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The measurement of the diffusion coefficient and the sorption isotherm of water in paint films
Chemical Engineering Science 58 (2003) 1521 – 1530
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The measurement of the di"usion coe#cient and the sorption isotherm of water in paint %lms A. J. J. van der Zanden∗ , E. L. J. Goossens Faculty of Architecture, Building and Planning, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands Received 2 April 2002; received in revised form 3 September 2002; accepted 1 October 2002
Abstract An experimental technique together with a numerical model is proposed with which the di"usion coe#cient and the sorption isotherm of water in paint can be measured. Inside a closed vessel, paint %lms are on stainless-steel plates. Water is present as water vapour in the air and in the paint. After blowing dry or wet air through the vessel for some time, the situation moves to a new equilibrium. The relative humidity of the air inside the vessel is measured as a function of time. From %tting the theoretical/numerical model against the experimental values, follow the di"usion coe#cient and the sorption isotherm of water in the paint. The results show large scattering. When the independently measured sorption isotherm is used as an input parameter in the model, the %tting procedure gives much smaller scattering for the di"usion coe#cient. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Di"usion coe#cient; Sorption isotherm; Paint; Drying; Mass transfer
1. Introduction Besides for their decorative function, paint %lms are also used to change the moisture transfer properties of materials. Thus, it is important to know the moisture transfer properties of the paint %lm itself. For an overview on measurement methods of di"usion in polymers (paint %lms most often contain polymers), see for instance Crank and Park (1968). For reviews on moisture transport in paint, see Blahn>?k (1983), Huld>en and Hansen (1985), and van der Wel and Adan (1999). Paint %lm properties are usually examined by making use of a free %lm or foil, which is not attached to a substrate. The mass transfer through such a %lm can be determined in various ways. Most often, this is done by measuring the mass increase or decrease of a cup %lled with water that is covered with the %lm. This technique is known as the cup method. More advanced techniques for measuring the change in the quantity of water are available, such as a chemical analysis or gas chromatography. Another type of technique uses a sorption–desorption method. In such a method,
∗ Corresponding author. Tel.: +31-40-247-37-21; fax: +31-40-243-85-95. E-mail address: [email protected] (A. J. J. van der Zanden).
the quantity of moisture in a sample subjected to an atmosphere, with which it is not in equilibrium, is registered as a function of time, most often by measuring the weight of the sample. The sorption technique is already an old technique for measuring di"usion and sorption in polymers, see for instance the articles of Lundberg, Wilk, and Huyett (1962) and Michaels, Vieth, and Bixler (1963). More recent examples, where the sorption technique is used, are the studies of Toi, Ito, Shirakawa, and Ikemoto (1992), Chan and Odlyha (1995), Higuchi et al. (1996), Shi and Economy (1998), and Perez, Collazo, Izquierdo, Merino, and Novoa (1999). In the present article, a sorption method in a %nite bath together with a numerical technique, with which the di"usion coe#cient and the sorption isotherm of water in paint %lms can be measured simultaneously, is proposed. Advantage of the use of a numerical model is that the technique does not depend on the availability of an analytical solution of the sorption process. Thus, the method can also be used for the situation that the di"usion coe#cient depends strongly on the concentration. As will be demonstrated in the article, other e"ects such as extra sources or sinks can be incorporated into the model. Also, more re%nements, such as non-Fickian di"usion, can be made in the model. The sorption method uses a closed vessel, in which one or more paint %lms on a substrate are subjected to an
0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0009-2509(02)00674-7
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atmosphere with which they are not in equilibrium. The result is a mass Kow in or out of the paint. Thus, the water concentration in the air phase in the vessel changes as a function of time until an equilibrium is reached. This concentration as a function of time is measured and compared with the theoretical/numerical model, of which the %tting parameters (di"usion coe#cient and sorption isotherm) are determined. Compared to other methods, this method is cheap and fast, and can be used for other materials as well. It is also an advantage that not a free paint %lm has to be produced. Detaching a paint %lm from a substrate could inKuence the moisture transfer properties. A disadvantage is that some computations have to be done. In Section 2, the set-up and the technique are described, including the mathematical equations. In Section 3, the implicit Crank–Nicholson method for solving the equations is described. The paint and the production of the paint %lms are described in Section 4. Sections 5 and 6 contain, respectively, the results and the discussion. 2. Measuring the diusion coecient and the sorption isotherm 2.1. The set-up A set-up is constructed inside a stainless-steel vessel as is given in Fig. 1. The cover of the vessel closes the vessel with the help of a rubber ring. Six stainless-steel plates are placed inside the vessel on a small support. The plates are on both sides partially covered with a layer of the paint that has to be examined. The vessel contains also a stand alone humidity sensor (Escort Junior temperature and humidity logger from the company Escort Data Logging Systems LTD). This programmable sensor has a battery and stores data in an internal memory. For mixing inside the vessel, an externally driven stirrer is present. The vessel has an inlet and an outlet for, respectively, blowing air into the vessel and letting air escape from the vessel.
2.2. Measuring procedure The vessel and the paint are left alone long enough to reach an equilibrium situation inside the vessel. An experiment then starts with blowing air into the vessel with a different humidity than the humidity of the air in the vessel. This is done with simultaneously opening the valves of the inlet and outlet of the vessel and blowing air in. After some time, the airKow is stopped and the valves are both closed. The situation inside the vessel now moves to a new equilibrium situation. From the air humidity inside the vessel as a function of time, the di"usion coe#cient and the sorption isotherm of water in the paint are calculated as explained further below. It appears that an extra source/sink is present. Even without plates with paint, the air humidity is not constant inside the closed vessel after stopping the blowing in of air. This change in humidity increases dramatically when a few extra rubber rings are added to the vessel. Clearly, the rubber ring absorbs/releases water during an experiment. Also other small leakages or sources can be present, such as moisture coming from the humidity sensor or its battery. This source/sink e"ect is measured. In the computations, from which the di"usion coe#cient and the sorption isotherm are calculated, this e"ect can be accounted for. 2.3. Governing equations 2.3.1. Paint 7lm The water mass Kux, n, inside a paint %lm is described with the gradient of the water concentration, C, in a one-dimensional situation as n = −D
@C ; @x
(1)
where D is the di"usion coe#cient and x the position coordinate. It is assumed that the behaviour of the paint %lm is described accurately enough with the equations for a non-shrinking system. In this case, Eq. (1) leads to @C @C @ D ; (2) = @t @x @x where t is time. The paint %lm has a thickness H . At position x = H , the paint %lm is attached to a plate. There can be no Kux there, which leads to the boundary condition @C =0 @x
at
x = H:
(3)
At the other side, where x = 0, the paint %lm is supposed to be in equilibrium with the perfectly mixed air. At this position, the concentration in the paint %lm depends on the air humidity in the vessel.
Fig. 1. Inside a closed stainless-steel vessel, paint %lms can adsorb/release water from/to the air phase.
2.3.2. Air humidity in the vessel At the interface between the air and the paint %lm, a water Kux, nx=0 , leaves the air phase with volume V . It is assumed
A. J. J. van der Zanden, E. L. J. Goossens / Chemical Engineering Science 58 (2003) 1521 – 1530
that there is a uniform water concentration, Cair , in the air in the vessel. Making a mass balance over the air phase in the case there is no inKow of dry or humid air, leads to − nx=0 A dt = V dCair ;
(4)
where A is the surface area of the paint and Cair the water concentration in the air. The maximum value of this concentration, the value at saturation, is computed with Antoine’s law and the ideal gas law as 3816:44 M Csat = exp 23:19695 − ; (5) RT −46:13 + T where M is the molar mass of water, R the gas constant, and T the temperature in Kelvin. In the experiments, the relative humidity, h, of the air is measured, and is expressed as a percentage taking a value between 0% and 100%. It is related to the concentration in the air as hCsat Cair = : 100
(6)
follows from a balance, using Eqs. (7) and (8), to be dh −nx=0 A × 100 + k(hs − h) + (hin − h) : = dt VCsat V
h = Cx=0 + ;
where and are parameters that can depend on the concentration. Linear equation (11) makes it possible to use a matrix solution technique to solve the equations. It is not assumed that the sorption isotherm is linear. Only the description for small intervals is a linear relation. Using Eq. (11) in Eq. (10), and with Eq. (1) for the Kux at x = 0, results
dCx=0 D (@C=@x)|x=0 A × 100 = dt VCsat
+(hin − Cx=0 − )
(7)
(8)
To get an idea of the magnitude of the parameters hs and k, an experiment is done with only the humidity sensor inside the vessel. This gives the relative humidity as a function of time. The solution of Eq. (8), with the initial condition of the relative humidity h0 at time t = 0, being h = hs − (hs − h0 ) exp(−kt);
(11)
+k(hs − Cx=0 − )
2.3.3. Source/sink e9ect As discussed in Section 2.2, an extra source/sink is present in the form of water coming from or being absorbed by the rubber ring. Now is described how this e"ect can be ‘subtracted’ intelligently from the measured relative humidities. The source/sink is assumed to be a unit having a constant relative humidity, hs . The driving force for transport between the source/sink and the vessel is assumed to be proportional with the di"erence in relative humidity between the source/sink and the vessel. When there is no paint inside the vessel, the change in relative humidity in the vessel is described with the proportionality constant, k, as dh = k(hs − h): dt
(10)
This relative humidity is related to the concentration in the paint %lm at its surface by the sorption isotherm. Here, a locally linear relation is assumed between the air humidity and the concentration at the %lm surface, as
Substitution of Eq. (6) into Eq. (4) leads to dh −nx=0 A × 100 = : dt VCsat
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(9)
is %tted against the experimental values, resulting in a value for hs and k. Also a value for h0 is obtained. This value (which may be negative or larger than 100%) is of no interest. 2.3.4. Boundary condition between air and paint 7lm When blowing air into the vessel with relative humidity hin and volumetric Kow , the relative humidity in the vessel
: V
(12)
Solving di"erential equation (2) with boundary conditions (3) and (12) gives implicitly the humidity in the vessel.
3. Numerical solution technique 3.1. Solving the di9erential equation In the beginning of an experiment, there may be fast changes in the paint %lm at the interface with the air phase. Thus, it is wise to have a grid for the numerical calculations that is %ne at the interface and coarser further into the paint %lm. Following van der Zanden (1998), such a grid is obtained with the coordinate transformation towards the dimensionless coordinate x=H
a − 1 ; aN − 1
(13)
where a is the factor one mesh larger than its smaller neighbour mesh, and N the number of grid points minus 1. In this study, for a and N is used, respectively, 1.06 and 30. The grid points have the values 0; 1; : : : ; N . Using this coordinate, Eq. (2) becomes 2 N @C @D @C a −1 = @t H ln a a @ @ N 2 2 a −1 @ C @C +D : (14) − ln a H ln a a @2 @ Because the grid is equidistant with interval 1, the %rst and the second derivative of C are easily approximated
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assuming a parabola through three grid points which results, respectively, in 1 1 @C = C+1 − C−1 @ 2 2
+C; t+Rt −2D; t+Rt
(15)
+C+1; t+Rt
+D; t+Rt − D; t+Rt (16) 2C; t =− Rt
where C is short for C(). Likewise for the %rst derivative of D. Again following van der Zanden (1998), Eq. (14) is developed in time according to the Crank–Nicholson procedure as
×
aN − 1 H ln a a
= ≈
1 − (D+1; t+Rt − D−1; t+Rt ) 4
+D; t+Rt + D; t+Rt
ln a 2
:
(18)
(19)
D(aN − 1) @C H ln a a @ =0
D0; t+Rt (aN − 1) H ln a 3 1 × − C0; t+Rt + 2C1; t+Rt − C2; t+Rt ; 2 2
(20)
where the last approximation results from the assumption of a parabola through the %rst three grid points, boundary condition equation (12) becomes, according to the Crank– Nicholson procedure,
(17)
where D+1; t+Rt , for instance, is short for D(C( + 1; t + Rt)), and Rt the time step in the computations. After rearranging the terms of Eq. (17), follows C−1; t+Rt
C−1; t − 2C; t + C+1; t
@C D @x x=0
C−1; t+Rt − 2C; t+Rt + C+1; t+Rt
1 − (D+1; t − D−1; t ) 4
With
2
;
2
CN; t+Rt − CN −1; t+Rt = 0:
ln a − (C+1; t+Rt − C−1; t+Rt ) 2
2
×(C+1; t − C−1; t ) − D; t
1 × (C+1; t+Rt − C−1; t+Rt ) 2 N 2 a −1 1 C−1; t − 2C; t + C+1; t + D; t 2 H ln a a
H ln a a aN − 1
Boundary condition equation (3) becomes
1 × (C+1; t − C−1; t ) 2 N 2 a −1 1 1 + (D+1; t+Rt − D−1; t+Rt ) 2 H ln a a 2
1 + D; t+Rt 2
H ln a a aN − 1
ln a 2
ln a − (C+1; t − C−1; t ) 2
C; t+Rt − C; t Rt N 2 a −1 1 1 = (D+1; t − D−1; t ) 2 H ln a a 2
ln a − (C+1; t − C−1; t ) 2
1 (D+1; t+Rt − D−1; t+Rt ) 4
and @2 C = C−1 − 2C + C+1 ; @2
2 − Rt
C0; t+Rt − C0; t Rt =
1 D0; t+Rt (aN − 1) 2 H ln a 3 A × 100 1 × − C0; t+Rt + 2C1; t+Rt − C2; t+Rt 2 2 VCsat 3 1 D0; t (aN − 1) 1 − C0; t + 2C1; t − C2; t + 2 H ln a 2 2 A × 100 kC0; t+Rt kC0; t − + k(hs − ) − VCsat 2 2 C0; t C0; t+Rt + ; +(hin − ) − V V 2 2
×
(21)
A. J. J. van der Zanden, E. L. J. Goossens / Chemical Engineering Science 58 (2003) 1521 – 1530
which, after rearranging the terms, gives 3 D0; t+Rt (aN − 1) A × 100 C0; t+Rt − 4 H ln a VCsat k − − − 2 Rt 2V +C1; t+Rt
to change, is supposed to be taken at time t=tlag . The relative humidity as predicted by the model is compared with the relative humidity as it is measured. The values of hs , k, , tlag and D are obtained by minimizing the sum of squares of the errors (SSE) SSE = (hmodel; i − hmeasured; i )2 ; (23)
D0; t+Rt (aN − 1)A × 100 H ln aVCsat
1 D0; t+Rt (aN − 1) A × 100 +C2; t+Rt − 4 H ln a VCsat 1 D0; t (aN − 1)A × 100 =− 2 H ln aVCsat 3 1 × − C0; t + 2C1; t − C2; t 2 2 −k(hs − ) +
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kC0; t 2
C0; t C0; t − (hin − ) + : (22) Rt V V 2 In making a time step, the values for the di"usion coe#cient at time t + Rt are not known, because the concentrations are not known a priori. The new values of the di"usion coe#cient could be approximated with the corresponding values at time t. In this study, however, a constant di"usion coe#cient is assumed in an experiment. Di"erential equation (18) and boundary conditions (19) and (22) are written in matrix form and solved numerically with the Gauss-elimination technique, as described for instance by Kreyszig (1988). −
3.2. Fitting procedure The sensor that records the relative humidities in the vessel is a stand alone device that samples the relative humidity with time intervals of 60 s. Thus, from the measured relative humidities it is not clear at what time exactly an experiment started. The %rst point of measurement, which is the %rst point where the relative humidity in the vessel has started
i
where i is the ith point of measurement. 4. Paint lms Two water-borne wall paints are used in this study. The %rst one is an alkyd emulsion wall paint with a pigment volume concentration of around 35.4%. The second one is a styrene acrylic dispersion wall paint (a latex paint) with a pigment volume concentration of around 39.8%. The composition of the paints are given in Tables 1 and 2. For creating a paint layer with a uniform thickness, a rod was scraped at a constant distance from the stainless-steel plates over wet paint with a constant velocity. The total surface area of the alkyd paint on the six steel plates is 0:1082 ± 0:0005 m2 , and of the latex paint 0:1080 ± 0:0005 m2 . The thickness of the paint %lms is calculated from the measured height of eight steel plates plus 12 paint %lms and the measured height of eight unpainted steel plates. The alkyd paint has a %lm thickness of 0:137 ± 0:010 mm. The latex paint has a %lm thickness of 0:116 ± 0:006 mm. 5. Results The set-up is placed in a room with a constant temperature of 23◦ C. At time t = 0, the Kushing of the vessel (with volume 5:12 × 10−3 m3 ), which is then in an equilibrium situation with humidity h0 , is started with a Kow rate of 8:33 × 10−5 m3 s−1 (5 l min−1 ). Every 60 s, the relative humidity of the air in the vessel is sampled. The Kushing is stopped at time tf . As an illustration, Fig. 2 gives the humidity as a function of time as measured in experiment 3 (symbols) (where the numbering is explained below) and as the result of the best %t. The agreement is good. The
Table 1 Composition of the alkyd emulsion wall paint Ingredient
Function
Manufacturer
Weight percentage
Water Disperbyk 190 Acticide BX Byk 024 Acrysol RM-8 Mikhart 5 Tioxide TR92 Uradil AZ554 Z50 Durham VX71
Solvent Wetting and dispersing additive In-can biocide Silicone defoamer Associative thickener Filler (chalk) Pigment (TiO2 ) Binder (alkyd emulsion) Drier
God BYK-Chemie Thor Chemicals BYK-Chemie Rohm & Haas Provencale Tioxide DSM Resins BV Harcros Chemical Group
16.09 2.06 0.15 0.09 2.40 15.21 23.96 38.13 1.91
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Table 2 Composition of the styrene acrylic dispersion wall paint
Ingredient
Function
Manufacturer
Weight percentage
Water SER-AD FX 504 Acticide MX Acrysol RM-8 Tegofoamex 1488 Tioxide TR92 Mikhart 5 Texanol Acronal 290D
Solvent Pigment dispersing agent In-can biocide Associative thickener Defoamer Pigment (TiO2 ) Filler (chalk) Coalescent Binder (styrene acrylic polymer)
God Servo Delden BV Thor Chemicals Rohm & Haas Tego Chemie Service Tioxide Provencale Eastman BASF
33.93 0.30 0.15 2.00 0.10 20.02 12.71 0.75 30.03
Fig. 2. Humidity as a function of time as measured in one of the experiments (symbols) and as the result of the best %t (solid line).
In Fig. 4, the di"usion coe#cient of water in the alkyd paint is given as a graph. In Fig. 5, the is given. The horizontal lines represent the relative humidities that would be in equilibrium with the concentrations that occurred in the paint %lm during one experiment. They also represent the relative humidity of the air during one experiment. The experiments are performed far from the saturation point to be sure not to have any small droplets in the set-up. In Figs. 6 and 7 are given, respectively, the di"usion coe#cient and for the latex paint. The dotted line in Fig. 7 will be explained below. It seems that there is a large scattering of results in all graphs. Also, the values for k (which are not given) have a large scattering. After doing four additional experiments with the alkyd paint, where the number of data points is increased to around 360 with the same sample time, it must be concluded that this does not bring any improvement. These experiments only con%rm the earlier experiments. Also, a teKon ring is tried between the vessel and its cover instead of the rubber ring. This brings no reduction of the source/sink term. In other research (van der Zanden & Goossens, 2003), the sorption isotherm of the latex paint was found to be described well with the relation C=
Fig. 3. The concentration pro%les in the paint in one of the experiments for t = 500, 1000, 1500, 2000, and 2500 s as computed in the best %t.
jumps in the experimentally obtained relative humidity are caused by the resolution of the sensor. The concentration pro%les in the paint are, for the same experiment, given in Fig. 3. In Table 3, the parameters of the experiments and the results of the %ts are given for the alkyd paint. The %rst column gives the experiment number. In every experiment, approximately 250 data points are taken. For the latex paint, the parameters of the experiments and the results of the %ts are given in Table 4.
98:2h2 + 16601h : h2 − 2359:7h + 239838
(24)
Using this sorption isotherm (and derive from it the (which is the dotted line in Fig. 7) and as a function of moisture content) as input in the computations, the experiments give the di"usion coe#cient of water for the latex paint as given in Fig. 8. When humid air is used to create a non-equilibrium situation in the vessel, the humidity of the Kushing air is used as a %tting parameter. In four of these experiments, k comes out to have a negative value. Because this is physically impossible, in a such case the value of k is put zero. The scattering in the results remains. One reason can be that the di"usion coe#cient is a function of the concentration. The horizontal lines in Fig. 8 have a large humidity range. It is possible that the di"usion coe#cient is not constant in one experiment. To verify if the last suggestion is correct, small concentration di"erences are used in a few experiments. To obtain this, the time of Kushing
A. J. J. van der Zanden, E. L. J. Goossens / Chemical Engineering Science 58 (2003) 1521 – 1530
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Table 3 Parameters of the experiments and results of the %ts for the alkyd paint
Exp.
T (K)
h0 (%)
hin (%)
tf (s)
D (m2 s−1 )
(% kg−1 m3 )
SSE (%2 )
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
296.7 295.6 295.6 295.6 295.7 295.9 295.9 295.7 295.8 295.9 295.9 295.7 295.6 295.8 295.7 295.6 296.1 295.9 296.0
50.5 47.5 37.0 28.0 23.0 49.0 58.0 63.0 56.5 62.5 53.5 52.0 56.5 53.0 63.0 65.5 53.0 42.5 34.0
0 0 0 0 0 79 79 79 79 79 0 79 0 82 84 0 0 0 78
300 300 300 300 300 300 300 300 300 300 60 120 60 300 300 300 300 300 300
0.77E-12 1.00E-12 1.25E-12 1.49E-12 1.23E-12 2.10E-12 2.51E-12 1.42E-12 1.83E-12 1.77E-12 0.81E-12 2.66E-12 1.91E-12 2.26E-12 1.29E-12 0.80E-12 1.18E-12 1.34E-12 2.70E-12
2.85 2.96 4.02 5.29 5.18 6.27 5.58 3.36 3.55 3.29 1.38 4.02 1.93 8.25 6.05 1.08 2.00 3.09 10.24
14.0 21.0 17.8 20.1 13.0 40.2 30.4 10.8 28.7 21.8 24.8 48.4 41.3 28.9 17.4 98.1 44.7 26.8 42.4
Table 4 Parameters of the experiments and results of the %ts for the latex paint
Exp.
T (K)
h0 (%)
hin (%)
tf (s)
D (m2 s−1 )
(% kg−1 m3 )
SSE (%2 )
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
295.7 295.9 295.9 295.6 295.7 295.8 295.7 295.7 295.6 295.8 295.9 296.1 296.0 296.1 295.9 295.9 295.8
52.5 35.5 25.5 49.0 31.5 19.0 17.0 10.0 7.5 9.5 52.0 66.0 72.0 76.5 54.0 57.5 50.5
0 0 0 0 0 0 0 0 0 0 80 83 85 87 74.5 11.5 74.5
300 300 300 300 300 300 300 300 300 300 300 300 300 300 60 60 60
3.13E-12 3.41E-12 3.32E-12 3.12E-12 3.30E-12 3.70E-12 3.21E-12 2.44E-12 2.39E-12 2.42E-12 3.18E-12 2.37E-12 2.00E-12 1.92E-12 2.69E-12 2.56E-12 3.83E-12
4.21 7.08 8.85 4.75 7.79 9.96 10.64 12.29 14.91 12.90 6.41 4.77 3.71 3.19 6.12 3.22 6.43
19.6 17.7 9.6 20.5 12.5 11.4 7.8 9.4 9.4 9.4 14.6 14.1 10.1 8.9 10.1 4.9 5.9
is reduced to 180 s and the Kushing Kow rate is reduced to 1:667 × 10−5 m3 s−1 (1 l min−1 ). To be certain of the humidity of the Kushing air, the completely dry air, as supplied by the university pressure net, is used without adding water vapour. The result of these experiments are given in Fig. 9. There is still a large scattering. Also, the value of k shows very large di"erences. Even values for k are obtained which are 10 times as large as the values that are measured in the beginning of the experiments. The %tting of k (and also hs ) does more harm than good. Without %tting these values, and ignoring the source/sink e"ect (putting k =0), the last experiments give the results as given in Fig. 10. The results do not link up with each other.
The results are grouped in a few ‘island’. The experiments of island 1 are done by %rst drying the paint completely and then adding some water to arrive at the desired air humidity. The experiments of island 2 are done after the paint has been in equilibrium with the climate room having a humidity of approximately 50%. The experiments of island 3 are done after the paint reached a high moisture content after a weekend in a climate room with high humidity. From these experiments, it seems that not only the humidity in the paint determines the di"usion coe#cient but also the history of the paint. To verify if the velocity of stirring is a parameter that inKuences the results, the experiments of Fig. 10 are repeated
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Fig. 4. The measured di"usion coe#cient of water in the alkyd paint as a function of the equilibrium humidity in air.
Fig. 5. The measured for the alkyd paint as a function of the equilibrium humidity in air.
Fig. 6. The measured di"usion coe#cient of water in the latex paint as a function of the equilibrium humidity in air.
with two di"erent stirring velocities. Also, the blades of the stirrer have been put in a 45◦ angle with the stirrer axis to increase the mixing. The results are given in Fig. 11. The experiments from the solid lines have a stirring velocity that is 3 times as large as the experiments from the dotted lines. The stirring velocity does not have an inKuence on the re-
Fig. 7. The measured for the latex paint as a function of the equilibrium humidity in air. The dotted line represents the values that come out of an independent study as will be explained below.
Fig. 8. The measured di"usion coe#cient of water in the latex paint as a function of the equilibrium humidity in air, where the sorption isotherm is used as input.
Fig. 9. The measured di"usion coe#cient of water in the latex paint for a short time of Kushing with a low Kow rate.
sults. The resistance against mass transfer is in the paint. The resistance against mass transfer in the air phase may be neglected, as is done in the model. The results agree with those from Fig. 10, except now there is not such evidently grouping of the results. Taking k = 1:5 × 10−5 s−1 and
A. J. J. van der Zanden, E. L. J. Goossens / Chemical Engineering Science 58 (2003) 1521 – 1530
Fig. 10. The measured di"usion coe#cient of water in the latex paint without %tting of the source/sink term (k = 0).
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parameters that inKuence the di"usion coe#cient. This is not a#rmed by the results given in Fig. 11. If the situation is more complex than the assumptions underlying the present work, the set-up used here might be not the best way to measure the moisture transfer properties. For instance, there is no guarantee that a solution/di"usion model is the best description of the process. Also, there could be hysteresis in the sorption isotherm. This would not be evident immediately from the results obtained from the method used here, where the water concentration at the surface of the paint goes down and up again in one experiment. If the assumptions underlying the present work are met, the method is suitable to measure di"usion coe#cients. Looking at Fig. 11, a linear relation could be supposed between the di"usion coe#cient and the equilibrium relative humidity as D = h + ;
(25)
where and are constants. The values of and could be obtained not from %tting one experiment but from %tting all experiments together. This has not been done because of the large computational e"ort, and because the result would probably be a non-surprising line in Fig. 11. Notation Fig. 11. The measured di"usion coe#cient of water in the latex paint. The solid and dotted lines are obtained from experiments with respectively high and low stirring velocity.
hs =50%, which are physically realistic values, gives results which are approximately 12% larger for low humidities and 2% smaller for large humidities than the results in Fig. 11. 6. Discussion A method has been used to measure the di"usion coe#cient and the sorption isotherm of water in paint %lms. The set-up is quite simple and cheap. An additional advantage of this method, compared to those found in the literature, where the %nite bath method is used, is that the changes in the relative humidity of the air are quite slow. Thus, the sensor, which has a response time of approximately 15 s can quite easily follow the changes in the relative air humidity. The di"usion coe#cient and the sorption isotherm are obtained from the best %t of the model against the experimental values. This gives results with large scattering. When the sorption isotherm is determined independently, the %ts give better results for the di"usion coe#cient. The inKuence of the rubber ring is not so large, but could be avoided by using another way to close the vessel. The model supposes the diffusion coe#cient to be a constant in one experiment. More experiments show that the di"usion coe#cient is a function of the water concentration in the paint. The results given in Fig. 10 seem to point out that there might be also other
a A C D h hin H i k M n N R SSE t tf tlag T V x
grid tuning factor, dimensionless surface area of the paint %lm, m2 concentration, kg m−3 di"usion coe#cient, m2 s−1 relative humidity, percentage relative humidity of the inKowing air, percentage thickness of the paint %lm, m number of measured point, dimensionless proportionality constant, s−1 molar mass of water, kg mol−1 mass Kux, kg m−2 s−1 number of grid points minus 1, dimensionless gas constant, J mol−1 K −1 sum of squares of the errors, percentage2 time, s time of Kushing, s time between the start of an experiment and the %rst point of measurement, s temperature, K volume of the air inside the vessel, m3 position coordinate, m
Greek letters
locally linear sorption isotherm parameter, percentage kg−1 m3 locally linear sorption isotherm parameter, percentage
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slope of di"usion coe#cient versus relative humidity, m2 s−1 percentage−1 constant in function between di"usion coe#cient and relative humidity, m2 s−1 dimensionless space coordinate, dimensionless volumetric Kow of air through the vessel, m3 s−1
Subscripts air s sat x=0 0
air source/sink saturation at x = 0 at time t = 0
Acknowledgements The authors are grateful to Roalt Bruininks and Guus Theuws for the fast technical support. They are also grateful to Michel van der Pal and Wim van der Spoel for lending the stainless-steel vessel. One of the authors (E.G.) is, thanks to prof. E.L.J. Bancken, partly supported by ‘NWO— Technologiestichting STW’ Project: DCT.4010 Subproject IV-b ‘Water balance of water-borne paint systems on plaster substrates in relation to fungal growth’. DSM Resins is thanked for making the alkyd paint available. References Blahn>?k, R. (1983). Problems of measuring water sorption in organic coatings and %lms, and calculations of complicated instances of moistening. Progress in Organic Coatings, 11, 353–392.
Chan, T. Y. A., & Odlyha, M. (1995). The e"ect of relative humidity and pigment type on paint %lms. Thermochimica Acta, 270, 755–767. Crank, J., & Park, G. S. (1968). Methods of measurement. In J. Crank, & G. S. Park (Eds.), Di9usion in polymers (pp. 1–39). London: Academic Press (Chapter 1). Higuchi, A., Nakajima, T., Morisato, A., Ando, M., Nagai, K., & Nakagawa, T. (1996). Estimation of di"usion and permeability coe#cients of CO2 in polymeric membranes by FTIR method. Journal of Polymer Science B, 34, 2153–2160. Huld>en, M., & Hansen, C. M. (1985). Water permeation in coatings. Progress in Organic Coatings, 13, 171–194. Kreyszig, E. (1988). Advanced Engineering Mathematics (6th ed.) (pp. 372–378). New York: Wiley. Lundberg, J. L., Wilk, M. B., & Huyett, M. J. (1962). Estimation of di"usivities and solubilities from sorption studies. Journal of Polymer Science, 57, 275–299. Michaels, A. S., Vieth, W. R., & Bixler, H. J. (1963). The measurement of di"usion constants of gases in polymers by sorption techniques. Journal of Polymer Science B, 1, 19–23. Perez, C., Collazo, A., Izquierdo, M., Merino, P., & Novoa, X. R. (1999). Characterisation of the barrier properties of di"erent paint systems. Part I: Experimental set-up and ideal Fickian di"usion. Progress in Organic Coatings, 36, 102–108. Shi, F. F., & Economy, J. (1998). Moisture transport studies of newly developed aromatic and aromatic/aliphatic copolyester thin %lms. Journal of Polymer Science B, 36, 1025–1035. Toi, K., Ito, T., Shirakawa, T., & Ikemoto, I. (1992). Sorption and di"usion of gases in a polyimide. Journal of Polymer Science B, 30, 549–556. van der Wel, G. K., & Adan, O. C. G. (1999). Moisture in organic coatings—a review. Progress in Organic Coatings, 37, 1–14. van der Zanden, A. J. J. (1998). An iterative procedure to obtain the concentration dependency of the di"usion coe#cient from the space-averaged concentration vs. time. Chemical Engineering Science, 53, 1397–1404. van der Zanden, A. J. J., & Goossens, E. L. J. (2003). The measurement of the sorption isotherm of water in paint %lms, submitted to Chemical Engineering and Processing.
www.elsevier.com/locate/ces
The measurement of the di"usion coe#cient and the sorption isotherm of water in paint %lms A. J. J. van der Zanden∗ , E. L. J. Goossens Faculty of Architecture, Building and Planning, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands Received 2 April 2002; received in revised form 3 September 2002; accepted 1 October 2002
Abstract An experimental technique together with a numerical model is proposed with which the di"usion coe#cient and the sorption isotherm of water in paint can be measured. Inside a closed vessel, paint %lms are on stainless-steel plates. Water is present as water vapour in the air and in the paint. After blowing dry or wet air through the vessel for some time, the situation moves to a new equilibrium. The relative humidity of the air inside the vessel is measured as a function of time. From %tting the theoretical/numerical model against the experimental values, follow the di"usion coe#cient and the sorption isotherm of water in the paint. The results show large scattering. When the independently measured sorption isotherm is used as an input parameter in the model, the %tting procedure gives much smaller scattering for the di"usion coe#cient. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Di"usion coe#cient; Sorption isotherm; Paint; Drying; Mass transfer
1. Introduction Besides for their decorative function, paint %lms are also used to change the moisture transfer properties of materials. Thus, it is important to know the moisture transfer properties of the paint %lm itself. For an overview on measurement methods of di"usion in polymers (paint %lms most often contain polymers), see for instance Crank and Park (1968). For reviews on moisture transport in paint, see Blahn>?k (1983), Huld>en and Hansen (1985), and van der Wel and Adan (1999). Paint %lm properties are usually examined by making use of a free %lm or foil, which is not attached to a substrate. The mass transfer through such a %lm can be determined in various ways. Most often, this is done by measuring the mass increase or decrease of a cup %lled with water that is covered with the %lm. This technique is known as the cup method. More advanced techniques for measuring the change in the quantity of water are available, such as a chemical analysis or gas chromatography. Another type of technique uses a sorption–desorption method. In such a method,
∗ Corresponding author. Tel.: +31-40-247-37-21; fax: +31-40-243-85-95. E-mail address: [email protected] (A. J. J. van der Zanden).
the quantity of moisture in a sample subjected to an atmosphere, with which it is not in equilibrium, is registered as a function of time, most often by measuring the weight of the sample. The sorption technique is already an old technique for measuring di"usion and sorption in polymers, see for instance the articles of Lundberg, Wilk, and Huyett (1962) and Michaels, Vieth, and Bixler (1963). More recent examples, where the sorption technique is used, are the studies of Toi, Ito, Shirakawa, and Ikemoto (1992), Chan and Odlyha (1995), Higuchi et al. (1996), Shi and Economy (1998), and Perez, Collazo, Izquierdo, Merino, and Novoa (1999). In the present article, a sorption method in a %nite bath together with a numerical technique, with which the di"usion coe#cient and the sorption isotherm of water in paint %lms can be measured simultaneously, is proposed. Advantage of the use of a numerical model is that the technique does not depend on the availability of an analytical solution of the sorption process. Thus, the method can also be used for the situation that the di"usion coe#cient depends strongly on the concentration. As will be demonstrated in the article, other e"ects such as extra sources or sinks can be incorporated into the model. Also, more re%nements, such as non-Fickian di"usion, can be made in the model. The sorption method uses a closed vessel, in which one or more paint %lms on a substrate are subjected to an
0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0009-2509(02)00674-7
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atmosphere with which they are not in equilibrium. The result is a mass Kow in or out of the paint. Thus, the water concentration in the air phase in the vessel changes as a function of time until an equilibrium is reached. This concentration as a function of time is measured and compared with the theoretical/numerical model, of which the %tting parameters (di"usion coe#cient and sorption isotherm) are determined. Compared to other methods, this method is cheap and fast, and can be used for other materials as well. It is also an advantage that not a free paint %lm has to be produced. Detaching a paint %lm from a substrate could inKuence the moisture transfer properties. A disadvantage is that some computations have to be done. In Section 2, the set-up and the technique are described, including the mathematical equations. In Section 3, the implicit Crank–Nicholson method for solving the equations is described. The paint and the production of the paint %lms are described in Section 4. Sections 5 and 6 contain, respectively, the results and the discussion. 2. Measuring the diusion coecient and the sorption isotherm 2.1. The set-up A set-up is constructed inside a stainless-steel vessel as is given in Fig. 1. The cover of the vessel closes the vessel with the help of a rubber ring. Six stainless-steel plates are placed inside the vessel on a small support. The plates are on both sides partially covered with a layer of the paint that has to be examined. The vessel contains also a stand alone humidity sensor (Escort Junior temperature and humidity logger from the company Escort Data Logging Systems LTD). This programmable sensor has a battery and stores data in an internal memory. For mixing inside the vessel, an externally driven stirrer is present. The vessel has an inlet and an outlet for, respectively, blowing air into the vessel and letting air escape from the vessel.
2.2. Measuring procedure The vessel and the paint are left alone long enough to reach an equilibrium situation inside the vessel. An experiment then starts with blowing air into the vessel with a different humidity than the humidity of the air in the vessel. This is done with simultaneously opening the valves of the inlet and outlet of the vessel and blowing air in. After some time, the airKow is stopped and the valves are both closed. The situation inside the vessel now moves to a new equilibrium situation. From the air humidity inside the vessel as a function of time, the di"usion coe#cient and the sorption isotherm of water in the paint are calculated as explained further below. It appears that an extra source/sink is present. Even without plates with paint, the air humidity is not constant inside the closed vessel after stopping the blowing in of air. This change in humidity increases dramatically when a few extra rubber rings are added to the vessel. Clearly, the rubber ring absorbs/releases water during an experiment. Also other small leakages or sources can be present, such as moisture coming from the humidity sensor or its battery. This source/sink e"ect is measured. In the computations, from which the di"usion coe#cient and the sorption isotherm are calculated, this e"ect can be accounted for. 2.3. Governing equations 2.3.1. Paint 7lm The water mass Kux, n, inside a paint %lm is described with the gradient of the water concentration, C, in a one-dimensional situation as n = −D
@C ; @x
(1)
where D is the di"usion coe#cient and x the position coordinate. It is assumed that the behaviour of the paint %lm is described accurately enough with the equations for a non-shrinking system. In this case, Eq. (1) leads to @C @C @ D ; (2) = @t @x @x where t is time. The paint %lm has a thickness H . At position x = H , the paint %lm is attached to a plate. There can be no Kux there, which leads to the boundary condition @C =0 @x
at
x = H:
(3)
At the other side, where x = 0, the paint %lm is supposed to be in equilibrium with the perfectly mixed air. At this position, the concentration in the paint %lm depends on the air humidity in the vessel.
Fig. 1. Inside a closed stainless-steel vessel, paint %lms can adsorb/release water from/to the air phase.
2.3.2. Air humidity in the vessel At the interface between the air and the paint %lm, a water Kux, nx=0 , leaves the air phase with volume V . It is assumed
A. J. J. van der Zanden, E. L. J. Goossens / Chemical Engineering Science 58 (2003) 1521 – 1530
that there is a uniform water concentration, Cair , in the air in the vessel. Making a mass balance over the air phase in the case there is no inKow of dry or humid air, leads to − nx=0 A dt = V dCair ;
(4)
where A is the surface area of the paint and Cair the water concentration in the air. The maximum value of this concentration, the value at saturation, is computed with Antoine’s law and the ideal gas law as 3816:44 M Csat = exp 23:19695 − ; (5) RT −46:13 + T where M is the molar mass of water, R the gas constant, and T the temperature in Kelvin. In the experiments, the relative humidity, h, of the air is measured, and is expressed as a percentage taking a value between 0% and 100%. It is related to the concentration in the air as hCsat Cair = : 100
(6)
follows from a balance, using Eqs. (7) and (8), to be dh −nx=0 A × 100 + k(hs − h) + (hin − h) : = dt VCsat V
h = Cx=0 + ;
where and are parameters that can depend on the concentration. Linear equation (11) makes it possible to use a matrix solution technique to solve the equations. It is not assumed that the sorption isotherm is linear. Only the description for small intervals is a linear relation. Using Eq. (11) in Eq. (10), and with Eq. (1) for the Kux at x = 0, results
dCx=0 D (@C=@x)|x=0 A × 100 = dt VCsat
+(hin − Cx=0 − )
(7)
(8)
To get an idea of the magnitude of the parameters hs and k, an experiment is done with only the humidity sensor inside the vessel. This gives the relative humidity as a function of time. The solution of Eq. (8), with the initial condition of the relative humidity h0 at time t = 0, being h = hs − (hs − h0 ) exp(−kt);
(11)
+k(hs − Cx=0 − )
2.3.3. Source/sink e9ect As discussed in Section 2.2, an extra source/sink is present in the form of water coming from or being absorbed by the rubber ring. Now is described how this e"ect can be ‘subtracted’ intelligently from the measured relative humidities. The source/sink is assumed to be a unit having a constant relative humidity, hs . The driving force for transport between the source/sink and the vessel is assumed to be proportional with the di"erence in relative humidity between the source/sink and the vessel. When there is no paint inside the vessel, the change in relative humidity in the vessel is described with the proportionality constant, k, as dh = k(hs − h): dt
(10)
This relative humidity is related to the concentration in the paint %lm at its surface by the sorption isotherm. Here, a locally linear relation is assumed between the air humidity and the concentration at the %lm surface, as
Substitution of Eq. (6) into Eq. (4) leads to dh −nx=0 A × 100 = : dt VCsat
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(9)
is %tted against the experimental values, resulting in a value for hs and k. Also a value for h0 is obtained. This value (which may be negative or larger than 100%) is of no interest. 2.3.4. Boundary condition between air and paint 7lm When blowing air into the vessel with relative humidity hin and volumetric Kow , the relative humidity in the vessel
: V
(12)
Solving di"erential equation (2) with boundary conditions (3) and (12) gives implicitly the humidity in the vessel.
3. Numerical solution technique 3.1. Solving the di9erential equation In the beginning of an experiment, there may be fast changes in the paint %lm at the interface with the air phase. Thus, it is wise to have a grid for the numerical calculations that is %ne at the interface and coarser further into the paint %lm. Following van der Zanden (1998), such a grid is obtained with the coordinate transformation towards the dimensionless coordinate x=H
a − 1 ; aN − 1
(13)
where a is the factor one mesh larger than its smaller neighbour mesh, and N the number of grid points minus 1. In this study, for a and N is used, respectively, 1.06 and 30. The grid points have the values 0; 1; : : : ; N . Using this coordinate, Eq. (2) becomes 2 N @C @D @C a −1 = @t H ln a a @ @ N 2 2 a −1 @ C @C +D : (14) − ln a H ln a a @2 @ Because the grid is equidistant with interval 1, the %rst and the second derivative of C are easily approximated
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assuming a parabola through three grid points which results, respectively, in 1 1 @C = C+1 − C−1 @ 2 2
+C; t+Rt −2D; t+Rt
(15)
+C+1; t+Rt
+D; t+Rt − D; t+Rt (16) 2C; t =− Rt
where C is short for C(). Likewise for the %rst derivative of D. Again following van der Zanden (1998), Eq. (14) is developed in time according to the Crank–Nicholson procedure as
×
aN − 1 H ln a a
= ≈
1 − (D+1; t+Rt − D−1; t+Rt ) 4
+D; t+Rt + D; t+Rt
ln a 2
:
(18)
(19)
D(aN − 1) @C H ln a a @ =0
D0; t+Rt (aN − 1) H ln a 3 1 × − C0; t+Rt + 2C1; t+Rt − C2; t+Rt ; 2 2
(20)
where the last approximation results from the assumption of a parabola through the %rst three grid points, boundary condition equation (12) becomes, according to the Crank– Nicholson procedure,
(17)
where D+1; t+Rt , for instance, is short for D(C( + 1; t + Rt)), and Rt the time step in the computations. After rearranging the terms of Eq. (17), follows C−1; t+Rt
C−1; t − 2C; t + C+1; t
@C D @x x=0
C−1; t+Rt − 2C; t+Rt + C+1; t+Rt
1 − (D+1; t − D−1; t ) 4
With
2
;
2
CN; t+Rt − CN −1; t+Rt = 0:
ln a − (C+1; t+Rt − C−1; t+Rt ) 2
2
×(C+1; t − C−1; t ) − D; t
1 × (C+1; t+Rt − C−1; t+Rt ) 2 N 2 a −1 1 C−1; t − 2C; t + C+1; t + D; t 2 H ln a a
H ln a a aN − 1
Boundary condition equation (3) becomes
1 × (C+1; t − C−1; t ) 2 N 2 a −1 1 1 + (D+1; t+Rt − D−1; t+Rt ) 2 H ln a a 2
1 + D; t+Rt 2
H ln a a aN − 1
ln a 2
ln a − (C+1; t − C−1; t ) 2
C; t+Rt − C; t Rt N 2 a −1 1 1 = (D+1; t − D−1; t ) 2 H ln a a 2
ln a − (C+1; t − C−1; t ) 2
1 (D+1; t+Rt − D−1; t+Rt ) 4
and @2 C = C−1 − 2C + C+1 ; @2
2 − Rt
C0; t+Rt − C0; t Rt =
1 D0; t+Rt (aN − 1) 2 H ln a 3 A × 100 1 × − C0; t+Rt + 2C1; t+Rt − C2; t+Rt 2 2 VCsat 3 1 D0; t (aN − 1) 1 − C0; t + 2C1; t − C2; t + 2 H ln a 2 2 A × 100 kC0; t+Rt kC0; t − + k(hs − ) − VCsat 2 2 C0; t C0; t+Rt + ; +(hin − ) − V V 2 2
×
(21)
A. J. J. van der Zanden, E. L. J. Goossens / Chemical Engineering Science 58 (2003) 1521 – 1530
which, after rearranging the terms, gives 3 D0; t+Rt (aN − 1) A × 100 C0; t+Rt − 4 H ln a VCsat k − − − 2 Rt 2V +C1; t+Rt
to change, is supposed to be taken at time t=tlag . The relative humidity as predicted by the model is compared with the relative humidity as it is measured. The values of hs , k, , tlag and D are obtained by minimizing the sum of squares of the errors (SSE) SSE = (hmodel; i − hmeasured; i )2 ; (23)
D0; t+Rt (aN − 1)A × 100 H ln aVCsat
1 D0; t+Rt (aN − 1) A × 100 +C2; t+Rt − 4 H ln a VCsat 1 D0; t (aN − 1)A × 100 =− 2 H ln aVCsat 3 1 × − C0; t + 2C1; t − C2; t 2 2 −k(hs − ) +
1525
kC0; t 2
C0; t C0; t − (hin − ) + : (22) Rt V V 2 In making a time step, the values for the di"usion coe#cient at time t + Rt are not known, because the concentrations are not known a priori. The new values of the di"usion coe#cient could be approximated with the corresponding values at time t. In this study, however, a constant di"usion coe#cient is assumed in an experiment. Di"erential equation (18) and boundary conditions (19) and (22) are written in matrix form and solved numerically with the Gauss-elimination technique, as described for instance by Kreyszig (1988). −
3.2. Fitting procedure The sensor that records the relative humidities in the vessel is a stand alone device that samples the relative humidity with time intervals of 60 s. Thus, from the measured relative humidities it is not clear at what time exactly an experiment started. The %rst point of measurement, which is the %rst point where the relative humidity in the vessel has started
i
where i is the ith point of measurement. 4. Paint lms Two water-borne wall paints are used in this study. The %rst one is an alkyd emulsion wall paint with a pigment volume concentration of around 35.4%. The second one is a styrene acrylic dispersion wall paint (a latex paint) with a pigment volume concentration of around 39.8%. The composition of the paints are given in Tables 1 and 2. For creating a paint layer with a uniform thickness, a rod was scraped at a constant distance from the stainless-steel plates over wet paint with a constant velocity. The total surface area of the alkyd paint on the six steel plates is 0:1082 ± 0:0005 m2 , and of the latex paint 0:1080 ± 0:0005 m2 . The thickness of the paint %lms is calculated from the measured height of eight steel plates plus 12 paint %lms and the measured height of eight unpainted steel plates. The alkyd paint has a %lm thickness of 0:137 ± 0:010 mm. The latex paint has a %lm thickness of 0:116 ± 0:006 mm. 5. Results The set-up is placed in a room with a constant temperature of 23◦ C. At time t = 0, the Kushing of the vessel (with volume 5:12 × 10−3 m3 ), which is then in an equilibrium situation with humidity h0 , is started with a Kow rate of 8:33 × 10−5 m3 s−1 (5 l min−1 ). Every 60 s, the relative humidity of the air in the vessel is sampled. The Kushing is stopped at time tf . As an illustration, Fig. 2 gives the humidity as a function of time as measured in experiment 3 (symbols) (where the numbering is explained below) and as the result of the best %t. The agreement is good. The
Table 1 Composition of the alkyd emulsion wall paint Ingredient
Function
Manufacturer
Weight percentage
Water Disperbyk 190 Acticide BX Byk 024 Acrysol RM-8 Mikhart 5 Tioxide TR92 Uradil AZ554 Z50 Durham VX71
Solvent Wetting and dispersing additive In-can biocide Silicone defoamer Associative thickener Filler (chalk) Pigment (TiO2 ) Binder (alkyd emulsion) Drier
God BYK-Chemie Thor Chemicals BYK-Chemie Rohm & Haas Provencale Tioxide DSM Resins BV Harcros Chemical Group
16.09 2.06 0.15 0.09 2.40 15.21 23.96 38.13 1.91
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Table 2 Composition of the styrene acrylic dispersion wall paint
Ingredient
Function
Manufacturer
Weight percentage
Water SER-AD FX 504 Acticide MX Acrysol RM-8 Tegofoamex 1488 Tioxide TR92 Mikhart 5 Texanol Acronal 290D
Solvent Pigment dispersing agent In-can biocide Associative thickener Defoamer Pigment (TiO2 ) Filler (chalk) Coalescent Binder (styrene acrylic polymer)
God Servo Delden BV Thor Chemicals Rohm & Haas Tego Chemie Service Tioxide Provencale Eastman BASF
33.93 0.30 0.15 2.00 0.10 20.02 12.71 0.75 30.03
Fig. 2. Humidity as a function of time as measured in one of the experiments (symbols) and as the result of the best %t (solid line).
In Fig. 4, the di"usion coe#cient of water in the alkyd paint is given as a graph. In Fig. 5, the is given. The horizontal lines represent the relative humidities that would be in equilibrium with the concentrations that occurred in the paint %lm during one experiment. They also represent the relative humidity of the air during one experiment. The experiments are performed far from the saturation point to be sure not to have any small droplets in the set-up. In Figs. 6 and 7 are given, respectively, the di"usion coe#cient and for the latex paint. The dotted line in Fig. 7 will be explained below. It seems that there is a large scattering of results in all graphs. Also, the values for k (which are not given) have a large scattering. After doing four additional experiments with the alkyd paint, where the number of data points is increased to around 360 with the same sample time, it must be concluded that this does not bring any improvement. These experiments only con%rm the earlier experiments. Also, a teKon ring is tried between the vessel and its cover instead of the rubber ring. This brings no reduction of the source/sink term. In other research (van der Zanden & Goossens, 2003), the sorption isotherm of the latex paint was found to be described well with the relation C=
Fig. 3. The concentration pro%les in the paint in one of the experiments for t = 500, 1000, 1500, 2000, and 2500 s as computed in the best %t.
jumps in the experimentally obtained relative humidity are caused by the resolution of the sensor. The concentration pro%les in the paint are, for the same experiment, given in Fig. 3. In Table 3, the parameters of the experiments and the results of the %ts are given for the alkyd paint. The %rst column gives the experiment number. In every experiment, approximately 250 data points are taken. For the latex paint, the parameters of the experiments and the results of the %ts are given in Table 4.
98:2h2 + 16601h : h2 − 2359:7h + 239838
(24)
Using this sorption isotherm (and derive from it the (which is the dotted line in Fig. 7) and as a function of moisture content) as input in the computations, the experiments give the di"usion coe#cient of water for the latex paint as given in Fig. 8. When humid air is used to create a non-equilibrium situation in the vessel, the humidity of the Kushing air is used as a %tting parameter. In four of these experiments, k comes out to have a negative value. Because this is physically impossible, in a such case the value of k is put zero. The scattering in the results remains. One reason can be that the di"usion coe#cient is a function of the concentration. The horizontal lines in Fig. 8 have a large humidity range. It is possible that the di"usion coe#cient is not constant in one experiment. To verify if the last suggestion is correct, small concentration di"erences are used in a few experiments. To obtain this, the time of Kushing
A. J. J. van der Zanden, E. L. J. Goossens / Chemical Engineering Science 58 (2003) 1521 – 1530
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Table 3 Parameters of the experiments and results of the %ts for the alkyd paint
Exp.
T (K)
h0 (%)
hin (%)
tf (s)
D (m2 s−1 )
(% kg−1 m3 )
SSE (%2 )
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
296.7 295.6 295.6 295.6 295.7 295.9 295.9 295.7 295.8 295.9 295.9 295.7 295.6 295.8 295.7 295.6 296.1 295.9 296.0
50.5 47.5 37.0 28.0 23.0 49.0 58.0 63.0 56.5 62.5 53.5 52.0 56.5 53.0 63.0 65.5 53.0 42.5 34.0
0 0 0 0 0 79 79 79 79 79 0 79 0 82 84 0 0 0 78
300 300 300 300 300 300 300 300 300 300 60 120 60 300 300 300 300 300 300
0.77E-12 1.00E-12 1.25E-12 1.49E-12 1.23E-12 2.10E-12 2.51E-12 1.42E-12 1.83E-12 1.77E-12 0.81E-12 2.66E-12 1.91E-12 2.26E-12 1.29E-12 0.80E-12 1.18E-12 1.34E-12 2.70E-12
2.85 2.96 4.02 5.29 5.18 6.27 5.58 3.36 3.55 3.29 1.38 4.02 1.93 8.25 6.05 1.08 2.00 3.09 10.24
14.0 21.0 17.8 20.1 13.0 40.2 30.4 10.8 28.7 21.8 24.8 48.4 41.3 28.9 17.4 98.1 44.7 26.8 42.4
Table 4 Parameters of the experiments and results of the %ts for the latex paint
Exp.
T (K)
h0 (%)
hin (%)
tf (s)
D (m2 s−1 )
(% kg−1 m3 )
SSE (%2 )
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
295.7 295.9 295.9 295.6 295.7 295.8 295.7 295.7 295.6 295.8 295.9 296.1 296.0 296.1 295.9 295.9 295.8
52.5 35.5 25.5 49.0 31.5 19.0 17.0 10.0 7.5 9.5 52.0 66.0 72.0 76.5 54.0 57.5 50.5
0 0 0 0 0 0 0 0 0 0 80 83 85 87 74.5 11.5 74.5
300 300 300 300 300 300 300 300 300 300 300 300 300 300 60 60 60
3.13E-12 3.41E-12 3.32E-12 3.12E-12 3.30E-12 3.70E-12 3.21E-12 2.44E-12 2.39E-12 2.42E-12 3.18E-12 2.37E-12 2.00E-12 1.92E-12 2.69E-12 2.56E-12 3.83E-12
4.21 7.08 8.85 4.75 7.79 9.96 10.64 12.29 14.91 12.90 6.41 4.77 3.71 3.19 6.12 3.22 6.43
19.6 17.7 9.6 20.5 12.5 11.4 7.8 9.4 9.4 9.4 14.6 14.1 10.1 8.9 10.1 4.9 5.9
is reduced to 180 s and the Kushing Kow rate is reduced to 1:667 × 10−5 m3 s−1 (1 l min−1 ). To be certain of the humidity of the Kushing air, the completely dry air, as supplied by the university pressure net, is used without adding water vapour. The result of these experiments are given in Fig. 9. There is still a large scattering. Also, the value of k shows very large di"erences. Even values for k are obtained which are 10 times as large as the values that are measured in the beginning of the experiments. The %tting of k (and also hs ) does more harm than good. Without %tting these values, and ignoring the source/sink e"ect (putting k =0), the last experiments give the results as given in Fig. 10. The results do not link up with each other.
The results are grouped in a few ‘island’. The experiments of island 1 are done by %rst drying the paint completely and then adding some water to arrive at the desired air humidity. The experiments of island 2 are done after the paint has been in equilibrium with the climate room having a humidity of approximately 50%. The experiments of island 3 are done after the paint reached a high moisture content after a weekend in a climate room with high humidity. From these experiments, it seems that not only the humidity in the paint determines the di"usion coe#cient but also the history of the paint. To verify if the velocity of stirring is a parameter that inKuences the results, the experiments of Fig. 10 are repeated
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Fig. 4. The measured di"usion coe#cient of water in the alkyd paint as a function of the equilibrium humidity in air.
Fig. 5. The measured for the alkyd paint as a function of the equilibrium humidity in air.
Fig. 6. The measured di"usion coe#cient of water in the latex paint as a function of the equilibrium humidity in air.
with two di"erent stirring velocities. Also, the blades of the stirrer have been put in a 45◦ angle with the stirrer axis to increase the mixing. The results are given in Fig. 11. The experiments from the solid lines have a stirring velocity that is 3 times as large as the experiments from the dotted lines. The stirring velocity does not have an inKuence on the re-
Fig. 7. The measured for the latex paint as a function of the equilibrium humidity in air. The dotted line represents the values that come out of an independent study as will be explained below.
Fig. 8. The measured di"usion coe#cient of water in the latex paint as a function of the equilibrium humidity in air, where the sorption isotherm is used as input.
Fig. 9. The measured di"usion coe#cient of water in the latex paint for a short time of Kushing with a low Kow rate.
sults. The resistance against mass transfer is in the paint. The resistance against mass transfer in the air phase may be neglected, as is done in the model. The results agree with those from Fig. 10, except now there is not such evidently grouping of the results. Taking k = 1:5 × 10−5 s−1 and
A. J. J. van der Zanden, E. L. J. Goossens / Chemical Engineering Science 58 (2003) 1521 – 1530
Fig. 10. The measured di"usion coe#cient of water in the latex paint without %tting of the source/sink term (k = 0).
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parameters that inKuence the di"usion coe#cient. This is not a#rmed by the results given in Fig. 11. If the situation is more complex than the assumptions underlying the present work, the set-up used here might be not the best way to measure the moisture transfer properties. For instance, there is no guarantee that a solution/di"usion model is the best description of the process. Also, there could be hysteresis in the sorption isotherm. This would not be evident immediately from the results obtained from the method used here, where the water concentration at the surface of the paint goes down and up again in one experiment. If the assumptions underlying the present work are met, the method is suitable to measure di"usion coe#cients. Looking at Fig. 11, a linear relation could be supposed between the di"usion coe#cient and the equilibrium relative humidity as D = h + ;
(25)
where and are constants. The values of and could be obtained not from %tting one experiment but from %tting all experiments together. This has not been done because of the large computational e"ort, and because the result would probably be a non-surprising line in Fig. 11. Notation Fig. 11. The measured di"usion coe#cient of water in the latex paint. The solid and dotted lines are obtained from experiments with respectively high and low stirring velocity.
hs =50%, which are physically realistic values, gives results which are approximately 12% larger for low humidities and 2% smaller for large humidities than the results in Fig. 11. 6. Discussion A method has been used to measure the di"usion coe#cient and the sorption isotherm of water in paint %lms. The set-up is quite simple and cheap. An additional advantage of this method, compared to those found in the literature, where the %nite bath method is used, is that the changes in the relative humidity of the air are quite slow. Thus, the sensor, which has a response time of approximately 15 s can quite easily follow the changes in the relative air humidity. The di"usion coe#cient and the sorption isotherm are obtained from the best %t of the model against the experimental values. This gives results with large scattering. When the sorption isotherm is determined independently, the %ts give better results for the di"usion coe#cient. The inKuence of the rubber ring is not so large, but could be avoided by using another way to close the vessel. The model supposes the diffusion coe#cient to be a constant in one experiment. More experiments show that the di"usion coe#cient is a function of the water concentration in the paint. The results given in Fig. 10 seem to point out that there might be also other
a A C D h hin H i k M n N R SSE t tf tlag T V x
grid tuning factor, dimensionless surface area of the paint %lm, m2 concentration, kg m−3 di"usion coe#cient, m2 s−1 relative humidity, percentage relative humidity of the inKowing air, percentage thickness of the paint %lm, m number of measured point, dimensionless proportionality constant, s−1 molar mass of water, kg mol−1 mass Kux, kg m−2 s−1 number of grid points minus 1, dimensionless gas constant, J mol−1 K −1 sum of squares of the errors, percentage2 time, s time of Kushing, s time between the start of an experiment and the %rst point of measurement, s temperature, K volume of the air inside the vessel, m3 position coordinate, m
Greek letters
locally linear sorption isotherm parameter, percentage kg−1 m3 locally linear sorption isotherm parameter, percentage
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slope of di"usion coe#cient versus relative humidity, m2 s−1 percentage−1 constant in function between di"usion coe#cient and relative humidity, m2 s−1 dimensionless space coordinate, dimensionless volumetric Kow of air through the vessel, m3 s−1
Subscripts air s sat x=0 0
air source/sink saturation at x = 0 at time t = 0
Acknowledgements The authors are grateful to Roalt Bruininks and Guus Theuws for the fast technical support. They are also grateful to Michel van der Pal and Wim van der Spoel for lending the stainless-steel vessel. One of the authors (E.G.) is, thanks to prof. E.L.J. Bancken, partly supported by ‘NWO— Technologiestichting STW’ Project: DCT.4010 Subproject IV-b ‘Water balance of water-borne paint systems on plaster substrates in relation to fungal growth’. DSM Resins is thanked for making the alkyd paint available. References Blahn>?k, R. (1983). Problems of measuring water sorption in organic coatings and %lms, and calculations of complicated instances of moistening. Progress in Organic Coatings, 11, 353–392.
Chan, T. Y. A., & Odlyha, M. (1995). The e"ect of relative humidity and pigment type on paint %lms. Thermochimica Acta, 270, 755–767. Crank, J., & Park, G. S. (1968). Methods of measurement. In J. Crank, & G. S. Park (Eds.), Di9usion in polymers (pp. 1–39). London: Academic Press (Chapter 1). Higuchi, A., Nakajima, T., Morisato, A., Ando, M., Nagai, K., & Nakagawa, T. (1996). Estimation of di"usion and permeability coe#cients of CO2 in polymeric membranes by FTIR method. Journal of Polymer Science B, 34, 2153–2160. Huld>en, M., & Hansen, C. M. (1985). Water permeation in coatings. Progress in Organic Coatings, 13, 171–194. Kreyszig, E. (1988). Advanced Engineering Mathematics (6th ed.) (pp. 372–378). New York: Wiley. Lundberg, J. L., Wilk, M. B., & Huyett, M. J. (1962). Estimation of di"usivities and solubilities from sorption studies. Journal of Polymer Science, 57, 275–299. Michaels, A. S., Vieth, W. R., & Bixler, H. J. (1963). The measurement of di"usion constants of gases in polymers by sorption techniques. Journal of Polymer Science B, 1, 19–23. Perez, C., Collazo, A., Izquierdo, M., Merino, P., & Novoa, X. R. (1999). Characterisation of the barrier properties of di"erent paint systems. Part I: Experimental set-up and ideal Fickian di"usion. Progress in Organic Coatings, 36, 102–108. Shi, F. F., & Economy, J. (1998). Moisture transport studies of newly developed aromatic and aromatic/aliphatic copolyester thin %lms. Journal of Polymer Science B, 36, 1025–1035. Toi, K., Ito, T., Shirakawa, T., & Ikemoto, I. (1992). Sorption and di"usion of gases in a polyimide. Journal of Polymer Science B, 30, 549–556. van der Wel, G. K., & Adan, O. C. G. (1999). Moisture in organic coatings—a review. Progress in Organic Coatings, 37, 1–14. van der Zanden, A. J. J. (1998). An iterative procedure to obtain the concentration dependency of the di"usion coe#cient from the space-averaged concentration vs. time. Chemical Engineering Science, 53, 1397–1404. van der Zanden, A. J. J., & Goossens, E. L. J. (2003). The measurement of the sorption isotherm of water in paint %lms, submitted to Chemical Engineering and Processing.