCl32- increase as the rotary speed is lowered. Assuming that the paint rotor used by Anderson24 has a diameter of about 0.5 m, a peripheral velocity of just 1.0 knots corresponds to a Reynold’s number (based on rotor diameter) of about 105. This high number is expected to cause turbulent flow around the rotor, and so, there is no obvious relation between the change in Xmax shown in Figure 12 and the flow regime (laminar or turbulent). As stated by Anderson24 and shown in Figure 12, the continual replenishment reaction of biocide at the paint surface can occur under static conditions, which is when the fouling is most severe. This feature is essential to maintain in novel self-polishing antifouling paints. Concentration Profiles and Biocide Release Rates. In Figure 13 are shown concentration profiles and pH of the leached layer at times where the rate of movement of the fronts have become constant. It can be seen that there is a large drop in biocide concentration and a decrease in pH of about 0.1 from the pigment to the polymer front. The higher value of pH at the pigment front is due to dissolution of Cu2O particles, which is a process consuming hydrogen ions. This kind of information can be important in the evaluation of novel self-polishing paints because the rate of hydrolysis of a polymer can be dependent on local values of pH. The rate of hydrolysis of the TBT polymer, for instance, is increased when the pH is raised (see eqs 3 and 4), though the effect is quite small. The rate of release of the various biocides and the thickness of the leached layer are shown in Figure 14 as a function of time. What is important to see here is that the rate of release reaches a constant value after

Figure 14. Simulated fluxes of biocides and thickness of the leached layer through the lifetime of paint P3 (Xmax ) 0.75). The temperature was 25 °C, the rotary speed 20 knots, and the initial dry-paint thickness 500 µm.

about one month. This delay is due to the polishing lag time. During this first month, the release rate of CuCl2-/ CuCl32- is higher than its constant value obtained later due to the very thin leached layer formed initially. The TBTCl release rate is lower, however, but it only takes 5-10 days to reach half the value of the constant release rate, and during these first days, Cu2O will probably provide sufficient fouling protection. Finally, it can be seen that once the pigment front reaches the last part of the antifouling, the TBT biocide release rate drops rapidly (from about 12.0 to 0.0 µg cm-2 day-1 in three weeks). This type of information can be important because “polish through” can occur in front parts of the ship where the seawater influence on the ship is severe and where damages to the paint more easily occur. Rotary Experiments and Modeling at an Elevated Temperature. To investigate the influence of temperature on the rate of polishing and extent of leaching, rotary experiments and simulations were conducted at 35 °C. In Figure 15, simulations of the polishing rate and the stable thickness of the leached layer are compared to experimental data for the paints P2 and P3. The experimental polishing rate at a given value of temperature was obtained from four individual measurements for each paint, corresponding to the slope of the data points of Figures 7-9. However, the total runtime was increased to 104 days for the run at 35 °C. The simulations for P2 and P3 were done using values of Xmax of 0.65 and 0.75, respectively. Good agreement between model simulations and experimental data can be seen, though the effect of temperature between 25 and 35 °C is almost within the uncertainty of the experiments. According to simulations (keeping in mind that the solubility product of CuOH and the equilibrium constant of reaction 11 were assumed independent of temperature), the polishing rate increases about 50% (for both P2 and P3) in the temperature range 25-35 °C. The hydrolysis and dissolution rates are about doubled, but

3918


Figure 15. Polishing rates (at conditions of a stable leached layer thickness) and leaching data for paints P2 (Xmax ) 0.65) and P3 (Xmax ) 0.75) at various temperatures and a rotary speed of 20 knots. Values estimated by the model are represented by lines and measured data by symbols.

diffusion coefficients are only increased about 10%. The polishing rate is influenced by both effective diffusion in the leached layer and the chemical reaction rates, and according to the model, this combined effect results in the 50% increase. Schatzberg14 measured the rate of release of TBT (µg cm-2 day-1) for a number of selfpolishing antifouling paints. He observed that a temperature increase from 10 to 25 °C influenced the release rate by a factor of 1.8-4.3. In comparison, simulations with paints P2 and P3 in the same temperature range show that the release rate of TBTCl increases by a factor of 2.5 and 1.8, respectively, indicating a good agreement. The model estimates that the stable thickness of the leached layer decreases with increasing temperature. The main reason for this is that the activation energy for the hydrolysis reaction is higher than that of the pigment dissolution. The latter results in a larger enhancement of the rate of movement of the polymer front when the temperature is increased compared to that of the pigment front. However, simulations show that the stable leached layer thickness of P2 and P3 only decreases 2.7 and 3.5 µm, respectively, when the temperature is increased from 25 to 35 °C. These changes are within the uncertainty of the experimental data, and the effect was not confirmed. An important observation from the above comparison between simulations and experimental data for P2 and P3 is that the model parameter Xmax appears to be independent of temperature (i.e., the same value of Xmax was used at 25 and 35 °C). The value of Xmax is expected to change as a function of temperature, mainly due to a change in the solubility of the hydrolyzed polymer, but the effect is probably so small, for a temperature increase of only 10 °C, that it is hidden within the uncertainty of the experimental data. Simulations with P4 (not shown) failed to match experimental data for the reasons discussed earlier. It

can be mentioned, however, that the experimental data recorded for this paint were almost identical at 25 and 35 °C. Thus, it is possible that the rate of polishing is controlled by diffusion of ions through the swelled polymer, but the data were too limited to confirm this hypothesis. Sensitivity Analysis. A sensitivity analysis of the model, keeping Xmax at 0.75 and the temperature at 25 °C (base case), with respect to all the physical and chemical parameters mentioned in the model development was performed. The following parameters were identified to influence the model simulations (i.e., rate of polishing and/or extent of leaching) with relative deviations from the base case between 5 and 13% when varied (20% around their estimated values: the rate constant for the Cu2O dissolution, the solubility product of CuOH, the tortuosity factor for the leached layer, the activation energy and surface rate constant for the hydrolysis reaction, the PSD of Cu2O particles in the paint, and the mean ion activity coefficient for seawater. Of these parameters, only the tortuosity factor, the activation energy, and the PSD are not known with a high accuracy. However, even with errors of up to 20% in the latter values, model predictions are still in good quantitative agreement with experimental data. Validation of Model Assumptions. The paint model is based on a number of simplifying assumptions that have not all been verified in the above discussion. Here, the most important ones are readdressed. It was assumed, that the oxidation of CuCl2- and CuCl32- to Cu2+ takes place in the bulk seawater only and not in the leached layer. If this was not the case, then according to model simulations, Cu(OH)2 and CuCO3 would precipitate in the leached layer. No precipitate has been observed experimentally (the leached layers were all transparent, whereas Cu(OH)2 and CuCO3 is light blue and green, respectively), and no reduction in the rate of dissolution was found, except for the paint with a large content of retarder. Thus, the rate of oxidation is too slow to influence the rate of dissolution in the leached layer. The assumption of pseudo-steady-state concentration profiles of all species in the leached layer was verified by the fact that the time constant for diffusion was much smaller than for the rate of conversion of the TBTpolymer. The absence of any water swelling of the polymer could not be confirmed, but the experimental data suggested that swelling occurs mainly in the retarderfree paint films, for which a very high polishing rate was observed, or at the surface of paints of very high retarder contents where the hydrolyzed polymer can get trapped in the retarder. The assumption of TBTCl being the only product of the hydrolysis is of minor importance because the model is practically insensitive to effective diffusion of TBTCl in the leached layer. The reason for this is, according to simulations, that the hydrolysis reaction is very close to being irreversible at the conditions considered. In this case, the assumption of the reverse hydrolysis rate being first-order in the concentration of TBTCl and the number of hydrolyzed TBT groups becomes insignificant. The influence of diffusion of residual solvent from the paint film and into the seawater on the rate of polishing and extent of leaching was assumed negligible. However, this assumption was not confirmed.


Conclusions A detailed model for a self-polishing antifouling paint was developed. The important rate-influencing steps, dissolution of Cu2O particles, hydrolysis and erosion of TBT polymer, effective diffusion in the leached layer, and external mass transport of relevant species, were all included. Simulations were in good quantitative agreement with experimental data obtained at various values of rotary speed and seawater temperature. The model has also been applied to perform an elaborate parameter study with respect to seawater conditions and paint composition, but results from this work will be published separately. The modeling tools developed, including the Xmax concept, may be adapted to other self-polishing antifouling paints employing different binder and pigment phases, but kinetic expressions for the relevant hydrolysis and pigment dissolution reactions, as well as solubility and diffusivity data, need to be determined from laboratory measurements if not available in the literature. Most (if not all) of the novel paints use Cu2O as one of the biocides and here the principles of this work are directly applicable. Future work includes investigations with other soluble (such as ZnO) and insoluble (such as TiO2) pigments present in the paint, as well as a model-based analysis of the novel self-polishing antifouling systems mentioned in the Introduction section. It would also be useful to test the models against experimental data obtained using real ships or at least rotors placed in the sea where the presence of fouling organisms may influence the data. Finally, long-term rotary experiments, lasting for up to 1 year or longer, was not covered in the present work. Acknowledgment S.K. is grateful to Michel Warnez for the many interesting discussions and advice on paint chemistry. Financial support of S.K. by J.C. Hempel’s Foundation is gratefully acknowledged. Nomenclature a ) reaction order with respect to the concentration of OH- (eqs 3 and 4) b ) reaction order with respect to the concentration of OH- (eqs 3 and 4) BMA ) butyl methacrylate dp ) particle diameter of a Cu2O particle, m De,i ) effective diffusivity of component i in the leached layer, m2/s fA ) ratio of surface area of Cu2O in total cross section area to the total surface area of cross section fR ) correction to the active binder surface area in the leached layer due to the presence of the retarder i ) component or species i kL,i ) external liquid-solid mass transport coefficient of component i, m/s kSH ) surface rate constant of hydrolysis reaction, (m3/ mol)1+a mol TBTCP/(m2 s) kSL ) surface rate constant of hydrolysis reaction, (m3/ mol)1+a mol TBTCP/(m2 s) k-S ) surface rate constant for the reverse hydrolysis reaction 1, m/s k1 ) surface rate constant for Cu2O dissolution reaction, mol Cu2O (mol/m3)-3 m-2 s-1 k-1 ) surface rate constant for reverse Cu2O dissolution reaction 10, m/s

k2 ) parameter in rate expression for hydrolysis reaction, (m3/mol)b Ki ) equilibrium constant for reaction i, various units l ) position in the paint film, m Li ) solubility product of species i, (mol/m3)2 Mi ) molar mass of species i, kg/mol MMA ) methyl methacrylate N ) number of different particle sizes PSD ) particle size distribution r ) rate of reaction, mol m-2 s-1 RL,eq ) defined in eq 9, (m3/mol)-a So ) initial specific surface area of the binder phase in the leached layer, m2/m3 t ) time, s TBTCP ) TBTM/MMA copolymer TBTM ) tributyltin methacrylate Vi ) volume fraction of species i xTBTM ) molar fraction of TBTM in copolymer XTBTCP ) conversion of active polymer Xmax ) value of surface conversion at which the active polymer is released into seawater [i] ) concentration of component i, mol/m3 Greek Letters ) porosity φi ) volume fraction of particle size i γ( ) mean ion activity coefficient of seawater Fi ) density of component i, kg/m3 σ ) parameter of eq 4 τ ) tortuosity factor Superscript o ) initial value Subscript eq ) equilibrium i ) species i I ) impurity LL ) leached layer s1 ) at the position of the moving pigment front s2 ) at the position of the moving polymer front o ) value of unreacted paint film or bulk value TBTCP ) TBTM/MMA copolymer unit ) repeating unit of TBTCP (shown in Figure 1) W ) water Mathematical Operations ∇ ) “del” or “nabla” operator Supporting Information Available: Physical and chemical constants used in the simulations. This material is available free of charge via the Internet at http:// pubs.acs.org. Literature Cited (1) Ayoub, M. M. H.; Abdel Malek, M. M.; Messiha, N. N. Laboratory and Ships Test of Modern Antifouling Paint Formulation. Pigm. Resin Technol. 1990, 19 (2), 4. (2) Gerigk, U.; Schneider, U.; Stewen, U. The Present Status of TBT Copolymer Antifouling Paints versus TBT-free Technology. Prepr. Ext. Abstr. ACS Natl. Meet. 1998, 38 (1), 91. (3) Weisman, G. R.; Sundberg, D. C.; Cimini, R. A.; Brown, M. G.; Beno, B. R.; Eighmy, T. T. Controlled Release Antifouling Coatings. I. Approaches for Controlled Release of 2,4-Dinitrophenolate and Benzoate into Seawater. Biofouling 1992, 6 (2), 123. (4) Swain, G. W. Biofouling Control: A Critical Component of Drag Reduction. International Symposium on Seawater Drag Reduction, Newport, RI, 22-24 July, 1998; The Naval Undersea Warfare Center, p 155.

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Received for review March 14, 2001 Revised manuscript received June 21, 2001 Accepted June 22, 2001 IE010242N

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