Formal Rheological Model of Acrylic Waterborne Paints - Springer Link

ISSN 0040-5795, Theoretical Foundations of Chemical Engineering, 2009, Vol. 43, No. 1, pp. 100–107. © Pleiades Publishing, Ltd., 2009. Original Russian Text © A.A. Bochkarev, P.I. Geshev, V.I. Polyakova, N.I. Yavorskii, 2009, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2009, Vol. 43, No. 1, pp. 105–113.

Formal Rheological Model of Acrylic Waterborne Paints A. A. Bochkarev, P. I. Geshev, V. I. Polyakova, and N. I. Yavorskii Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Lavrent’eva 1, Novosibirsk, 630090, Russia e-mail: [email protected] Received April 29, 2008

Abstract—A rheological model has been constructed, which has revealed, against the background of the viscous rheological properties, the role of elasticity in the orientation of pigment particles in painting by pneumatic spraying and aerosol deposition onto the surface. The model additively takes into account the elastic and viscous components of the total stress. DOI: 10.1134/S0040579509010138

INTRODUCTION As compared to solvent paints, waterborne paints have more pronounced viscoelastic properties. The high viscoelasticity of the waterborne paints is due to their colloidal state [1, 2]. Elasticity is an important property useful in painting technology because it prevents the paint on vertical surfaces from sagging [3]. For this reason, the increasing use of waterborne paints is accompanied by the development of new rheological concepts for them [4−7] and methods for measuring the response of a liquid to periodic shear strain [8–10]. However, although much information concerning the viscoelasticity of paints has been accumulated, the studies of the formation of optical properties of painted surfaces due to the orientation of pigment particles have been limited to qualitative consideration of the general effect of the rheological properties of paints [11, 12]. As the viscous and elastic components of rheological properties are inseparably connected in the existing rheological models of paints, it is impossible to judge the role of either of them. Our earlier model of pigment particle orientation [13] takes into account elasticity alone in terms of limiting stresses above which shear strain is possible. The purpose of this study is to construct a formal rheological model of waterborne paints in order to elucidate, against the background of the viscous component of rheology, the role of elasticity in pigment particle orientation in painting by pneumatic spraying and aerosol deposition onto the surface. Taking into account the role of elasticity is expected to provide a better fit to the observed reflective properties of painted surfaces, as compared to the fit achieved in [13], and to predict these properties from the rheological properties of the paint. The rheological model presented here has some quite disputable points. For this reason, we call it a formal model intended only for approximation of the

observed rheological properties of paints. Nevertheless, this model has provided a deeper insight in the role of elasticity. The starting data for this study are rheological data for a series of acrylic paint samples and reflection data for surfaces covered with these paints. The general scheme of this work was as follows. A rheological model in which the elastic and viscous stresses are additive was suggested for paints. An algorithm was developed for calculating the elastic and viscous rheological parameters by comparing the model with experimental rheological data. Experimental reflection data for painted surfaces were generalized taking into account or ignoring elastic rheological parameters. A comparison of these generalization variants suggested useful inferences as to the role of elasticity. RHEOLOGICAL MODEL In Kelvin’s rheological model, the general form of the shear stress appears as τ(ε, ε') = G(ε)ε + η(ε')ε'. (1) This is a model of a viscoelastic liquid in which the shear stress τ is the sum of two components, namely, the elastic stress G(ε)ε, which depends only on the strain ε, and the viscous stress η(ε')ε', which depends only on the strain rate ε'. The totality of experimental data available on shear viscosity is well described using the rheological law of shear viscosity as the two-term formula α

β

η ( ε' ) = η α ( ε'/ε '1 ) + η β ( ε'/ε '6 ) ,

(2)

where ε '1 is the strain rate at low shear rates and ε '6 is the strain rate at high shear rates. The first term of this formula describes the behavior of viscosity at low shear rates, and the second describes the behavior of viscosity

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FORMAL RHEOLOGICAL MODEL OF ACRYLIC WATERBORNE PAINTS

at high shear rates. The ηα value is unknown, but it is assumed that it is of the same order of magnitude as the viscosity ηLSV measured at a certain high strain rate. A more complicated problem is to find an approximating relationship for the rheological law governing the changes of the shear modulus ηβ as a function of strain. The available experimental data do not allow an appropriate physical model to be constructed for elastic stresses. For this reason, we will limit out consideration to the simplest hypothetic ηHSV relationship that formally describes the shear modulus throughout the strain range and satisfies the following asymptotic relations: (1) At small strains, the paint behaves as an elastic medium and obeys Hooke’s law. This means that, in the infinitely small strain limit, the shear modulus takes a finite value of G(0) > 0. In this case, as the strain increases, departing from zero, the shear stress initially increases. This assumption seems natural, since it means that G(ε) is an analytical function of the variable ε and Hooke’s law is the first term in the Taylor expansion of this function. (2) The second relation follows from the assumption that increasing the strain to infinitely large values will raise the elastic shear stress not to infinity, but only to a certain limit. In particular, this assumption is based, inter alia, on the supposition that the shear stress can be measured with a viscometer and that the result of these measurements depends only slightly on time within a rather wide range of shear viscosity measurement dura∞. Hence, tions. In this case, G(ε)ε τ∞ > 0 at ε the shear modulus at very large strains is infinitely small: G(∞) = 0. Thus, we assume that, for any flow of the paint, the elastic forces make a finite contribution to the shear stress, no matter how long the time of motion at a finite strain rate. The following simple formula corresponds to the above asymptotic relations: G(ε) = G0/(1 + aε), (3) where G0 = G(0) and a = G0/τ∞ is the dimensionless ratio of the elastic shear stress at unit strain according to Hooke’s law to the elastic shear stress at infinite strain. Obviously, these asymptotic relations are insufficient for complete description of the rheological law G(ε); however, it is apparently sufficient for approximately describing the reflectivity of the painted surface and for establishing a relationship between the reflectivity and dimensionless criteria—the purpose of this part of our study. In particular, this approach is expected to separate the contribution from the elastic forces to the experimentally measured shear viscosity coefficients. This optimism is based on the following: (1) viscosity measurements in a rotary viscometer the strain rate is fixed. The strain in these experiments increases linearly with time to reach very large values depending on the measurement duration. Therefore, there is good reason to believe that the second asymp-

101

totic relation is satisfied in these experiments. In further analysis, we will assume that all experimental shear viscosity data obtained using a viscometer refer to very large strain values and that the second asymptotic relation is valid for them. (2) In complex viscosity measurements by vibrational techniques, one usually operates in a small-strain region in which the strain dependence of the shear modulus can be neglected in the first approximation. This is the region where Hooke’s law is valid. Therefore, we will assume that all experimental data obtained by this method satisfy the first asymptotic relation. Summing up these assumptions, we arrive at the following rheological model of the shear stress: τ ( ε, ε' ) = G 0 ε/ ( 1 + aε )

(4)

α β + [ η α ( ε'/ε '1 ) + η β ( ε'/ε '6 ) ]ε'.

This rheological model has six yet unknown parameters: G0, a, ηα, ηβ, α, and β. These parameters can be derived from available experimental data. DETERMINING THE PARAMETERS OF THE RHEOLOGICAL MODEL In the determination of the parameters of the rheological law (4), we used experimental shear viscosity (ηi) data at strain rates of ε 'i = 0.1, 0.6, 1, 10, 103, and 15 × 103 s–1 obtained using a viscometer. These data can be used to calculate the left-hand side of Eq. (4): τ(ε, ε') = ηi ε 'i . Here, i is the conditional number of the shear viscosity determination experiment. These measurements were fitted to formula (4) for infinitely large strains in the form of α β τ ( ∞, ε' ) = τ ∞ + [ η α ( ε'/ε '1 ) + η β ( ε'/ε '6 ) ]ε',

(5)

where τ∞ is the elastic stress limit at infinite shear stress: G0ε/(1 + aε) τ∞ at ε ∞. There are initial shear stress data obtained at a strain rate of ε 0' = 0.01 s–1. These data most likely refer to the point in time immediately following the application of a constant strain rate of 0.01 s–1 to the viscous liquid. Therefore, these measurements were taken under conditions such that the strain rate has reached a value of 0.01 s–1 and the shear strain is still insignificant. For this reason, these experimental data were processed under the assumption that ε = 0 and ε '0 = 0.01 s–1. Accordingly, it follows from Eq. (5) that the initial shear stress is α

β

τ ( 0, ε '0 ) = [ η α ( ε '0 ⁄ ε '1 ) + η β ( ε '0 /ε '6 ) ]ε'.

(6)

Thus, we assumed that the initial shear stress is determined solely by the viscous strain. We realize the sub-

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Table 1. Coefficient of the approximation formula (5) for the shear stress τ(∞, ε') for various paints N

1

2

3

4

5

6

7

8

9

10

11

τ∞, N/m2 ηα, N s/m2 ηβ, N s/m2 α β ∆, %

4.31 34.0 0.0297 –0.772 –0.057 3.7

1.71 11.9 0.0250 –0.601 0.122 3.9

0.059 5.20 0.0670 –0.624 0.112 3.6

0.222 23.8 0.0355 –0.729 0.363 6.5

0.413 10.9 0.0186 –0.567 1.127 5.2

0.588 12.7 0.0246 –0.605 0.190 2.4

0.393 17.1 0.0289 –0.677 0.031 1.7

0.347 20.3 0.0307 –0.716 –0.020 1.9

2.18 41.6 0.0274 –0.811 –0.040 4.5

0.015 6.09 0.0288 –0.571 –0.139 5.3

0.538 9.30 0.0163 –0.548 3.911 4.5

tlety of this assumption, but we know that it will be very helpful in our further analysis. Suppose there are seven experimental data points, including six points for the shear stress described by formula (5), which has five parameters (τ∞, ηα, ηβ, α, and β), and one point for the shear stress described by formula (6), which has four parameters (ηα, ηβ, α, and β). The set of the corresponding seven equations does not make a system; therefore, it cannot have a single solution. However, it can be used to determine the unknown parameters. The parameters τ∞, ηα, ηβ, α, and β were determined simultaneously by minimizing the normalized standard deviation of the shear stress predicted by formulas (5) and (6) from the shear stresses observed at seven experimental points. For this purpose, the shear stress was calculated from experimental viscosity and shear strain rate data using the formula τi = ηi ε 'i ,

i = 1, …, 6

for six viscosity measurements at different strain rates. Next, we determined the normalized standard deviaτ, Ns/m2 103

102

1 2

101

100 10–3 10–2 10–1 100

101 102 ε', s–1

103

104

105

Fig. 1. Calculated dependence of the shear stress τ(∞, ε') on the shear rate ε' (formula (5) with coefficients from Table 1) versus experimental data for paint 4 (τ(ε, ε') = ηi ε 'i ): (1) experimental data and (2) approximation.

tions of τ(∞, ε 'i ) values calculated using formulas (5) and (6) from the experimental τi values obtained in the series of seven measurements: M ( τ ∞ , η α, η β, α , β ) τ ( 0, ε '0 ) = 1 – ----------------τ0

6

2

+

∑ i=1

τ ( ∞, ε 'i ) 2 1 – -----------------. τi

(7)

Here, the squared deviation is normalized to the square of the corresponding experimental shear stress value. With this normalization, the roles of all experimental points are equal. In formula (7), τ0 is the experimental value of the initial shear stress. The values of τ∞, ηα, ηβ, α, and β were determined by minimizing the functional M(τ∞, ηα, ηβ, α, β) (formula (7)), which has the meaning of the standard deviation of the shear stress values calculated using the rheological model from the experimental data. The calculated values of the parameters τ∞, ηα, ηβ, α, and β are such that the standard error of approximation of the shear viscosity does not exceed 7% throughout the viscosity variation range. For most paints, the standard deviation is within 2–4%. The calculated data are presented in Table 1. By way of example, Fig. 1 presents a comparison between the calculated τ(∞, ε 'i ) dependence (formula (5)) and experimental data for paint 4, for which the worst approximation was obtained (standard deviation of 6.5%). In order to determine the parameters G0 and a completely, it is necessary to know the shear modulus G0 in model (4). The shear modulus was determined using complex viscosity (η*) and phase difference (δ) data. The shear modulus and complex viscosity are known to be related by the formula G' = |η*|cosδω. Complex viscosity is measured by vibrational methods, so, as was mentioned above, it is assumed the thusmeasured shear modulus is equal to the shear modulus G0 in the validity region of Hooke’s law. Thus, we assumed that G0 = |η*|cosδω.


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Table 2. Cyclic frequency ω and shear modulus G0 calculated from the mean values of the complex viscosity η* and loss angle δ for 11 paint samples N ω, s–1 G0,

N/m2

1

2

3

4

5

6

7

8

9

10

11

1.75

2.72

1.37

0.89

2.44

3.04

3.32

3.04

2.62

3.32

2.64

6.62

4.46

1.42

4.29

4.35

4.92

5.37

5.50

7.84

3.04

4.08

Unfortunately, the cyclic vibration frequency ω at which the measurements were taken are unknown to our research team. For this reason, ω was calculated using the Cox-Merz empirical rule [14], according to which the following equality is true at ε' = ω: η(ε') = |η*|. (9) By solving Eq. (9) for ε', with η(ε') represented by formula (2) with coefficients taken from Table 1, we find the desired ω value. The results of these calculations are presented in Table 2. CRITERION APPROXIMATION OF REFLECTIVITY TAKING INTO ACCOUNT THE RHEOLOGICAL MODEL OF THE VISCOELASTIC LIQUID The dimensionless reflectivity of a painted surface is exp calculated using the formula F p = (I–30 – I15)/I0, in which I–30, I15, and I0 are the experimental light intensities measured at angles of –30°, 15°, and 0° from the normal at a light incidence angle of 45° [15, 16]. At exp large F p values, the painted surface is visually perceived as bright. It is necessary to find the most exact exp and universal approximation for experimental F p data in order to be able to predict, when developing new paints, the brightness of painted surfaces from the rheological properties of these paints. As was mentioned above, the rheological model of paints presented here, is essential for determining the role of the elastic stress in paint aerosol deposition onto the surface being painted. It was necessary to correct the form of the rheological relationship between the viscous stress and the strain rate, as was discussed in the previous section. The above rough rheological model suggests that it is necessary to take into account the parameters G0 and a. In this connection, it is noteworthy that a real physical model of paint rheology will likely contain similar parameters because they come from asymptotic relations representing rather general constraints imposed on any physically substantiated description of rheological media. The value of G0 can be determined from the complex viscosity provided that the loss angle (phase difference) δ is known. Therefore, for taking into account the elastic forces, the relationship between reflectivity and criteria must contain the angle δ and the parameter a. The simplest correction to the criterion relationship for reflectivity would be the

introduction of a power-law factor ad into the criterion formula. The general form of the resulting formula is representable as F p ( d0, d1, d2, d3, d4, d5, d6, d7 ) = exp ( d0 )Ma ( η LSV /η HSV ) ( 1 + α ) d1

d2

d3

(10)

× ( 1 + β ) ( S C ) tan δ a , d4

d5

d6 d7

where d0, d1, d2, d3, d4, d5, d6, and d7 are approximation coefficients; Ma = σ/(ηHSVU) is the Maron number; σ is the surface tension; U is the characteristic velocity of the aerosol drops falling onto the surface being painted; and SC is the fraction of nonvolatile components in the paint. However, this formula has a serious flaw. Experimental viscosities are determined by dividing of measured shear stresses by the shear strain rate. When the elastic stress makes an essential contribution to the total shear stress, the viscosity determined in this way will have a rather large error. For the dimensionless criteria in the approximation formula to be independent, it is necessary to eliminate these errors. One of the basic physical criteria of significance in paint aerosol deposition includes the ratio of the elastic stress to the viscous stress (K). This ratio is given by the quantity 1/ tan δ . However, our calculations of the shear rate, which is equal to the cyclic frequency ω (according to the Cox–Merz empirical rule), demonstrate that the shear rates in these measurements exceed, by one order of magnitude, the shear rate at which the viscosity ηLSV is measured. In the deposition of aerosol drops onto the surface being painted, an important role in the orientation of aluminum particles may be played by the last stage of paint drop spreading, at which the strain rate is low and the viscosity is similar in its order of magnitude to ηLSV. In this case, the proportion of the viscous stress in the total shear stress at low shear rates can be a significant parameter. Table 3 lists values of the ratio of the viscous stress calculated using the viscosity approximation formula (2) at a strain rate of ε '1 = 0.1 s–1 (which corresponds to the viscosity value ηLSV) to the total shear stress τ1 = ηLSV ε 1' , derived from experimental data. This ratio is [η( ε '1 ) ε '1 ]/τ1 = η( ε '1 )/ηLSV. In a similar way, it is possible to determine the proportion of the viscous stress at high shear rates. In this case, the contribution from the elastic forces turned out


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Table 3. Ratio of the viscous stress to the total shear stress, [η( ε '1 ) ε '1 ]/τ1 = η( ε '1 )/ηLSV, at a strain rate of ε '1 = 0.1 s–11 N η( ε 1' )/ηLSV

1

2

3

4

5

6

7

8

9

10

11

0.463

0.432

0.883

0.979

0.761

0.703

0.819

0.850

0.703

0.939

0.663

to be insignificant. The ηHSV value is involved in two criteria of formula (10), namely, the Maron number and the viscosity ratio. For this reason, as the first approximation, we will use the ratio of the viscous stress to the total stress (Table 3) instead of the ηLSV/ηHSV ratio. The effect of the viscosity ηHSV will be taken into account in terms of the Maron number. This will bring formula (10) into the following form: 11

M ( d0, d1, d2, d3, d4, d5, d6, d7 ) =

∑

which has the meaning of the standard deviation of the logarithms of reflectivity from experimental data exp

( F p , ) obtained for 11 paint samples. The choice of this form of the functional is dictated by the necessity of having a functional that has a single global minimum in the eight-dimensional space of the parameters d0, …, d7. The thus-determined function M(d0, …, d7) is a positively defined quadratic form in the variables d0, …, d7. Thus, it is assured that this function has a single minimum. This is essential for finding the extremum by numerical methods. The results of these calculations are represented by the formula – 0.01

× (1 + β)

– 1.8 – 0.37 [ η ( ε '1 )/η LSV ] ( 1 + α )

0.18

( SC )

0.08

tan δ

– 0.41 0.3

(13)

a .

Fp 1 2

4 3

4

5

6

d4

7

8

9

10 11 N

Fig. 2. Reflectivity Fp of various paint samples: (1) observed and (2) calculated using formula (13).

(11)

d5

d6 d7

The exponents d0, …, d7 were determined by minimizing the functional

(12)

A comparison between experimental reflectivity data (Fp) and the data calculated using formula (10) are presented in Fig. 2. The standard deviation in this case is 2%. This is radically better than in the case of approximation made for the samples in an earlier work [13]. It is clear from formula (13) that the effect of the Maron number is insignificant when experimental data are processed in this way. The absence of an effect of this number can be attributed to experimental conditions such that some regime parameters are fixed and the effects of important criteria, such as the Reynolds and Maron numbers, are not manifested. To determine the contributions from these criteria to the approximation formula for reflectivity, it is necessary to carry out additional experimental studies going beyond the set of paint samples examined here. Using approximation formula (13), it is possible to obtain a rougher, but well-working, formula with rounded exponents:

0.2

6

d3

× ( 1 + β ) ( S C ) tan δ a .

– 1.8

× ( 1 + β ) ( S C ) tan δ

8

2

d1

F p = 0.63 [ η ( ε '1 )/η LSV ]

10

1

d2

= exp ( d0 )Ma ( η ( ε '1 )/η LSV ) ( 1 + α )

log [ F p n ( d0, d1, d2, d3, d4, d5, d6, d7 ) ] 2 - , 1 – --------------------------------------------------------------------------------------------------exp log [ F p n ]

n=1

F p = 0.71Ma

F p ( d0, d1, d2, d3, d4, d5, d6, d7 )

0.1

(1 + α)

– 0.4 0.3

– 0.4

(14)

a .

The standard deviation for this formula is 3%. A comparison between experimental reflectivity data and the data calculated using formula (14) is presented in Fig. 3. Thus, taking into account the elastic stress determined using the above rheological model allows one to qualitatively improve the approximation of experimental reflectivity data without increasing the number of dimensionless parameters. There are six criteria in formula (11). It is quite possible that the number of criteria can be further reduced. Indeed, in formula (14), with roughened exponents, some criteria can be grouped


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into complexes. For example, this formula can be rewritten as a function of four criteria, F p = 0.63 [ η ( ε '1 )/η LSV ]

0.4

1/2

Fp 10 1 2

– 1.8

(14')

× { ( 1 + β ) / [ ( 1 + α ) tan δ ] } ( S C ) a

0.1 0.3

8

or even two ones,

6

F p = 0.63 [ η ( ε '1 )/η LSV ]

– 1.8 4 0.1

× { S C a ( 1 + β ) / [ ( 1 + α ) tan δ ] } . 3

105

2

(14")

4 1

However, for safely claiming that a reliable criterion approximation has been found for reflectivity, is necessary to carry out a more detailed experimental study for a larger number of diverse paints. Note that the contribution from the elastic forces to the shear stress depends strongly on the kind of paint. By way of example, we list contributions from the elastic stress at a strain rate of 0.1 s–1 (Table 4). These data were calculated using the rheological model (5). It is clear from Table 4 that the contribution from the elastic stress is particularly large for paints 1 and 2. At the same time, an important parameter of the rheological model is a, which is the ratio of the elastic shear stress at unit strain according to Hooke’s law to the elastic shear stress at infinite strain. Values of this dimensionless parameter for various paints are listed in Table 5. From formula (3) for the elastic stress, it is clear that, the larger the parameter a, the lower the elastic stress, the other conditions being equal. Therefore, the smallest value of a will correspond to the largest value of the elastic stress. It is clear from Table 5 that this similarity is the smallest for paints 1 and 2. This confirms the above inference that the elastic stress plays the most significant role in paints 1 and 2. An analysis of the exponents of the terms appearing in formula 3 provides an understanding of its physical

2

3

4

5

6

7

8

9

10 11 N

Fig. 3. Reflectivity Fp of various paint samples: (1) observed and (2) calculated using formula (14).

meaning. Reflectivity increases with a decreasing Maron number (i.e., surface tension), with decreasing dynamic viscosity, and with an increasing steepness of decrease of shear viscosity as a function of the strain rate. These relationships are fully consistent with the opinion formed by the authors when they worked on this publication. However, this formula has some selfinconsistencies. A decrease in the dynamic-to-shear viscosity ratio and a decrease in the tangent of the loss angle imply an increase in the roles of the elastic stress and in reflectivity. However, the increase in the role of the elastic stress is accompanied by a decrease in a. As judged from the positive value of the exponent of the corresponding term of the formula, this will reduce the Fp value. This inconsistency is likely caused by the fact that the form of the approximation formula was chosen arbitrarily under the assumption that all of its components are independent. In fact, this is apparently untrue; therefore, formula (10) should not be regarded as a final result. It should be additionally tested against other experimental data and then corrected.

Table 4. Proportion of the elastic stress calculated using formula (5) for 11 paint samples N (G(ε)ε/τ) × 100, %

1

2

3

4

5

6

7

8

9

10

11

53.7

56.8

11.7

2.1

23.9

29.8

18.1

15.0

29.7

6.1

33.7

Table 5. Dimensionless parameter a in the rheological model (4) for 11 paint samples N

1

2

3

4

5

6

7

8

9

10

11

a

1.54

2.61

24.2

19.3

10.5

8.37

13.7

15.7

3.60

196.4

7.58


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CONCLUSIONS A formal rheological model of paints has been developed in order to analyze the roles of the viscous and elastic rheological forces. This model is based on Kelvin’s classical model and on the additivity of the contributions from the viscous and elastic stresses. The classical model is supplemented with the shear modulus as a formal function of strain. This made it possible to use experimental shear viscosity data obtained at different strain rates and to calculate all parameters of the rheological model for 11 paint samples. It was also possible to calculate the proportion of the viscous stress in the total shear stress and to interpret this quantity as a criterion important for the orientation of pigment particles and for generalization of experimental reflectivity data. One more important parameter of the rheological model was obtained, whose physical meaning is the ratio of the elastic shear stress at unit strain to the shear stress at infinite strain. The approximation of experimental reflectivity data taking into account these two extra rheological parameters is characterized by an error of several percent. The resulting approximation formulas are applicable to estimating the reflectivity of paints whose rheological properties are known. Based on the formal rheological model, we calculated the contribution from the elastic stress to the total shear stress. The elastic stress fraction is higher for paint samples with a definitely high reflectivity (samples 1, 2, and 9). This fact is of great significance because it shows the way to the developers of new paints. It is possible that the positive contribution from the elastic stress to the orientation of paint particles shows itself at the deposition stage. The elastic forces prevent the relaxation of drops after their impact on the surface being painted. However, this tentative deduction needs verification. The important result concerning the role of the elastic stress in the rheology of promising paints should be checked by numerous experiments involving a wide variety of samples. This check is quite necessary because any formal mathematical model whose parameters are determined by comparison with experimental data, plausible as it may be, may fail to reveal the real physical meaning of the physical phenomena that it describes. Therefore, the results of this study should be viewed as promising, but the study itself should not be considered complete. NOTATION a = G0/τ∞—dimensionless quantity; d0, d1, d2, d3, d4, d5, d6, d7—approximation coefficients; Fp—dimensionless reflectivity in approximation formulas; exp F p —dimensionless reflectivity calculated using the formula Fp = (I–30 – I15)/I0, where I–30, I15, and I0 are the reflected light intensities measured at angles of

−30°, 15°, and 0° from the normal at a light beam incidence angle of 45°; G(ε)—shear modulus measured by a rotary viscometer, N/m2; G0 = G(0) —elastic shear stress at unit strain, N/m2; G'—shear modulus (G' = G0 for complex viscosity measurements by vibrational methods), N/m2; K—criterion equal to the ratio of the elastic stress to the viscous stress; M—functional; Ma = σ/(ηHSVU)—Maron number; N—paint number; SC—fraction of nonvolatile components in the paint; U—characteristic velocity of the aerosol drops falling onto the surface being painted, m/s; α, β—exponents in the rheological law for the shear viscosity; δ—phase difference or loss angle, rad; ∆—error, %; ε—strain, rad; ε'—strain rate, rad/s; ε 1' —characteristic low strain rate, rad/s; ε 6' —characteristic high strain rate, rad/s; η*—complex viscosity, N s/m2; ηα ≈ ηLSV, where ηLSV is viscosity measured at a certain low strain rate, N s/m2; ηβ ≈ ηHSV, where ηHSV is viscosity measured at a certain high strain rate, N s/m2; σ—surface tension, N/m; τ—shear stress, N/m2; τ0—experimental value of the initial shear stress, N s/m2; τ∞—elastic shear stress at infinite strain; N/m2; ω—cyclic vibration frequency, s–1. SUBSCRIPTS AND SUPERSCRIPTS 0—initial; 1—parameter measured at a low strain rate; 6– parameter measured at a high strain rate; α, β—exponents; ∞—infinity, infinite strain; ë—solid content; exp—experimental; i—conditional number of the shear viscosity determination experiment; n—number of samples; —painted; *—complex.


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Vol. 43

No. 1

2009