Experimental Study on the Effect of Polyol Admixtures on the

Ind. Eng. Chem. Res. 2009, 48, 2659–2670

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GENERAL RESEARCH Experimental Study on the Effect of Polyol Admixtures on the Interfacial Tension of Potassium Dihydrogen Phosphate in Aqueous Solutions: Induction Time Experiments Versus Drop Shape Analysis Stephan Machefer*,† and Klaus Schnitzlein‡ Department of Chemical Reaction Engineering, Brandenburg UniVersity of Technology, Cottbus, Burger Chaussee 2, 03046 Cottbus, Germany

The interfacial tension (IFT) between potassium dihydrogen phosphate (KDP) and solutions of water with a polyol admixture is determined with two experimental techniques. First induction time experiments have been carried out to evaluate the IFT based on the thermodynamic theory of nucleation. Published IFT data is carefully reviewed and compared to the experimental results obtained in this work. Special attention is paid to reproducibility of induction time measurements and to the impact of agitation. The second experimental technique directly determines the IFT on the KDP prismatic face via drop shape analysis (DSA). Provided that the solution does not spread on the crystal surface, drop shape analysis gave reasonable IFT approximations for elevated fractions of the polyol, offering the potential of being a quick alternative method to the timeconsuming induction time measurements that typically are of poor reproducibility. However, the most critical aspect about DSA has shown to be the determination of the crystals’ surface energy (SFE). Furthermore, for systems with low-IFT values the curvature dependence must be considered, which can be taken into account with the simple Tolman equation. 1. Introduction In classical thermodynamic theory of nucleus formation, the interfacial tension (IFT) between the nucleus and the surrounding solution (index nl) is the key parameter.1,2 The rate of nuclei formation is extremely sensitive to the value of the IFT. This is why a reliable determination of the IFT is highly important. However, a direct measurement of the IFT is not possible. Instead, kinetic experiments, which are essentially the measurement of nucleation induction time or crystal growth rate are performed to obtain a value for the IFT (index cl) indirectly based on a presupposed model.3–5 The most common experimental method for the determination of the IFT is the measurement of the induction time of nucleation.6 Because of its stochastic nature and the vast variety of conditions influencing nucleation,3 reproducibility of this kind of experiments is poor. In practice, data scattering of up to 2 orders of magnitude can be observed. As a result, published IFT values for one and the same system can vary dramatically. It is so much more incomprehensible that experimental data regarding IFT are mostly published without any indication of reproducibility. Systems with low IFT values seem to be least reproducible.7 In fact, published IFT values for a salt with comparatively low IFT such as potassium dihydrogen phosphate (KDP) differ considerably for induction time experiments (nucleus-water interface).7–11 This finding is even more meaningful, considering the high * To whom correspondence should be addressed: Phone: +33 383 322975. Fax: +33 383 175104. E-mail: [email protected]. † Current address: ENSIC, Laboratoire des Science du Génie Chimique, 1 rue Grandville BP 20 451 F-54001 Nancy cedex, France. ‡ Current address: Department of Chemical Reaction Engineering, Brandenburg University of Technology, Cottbus, Burger Chaussee 2, 03046 Cottbus, Germany.

impact of the IFT on nucleation kinetics. It is therefore sufficiently reasonable to take a closer look at these results and the impact of reproducibility when applying induction time experiments. The lack of reproducibility requires many repetition experiments. Induction times can last up to several weeks when low supersaturations are to be investigated. Hence, the requirement for repetition experiments is critical when they cannot be performed simultaneously but only consecutively. This is especially true when complex detection methods such as light scattering are required or when mixing has to be applied, that is in case that supersaturation is created by precipitation reactions or the salting-out effect.7,12,13 If temperature dependency or the effect of additives on the IFT has to be investigated additionally, induction time experiments are associated with an unjustifiable expenditure of time. Hence, it would be highly desirable to improve reproducibility of induction time experiments or to look for approximative methods as reasonable alternatives. This article deals with the evaluation of the IFT determined from induction time experiments in stirred and unstirred experiments. Special attention is paid to the impact of agitation on the IFT and experimental reproducibility. The KDP/water system is chosen as a representative test case for systems with low IFT. The results are compared to reviewed and re-evaluated IFT values in literature. A special difficulty regarding induction time experiments can be often found when organic admixtures are present: Liquid phase splitting occurs especially for low to intermediate fractions of organic cosolvent where electrolyte solubility is sufficiently high.14 Hence, the supersaturations that can be applied are very limited and mostly fall into the zone of metastability where induction time is infinite. Alternative experimental techniques would be therefore favorable for this special case. This is why drop shape analysis (DSA) is applied to directly measure the

10.1021/ie800327w CCC: $40.75  2009 American Chemical Society Published on Web 01/30/2009

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IFT between the flat prismatic KDP face and the saturated solution. The results are discussed and compared to the IFT values obtained from induction time experiments. Additionally, the impact of an organic cosolvent is investigated with both methods, induction time experiment and drop shape analysis. In contrast to previous works, which solely used low molecular solvents,13,15 a polyether-polyol is used in this study as organic admixture.14,16,17 2. Induction Time Experiments 2.1. Theory. The induction time tind marks the time elapsed between the creation of supersaturation and the detection of crystals. Consequently, the actual time for the formation of critical nuclei tn is well represented by tind when the transition period into steady state growth (ttr) and subsequent growth to a detectable size (tG) can be neglected.18 tn ≈ tind

∀ (ttr + tG) , tn

(1)

Provided that this condition is true for the present work (a discussion follows), tind is inversely proportional to the nucleation rate:6 B ∝ tind-1

(2)

According to the thermodynamic theory of nucleus formation, B can be expressed as follows:3

{

B ) C1 exp -f(θ)

βσ3nlV2

1 ν2(kT)3 ln2 Sa

}

(3)

A combination of eq 2 with eq 3 gives a linear relationship between tind and 1/(ln2 Sa): ln tind ) C2 + f(θ)

βσ3nlV2

1 ν2(kT)3 ln2 Sa

(4)

Note that the relative supersaturation is usually defined on a concentration basis (i.e., γ( ) 1): Sa )

γ(〈c〉c a c ) ≈ )S * a γ(〈c* 〉c* c*

(5)

The shape of the nucleus is accounted for with the factor β: β)

4kA3 27kV2

(6)

Spherical nuclei are most commonly assumed, resulting in β ) 16π/3. Other shapes can be approximated by a prism of length x, width y, and height z:3 xyz 2(xy + xz + yz) , kA ) (7) 3 y y2 Heterogeneous nucleation is usually considered by the correction factor f(θ) varying between 0 and 1. When the nucleus is assumed to be cap-shaped, the correction factor is a function of the contact angle with the foreign body: kV )

(2 + cos θ)(1 - cos θ)2 (8) 4 The value of σnl is usually evaluated from a graphical analysis of eq 4. Typically, two linear, intersecting gradients can be observed when evaluating the ln(tind) - 1/(ln2 S) relationship for a broad range of supersaturations6 (also Figure 3). Nucleation at high supersaturations is assumed to be independent from foreign surfaces in the solution (θ ) 180, or f(θ) ) 1). f(θ) )

Consequently, the slope at high values of S directly provides σnl. For low values of S, the gradient is smaller, indicating heterogeneous nucleation or f(θ) < 1. Hence, θ can be evaluated from the slope at low supersaturations provided that σnl is known. 2.2. Experimental Procedure. Induction time experiments have been carried out with two different experimental setups. First a static, unstirred procedure is applied. In a second setup, induction times are measured under agitated conditions. The solutions used throughout this work were prepared with analytical grade KDP (Merck,>99.5%) and analytical grade water (Merck). An industrial grade polyether-polyol has been used for the experiments with organic additive. Volatile compounds of the poylol have been removed via vacuum evaporation. The composition of the supersaturated KDP/water/polyol solutions were calculated from solubility measurements recently published for this system.14 In both setups, induction times are detected visually as was done in most of the works that deal with the aqueous KDP system.9–11,13 This means that crystals of macroscopic size are observed rather than nuclei. Accordingly, the question arises whether eq 1 is met. ttr can generally be neglected except for extremely low supersaturations or low diffusivities, which both do not apply for this work.18 From growth experiments conducted for the present system, we also assume that tG is sufficiently low.19 On the basis of the measured growth rates of single KDP crystals (1 mm), and assuming a reduction of these macroscopic growth velocities by 1 order of magnitude caused by size-dependent growth,20 tG would be in between 20 and 500 s (high and low supersaturations respectively) to attain a visually observable size. The induction times measured in this work range from 1000 to 500 000 s, which corresponds to an error of 0.01 to 2% for not considering tG. This error is far less than the overall error of induction time experiments as will be shown in the following. In fact, the detection method does not affect the derivation of σnl because visual detection has shown to give similar results to investigations with more sensitive detection techniques based on light scattering for the aqueous KDP system.7 All aspects considered, we believe that condition 1 holds for this work. 2.2.1. Static System. The static experiments are similar to the procedure published by other authors.10 Twenty sealed test tubes containing the solutions are inserted into a transparent, tempered water bath. The salt is dissolved in a test tube shaker until the solution is visually clear. To make sure that dissolution is complete, that is all nuclei are desintegrated, the test tubes are kept for 45 min and 20 K above the saturation temperature before the experiment is started. The supersaturation is created by natural cooling. The target temperature is attained after 60-90 min. The nucleation is detected visually with serial photography. Half-time between saturation point and achievement of the target temperature has been defined as starting point of the induction period. 2.2.2. Stirred System. Figure 1 illustrates the experimental setup of the agitated induction time experiments. Ten sealed long-neck flasks (100 mL) are arranged on two vertically displaced rows in a custom-made transparent water bath (called nucleation bath in the following). The nucleation bath is tempered with an internal thermostat and cooling coils attached to the external circulation of a chiller. Because of the special shape and the large volume of the bath (40 L), the water circulation has been optimized ensuring identical heat transfer conditions for each flask. This way the cooling curves can be controlled with a single temperature probe inserted in one of

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2661

Figure 1. Setup for the agitated induction time experiments.

the flasks being representative for all flasks. The flasks are mixed with two multiposition magnetic stirrers using high-performance Teflon-coated stir bars. Nucleation within the flasks is detected via serial photography. To optimize image quality, the whole setup has been installed in a completely shaded chamber. The flasks are illuminated by a white-light LED perpendicular to the camera. A semi-fish-eye lens allows simultaneous photography of all flasks. The solutions were dissolved for 1 h and 5 K above the saturation temperature at medium stirring intensity (600 rpm). The temperature difference has been chosen to avoid incomplete dissolution or some kind of memory effect due to prevailing nuclei or fragments, even for higher polyol contents. For comparison, a temperature differences of only 1 K has been applied in literature for KDP in water.9 However, because of the low IFT of KDP (in water), nuclation clusters are small and desintegration should be fast. This may be the reason why the role of dissolution temperature is mostly disregarded for this particular system. Consequently, most of the references dealing with induction time experiments of KDP in water do not mention the dissolution temperature difference. Following dissolution, the ten flasks are simultaneously removed from the bath with a special carrier construction and stored in a second standard water bath tempered at dissolution temperature. Then the nucleation bath is cooled. After having reached the target temperature, the flasks are inserted back into the nucleation bath, the stirrers are started, and the induction

time measurement starts when the temperature inside the flasks falls below saturation temperature. This shock cooling allows a cool-down to the target temperature within 2-3 min. Hence, the definition of the starting point of the induction period is less problematic than that in the static system. This is also supported by another study that suggested that the reproducibility of induction time experiments is increased by reducing the time required for creating the supersaturation.7 Another advantage of the agitated setup in comparison to static systems is its higher detection sensitivity. In the static system, nucleation manifests itself in the formation of nuclei that sink inside the test tube, grow, and agglomerate further on the bottom. The camera must be close to the test tubes for detection. In stirred systems, multiple nucleation events turn the whole solution turbid. This way even sparingly soluble systems can be investigated in a much more reliable way and experiments could be carried out in the stirred system for water/ * ) 2.1 g/L). In polyol mixtures containing 70% polyol (cKDP the static system it was already highly difficult to detect nucleation for 55% polyol (c*KDP ) 12.6 g/l). Further advantages of the stirred setup are the reduction of temperature gradients inside the nucleation vessel and the reduction of the overall induction times. 2.3. Results. 2.3.1. Reproducibility. To compare experiments at different supersaturations, a dimensionless induction

2662 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009

Figure 2. Standard deviation of induction time experiments in terms of dimensionless induction time. The error bars of the individual points have been omitted for better readability. The average error range is στ ) (0.5.

Figure 3. Plot of ln(tind) versus 1/(ln2 S) for the induction time measurements of KDP in water with the static setup (filled circles) and the stirred setup (open symbols). The error bars represent the overall range of data scatter. The plain linear regression curves mark the homogeneous nucleation region, whereas the dashed lines mark heterogeneous nucleation.

time is defined as ratio of actual to mean induction time of all test tubes (static setup) or flasks (stirred setup), respectively. τind )

|

tind jt ind S,n)const.

(9)

Figure 2 shows the standard deviation of τind regarding all test tubes/flasks of one experiment as a function of supersatu-

ration for the induction time measurements in water. An impact of supersaturation is observed. However, the trend is opposite for the static and the stirred systems. In the static system, the experimental error increases with supersaturation. The stirred setup shows largest standard deviations for low supersaturations. When comparing the agitated experiments, the standard deviation slightly decreases with increasing stirrer speed. We conclude that the static system shows higher reproducibilities for low supersaturations, whereas stirred systems may be favorable for higher supersaturations. A reasonable interpretation of this observation is difficult and is not discussed further in this part. Furthermore, there are other differences of both setups other than agitation (i.e., temperature profiles) so that it remains unclear if the different trends regarding reproducibility can be attributed to stirring only. 2.3.2. Interfacial Tension - KDP/Water. The results of the induction time experiments in pure water are illustrated in Figure 3. The induction times in the static experiment are higher than those observed in the stirred setup. This is why low supersaturations S < 1.3 could not be investigated in the static case because induction times of several weeks or even months were expected. Induction times are up to 2 orders of magnitude lower in the stirred system. Hence, the supersaturation could be decreased to values where heterogeneous mechanisms dominate the nucleation process. The intersection point between homogeneous and heterogeneous nucleation can be found at S ) 1.25-1.3, which agrees well with the observations made by other authors7,9 (Table 1). From the slopes in the heterogeneous region, contact angles between 30° and 40° were calculated. The error bars given in Figure 3 represent the entire range of induction times measured for the corresponding supersaturation. In the most extreme cases, that is for low supersaturations in the stirred and for high supersaturations in the static system, induction times measured under identical conditions can differ by nearly 2 orders of magnitude. Figure 4 gives the calculated values for σnl as function of stirring speed. σnl systematically decreases with increasing agitation intensity. The observation that σnl values for KDP from induction time experiments varies with agitation intensity has also been made by other authors (σnl ) 7.3-9.2 mN/m).15 However, no clear trend could be found. From a theoretical point of view, σnl should be independent of the mixing conditions unless viscous forces play an important role.21 Therefore, a possible explanation for this trend may be attributed to the increasing fraction of impurities stemming from abrasive interactions between the crystals and the stirrer or the vessel wall. This hypothesis will be discussed later. Within the many works published for KDP/water, values between 1.8 and 546 mN/m have been reported for σnl.7–11 Although the error of induction time experiments is quite high indeed, the observed scattering of data can hardly be due to experimental errors. This is why we took a closer look at those research works, realizing that the differences in σnl mainly stem from different parameters and assumptions that have been used (eq 4). Various values for β or ν have been applied, and in one case the natural logarithm has presumably been replaced by the decade logarithm without introducing any conversion factor. This is why all published data, if necessary, have been re-evaluated with eq 4 for Sa ) S, β ) 16π/3, and ν ) 2 as standard case. The results are shown in Table 1 giving interfacial tensions between 5.9 and 14.3 mN/m. The values measured in this work lie well inside the range of the published values. Table 1 also includes IFT obtained from growth rate experiments.4–6 Some of these authors claim that induction time

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2663 a

Table 1. IFT for the System KDP/Water at Various Temperatures

σnl, σcl [mN/m] method IT IT IT IT IT IT IT GR GR GR GR P

T ) 25 °C

T ) 28 °C

T ) 30 °C

T ) 32 °C

7.9

T ) 35 °C

T ) 40 °C

T ) 45 °C

8.9

7.4 10.1

6.7

T ) 50 °C

8.8 5.9 14.3 10.6 6.8 15.0 11.5 16.7 22.5 13-30

10.1

9.6

source 9b 7c 11d 10e 8f this workg this workh 22i 4j 5k 5l 28–30m

a

Comparison of different literature sources. The values were obtained from growth rate experiments (GR, σcl), induction time measurements (IT, σnl) or different predictive methods (P, σnl, or σcl). All values refer to eq 4 (Sa ) S, ν ) 2, β ) 16π/3). Transition from heterogeneous to homogeneous nucleation are provided if available (S(tr)). b IT, S ) 1.3-1.8, static, S(tr):1.25. c IT, S ) 1.4-2.0, static, S(tr):1.3. d IT, S ) 1.55-1.85, static. e IT, S ) 1.45-1.8, static, S(tr):1.45. f IT, S ) 1.55-1.7, static. g IT, S ) 1.3-1.7, static. h IT, S ) 1.25-1.5, 1200 rpm. i GR, dislocation growth analysis, [100] prismatic face. j GR, NAN-fit, [100] prismatic face. k GR, dislocation growth analysis, [100] prismatic face. l GR, dislocation growth analysis, [001] prismatic face. m Predictions/correlations based on physical properties (c*, γ(,...).

Figure 4. IFT of KDP in water from induction time experiments at different stirrer speeds.

Figure 5. Plot of ln(tind) versus 1/ln2S for the induction time measurements of KDP in water with polyol as additive.

experiments underestimate the value of σnl because of heterogeneous processes, which would be of special relevance for systems with a low IFT such as KDP in water.22 Instead, IFT values are proposed to be determined from growth rate experiments. Other than in case of homogeneous nucleation, where the Gibbs-Volmer theory is widely accepted,18 a case specific mechanism has to be postulated for crystal growth. In most of the growth theories, the origin of the growth source is the mainly characteristic difference. In case of spiral dislocation growth, the BCF theory is applied, whereas in the case of surface

nucleation the nuclei above nuclei model (NAN, also called Birth and Spread model) is used. In BCF theory, σcl determines the steepness of a dislocation spiral and dislocation-source activity, whereas in the NAN model σcl is a nucleation characteristic property. For the present system, different faces have shown different growth characteristics. For example, Rashkovich showed in an extensive study of dislocation spiral hillock growth that prismatic faces of KDP-family crystals (KDP, DKDP, ADP) grow by the dislocation mechanism.5 This mechanism has also been observed for the gravimetric overall growth of ADP23 whereas for the overall growth of the pyramid faces of KDP and ADP the NAN model applied best.24 From the corresponding edge energies, a σcl value of 10.7 mN/m (KDP, 35 °C) can be calculated for the pyramid face using Nielsens relation.22 This would be a significantly lower IFT than the values reported for the prismatic faces in Table 1. Later on it was found that the mechanism depends on supersaturation for the prismatic faces.25 At S < 1.07, dislocation growth was observed changing to nucleation-induced growth for higher supersaturations. From the provided edge-energies, we calculated σcl values between 10.5 and 8.8 mN/m for both regimes respectively, which comes close to the σnl values in Table 1. We conclude that scattering of IFT values obtained from growth experiments are comparable or even higher when compared to induction time experiments. The σcl calculation is sensitive to model postulations, approximations, and the range of supersaturations that are investigated. Furthermore, σcl values are face-specific. It was shown that not only prismatic and pyramid faces but also for different prismatic faces5 different values of σcl were measured. Another reason that may favor induction time experiments is the well-known curvature dependence of σnl.26 In contrast to growth experiments, induction time measurements deal with the nucleation phenomenon, hence implicitly consider curvature dependence. Especially for systems with low interfacial tensions such as KDP/water, the size of the (critical) nuclei is very small, hence curvature dependence gains importance. σnl tends to be smaller than the corresponding value for a flat crystal surface.27 In case of curvature dependence, σnl would be, strictly spoken, a function of supersaturation or critical nucleus size, respectively. This is why the somewhat lower values for σnl obtained from induction time experiments may reflect an averaged curvature dependence for the range of supersaturations investigated. This

2664 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 Table 2. IFT (σnl) in the KDP/Water System at 30 °C; Effect of Organic Additivesa σnl [mN/m] solvent (2) w/ethanol w/1-propanol w/2-propanol w/polyol w/polyol

w2 ) 0 g/g

w2 ) 0.05 g/g

w2 ) 0.10 g/g

w2 ) 0.15 g/g

10.4

9.5 8.7 9.5

11.0 13.4

8.3 10.6 (7.3)c 6.8 (4.7)e

w2 ) 0.55 g/g

w2 ) 0.70 g/g

source 13b 13b 13b this work

14.6 (12.2)d 10.5 (8.7)f

13.4 (11.6)g

this work

a Literature values have been recalculated for β ) 16π/3. The values in brackets represent values that would have been obtained with Sa. b IT, salting out precipitation, Sa ) 1.15-1.35, 400 rpm. c IT, S ) 1.3-1.7 (Sa ) 1.15-1.30), static. d IT, S ) 1.65-1.8 (Sa ) 1.45-1.55), static, 20 °C. e IT, S ) 1.25-1.5 (Sa ) 1.13-1.22), 1200 rpm. f IT, S ) 1.3-1.5 (Sa )1.22-1.35), 1200 rpm. g IT, S ) 1.5-1.6 (Sa )1.38-1.45), 1200 rpm.

Figure 6. Drop shape analysis: (a) pendant drop experiment, (b) sessile drop experiment.

may also contribute to the deviations in σnl observed for the literature values. Finally, the results of several predictive methods are also given in Table 1. Most of these methods relate the interfacial tension and the solubility in the framework of a thermodynamic model, giving values between 17 and 30 mN/m for the KDP/ water system.28–30 These models were fitted to experimental data for various aqueous electrolyte systems. Although a correlation between IFT and solubility clearly exists, the estimation of these equations is rather crude. Another predictive method considers the ion hydration along with solubility.31 This way a value of 13 mN/m is obtained. However, the evaluation of the ion hydration number is difficult and depends on the method applied.4 2.3.3. Interfacial Tension - KDP/Water/Polyol. The impact of organic additives on the interfacial tension of KDP in water has previously been investigated for low molecular alcohols and low additive mass fractions.15 However, no clear conclusion could be drawn either regarding the impact of additive concentration or regarding the influence of the type of solvent. In this work, we investigate the impact of a polyetherpolyol (molar mass of 500 g/mol)16 at higher mass fractions. Because of the formation of two liquid phases, reliable induction time experiments could only be carried out for solutions higher than 55% polyol14 and an agitation intensity of 1200 rpm. The supersaturations applied were limited to S e 1.6 because gas bubbles forming in the cooling process disturbed the detection of nuclei. Because of the elevated viscosity of the solution, the bubbles remained in solution for several minutes (55% polyol) up to hours (70% polyol). Hence, only supersaturations could be investigated that lead to an induction time longer than the time bubbles disappear. The results are given in Figure 5. The plot indicates increased slopes or interfacial tension with increasing polyol content. Values for σnl are listed in Table 2. Again the IFT obtained for the static system is higher than the corresponding value of the stirred system. Before we compare these values to the reference data, we must consider that the literature values have been obtained using activity-based su-

persaturations (Sa). Therefore, our measurements have been reevaluated using Sa by means of mean ionic activity coefficients (eq 5) obtained from a thermodynamic model recently published.14 The σnl values for the polyol on a Sa basis are of the same order of magnitude like the literature values. However, the transition from heterogeneous to homogeneous mechanisms is shifted to higher supersaturations with increasing content of organic admixture.13 This promotion of heterogeneous nucleation must also be considered for the polyol. From the literature values, we expect that for 55% and 70% polyol the transition would be around Sa ) 1.8 and Sa ) 2.0, respectively. All supersaturations applied in this work are well below these values. The observed nucleation for the experiments with polyol addition are therefore very likely of heterogeneous nature. Hence, the true values for σnl are expected to be higher. However, induction time measurements for high amounts of organic additive in the homogeneous region are practically impossible with our apparatus since nucleation would be instantaneous. 3. Drop Shape Analysis 3.1. Theory. Instead of calculating the IFT indirectly from induction time or growth rate experiments, we will attempt a direct measurement of this physical property in the following. From wetting experiments of saturated KDP/water/polyol solutions on KDP crystals, the interfacial tension (IFT) between the prismatic crystal surface and the solution σcl can be obtained. The prismatic faces of KDP have shown to be terminated by both K+ and H2PO4- ions in alternation, whereas the pyramid face (101) is terminated with K+ ions only.32 This is why wetting experiments on the prismatic face are assumed to be more representative for the IFT of the nucleus/solution interface if curvature-dependence can be neglected: σcl ≈ σnl

(10)

In fact, the curvature of small nuclei normally reduces σcl by not more than approximately 25% provided that the Tolmans theory holds.27 This deviation would be tolerable considering the error expected for the experimental determination of σnl. A major prerequisite is that the solution only partly wets the crystal, hence forming a contact angle on the surface (Figure 6). In this case, the physical relationship between the contact angle θ and σcl can be expressed by the Young equation:33 σcl ) σcg - cos θσlg

(11)

Surface energies (SFE) of the crystal-gas (σcg) and the surface tensions (SFT) of the liquid-gas (σlg) interface as well as the contact angle (θ) can be evaluated directly from drop shape analysis (DSA). σlg can be calculated from the shape of a pendant drop (Figure 6) with the Young-Laplace method.33 θ

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2665

Figure 7. Left: Solubility of KDP in polyol/water mixtures at 30 °C. Right: Density of saturated KDP/water/polyol solutions at 30 °C.

because of the small contact angles observed for the present system the error in cos θ may be negligible because d(cos θ)/ dθ is small. The determination of σcg is somewhat more complicated and is usually derived from so-called component approaches. The most common methods postulate interfacial tensions to be composed of dispersive (superscript D) and polar (superscript P) components. σij ) σDij + σPij

Figure 8. Top: Liquid surface tension of saturated KDP/water/polyol solutions at 30 °C. Bottom: Contact angles of saturated KDP/water/polyol/ saturated solutions on a planar prismatic KDP crystal surface at 30 °C.

is also directly measured via image analysis of the sessile drop on the crystal surface. However, in presence of a surface-active organic additive the question arises whether adsorption effects may disturb contact angle measurements. In fact, the polyol forms hydrogen bonds with the H2PO4- ions of the prismatic faces (above), which leads to retardation of growth units.19 For contact angle measurements, the most interesting question is whether polyol is preferably adsorbed as compared to the water molecules. Consequently, the concentration difference between bulk solution and the adsorbed layer may affect the observed overall contact angle. However, because both water and polyol are bound to the KDP prismatic surface via hydrogen bonds, preferential adsorption may not be as significant. Furthermore,

(12)

From contact angle experiments of two or more liquids with D P known σlg and σlg , the components σDcg and σPcg can be found 33 analytically. 3.2. Experimental Procedure. The pendant and sessile drop experiments have been carried out with the drop shape analysis system Kru¨ss DSA10 Mk2 at 30 °C. This apparatus provides live images of pendant or sessile drops, hence allowing for comfortable time dependent analysis. The liquid surface tension or contact angles are calculated from the contours of these images with the software Kru¨ss DSA1 v 1.90 including contour recognition and fitting as well as the Young-Laplace method. 3.2.1. Pendant Drop Experiments. The Young-Laplace evaluation of the pedant drops requires the density of the saturated solution. These were calculated from experimental density data for polyol/water mixtures,16 and the impact of the KDP has been adopted from data measured in water.5 KDP solubilities in water/polyol mixtures and the densities of saturated water/polyol/KDP solutions are illustrated in Figure 7. Needle diameter and drop volume have been adapted casespecifically to obtain long and slender drop shapes as recommended in literature.34 Needle diameters of 1.25 and 0.9 mm have been used for low and high polyol fractions respectively. The corresponding drop volumes were 14 µl and 7-10 µl, respectively. The surface tension is monitored time dependently. Experiments have been carried out in both normal and (water-) saturated air atmosphere. In both cases, the drop shape remains relatively stable ((0.3 mN/m) for the first couple of minutes. After that time, a systematic drop in mixture surface tensions can be observed in most cases due to preferential evaporation of water in normal atmosphere. Under humid conditions, a systematic rise can sometimes be observed, which is most likely related to water uptake. The extent of both gradients depends on the actual polyol/water mixture composition. A good representative procedure turned out to take a 20 s averaged SFT value after 90 s, which assured to be in the stable phase for all experiments.

2666 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009

The B-shape parameter, which is often used as a quality value for pendant drop experiments, varied between 0.57 and 0.61 for the different compositions applied, which is very close to the optimum range (0.6-0.7). These B values correspond to errors of 0.12 and 0.06% respectively regarding the SFT.34 For the statistical evaluation, pendant drop experiments were carried out in two independent experimental series with five repetition drops for every composition. 3.2.2. Sessile Drop Experiments. The KDP crystals used for the contact angle experiments have been carefully grown at low supersaturations from aqueous solutions. This way crystals with a width of 1-2 cm and a length of 2-4 cm could be obtained. The larger of the two prismatic faces has been used for the contact angle experiments. Crystals with a maximum of surface smoothness have been selected. However, on a microscopic level water-grown KDP prismatic surfaces are rather rough, covered with spiral hillocks.5 Static sessile drop experiments have been performed. From different contour fitting procedures tested, circle fitting applied best for the current system. The fitting error expressed in mean deviation between the observed and the fitted drop contour was in between 5-15 µm, which is small considering the sessile drop having a base diameter of 3.5-4.5 mm. For the evaluation of the static contact angle, the superimposed time-dependent effects of sessile drop formation and concentration changes of the mixture (above) must be considered. After charging the drop on the surface, the contact angle decreases rapidly and then slowly approaches an asymptotic value. Because of those concentration effects, however, the slope changes after 5-10 min (e.g., due to water evaporation). To obtain the asymptotic value of static contact angle, the curve is fitted until this inflection point with the empirical equation: θ(t) ) (1 + At)/ (B + Ct), R ) 0.99-0.999 and the value limtf∞ ) A/C is taken as static contact angle. In two experimental series, six different crystals have been applied with five repetition sessile drop experiments on different positions of each crystal. The drop volumes varied between 15 and 25 µL depending on the size of the crystal. In preliminary experiments, the impact of drop volume (10-100 µL) on the contact angle turned out to be negligible when compared to the impact of the different crystals. Because of the different microscopic surface structures, different crystals or different positions on the same crystal lead to strong variations of the contact angles (Figure 8). 3.3. Results. 3.3.1. Surface Tension and Contact Angle. Figure 8 illustrates the results of the pendant and sessile drop experiments for the KDP/water/polyol system. σlg drops steeply when small amounts of polyol are added. Hence, the polyol seems to adsorb on the gas-liquid interface. This behavior is comparable and the steepness somewhat higher when compared to other organic additives such as alcohols or carboxylic acids.35 The surface tension for the saturated KDP/water solution (78 mN/m) is higher than the value without electrolyte (71.2 mN/ m), which is very similar to other aqueous electrolytes such as NaCl/water.35 The slightly winding shape of curve for σlg as function of polyol content has also been observed for other organics in water such as ethylene glycol ethers.36,37 The interplay between KDP solubility (Figure 7) and polyol content may also contribute to this effect. The contact angle measurement of the sessile drops show increased experimental deviations. The main reason for data scatter are very likely small differences in surface smoothness and structure formed during the growth process of each individual crystal. The contact angle almost rises linearly with

increased polyol concentration. For mass fractions of polyol below 20%, the contact angles get very small making reliable measurements infeasible. The saturated KDP/water solution without polyol tends to spread spontaneously. However, the surface is not covered entirely. A so-called pancake is formed, which is characteristic for completely wetting liquids.43 However, the distinction between a real sessile drop and a pancake is sometimes difficult. The experimental SFT data have been fitted with RedlichKister type equations38

[∑

σlg,12 ) σlg,2X2 + σlg,1(1 - X2) + Y2(1 - Y2)

(1 - 2Y2)nan

n

n ) 0, 1, 2, 3, 4

] (13)

and flexible fractions X2 )

w2 w2 + (1 - w2)R1

Y2 )

w2 w2 + (1 - w2)R2

(14)

Thus, seven parameters (R1, R2, a0-a4) have been adjusted to the experimental data showing a reasonable fit (Figure 8). The contact angles have been fitted with a linear relationship on a mass fraction basis: θ12 ) θ2w2 + θ1(1 - w2)

(15)

According to our experimental observations, θ1 is set to zero and θ2 is fitted to the experimental data (Figure 8). 3.3.2. KDP Surface Energy. A difficult task turned out to be the retrieval of suitable reference liquids for the determination of the SFE (σcg). Three prerequisites must be fulfilled for suitability: The liquid must form a contact angle on the KDP surface, the dispersive and the polar part of the liquids surface tension must be known and the liquid must not dissolve the crystal. Among many liquids tested most of them spread on the KDP surface. For example, practically all (mono-) alcohols and alkanes have shown to spread spontaneously. Finally, four liquids have been identified that form small but measurable contact angles on the KDP surface: di-iodomethane (DIM), formamide (FA), glycerol (GCL), and ethylene glycol (EG). Their dispersive and polar SFT have been derived from different sources.39–41 The method of Owens, Wendt, Rabel, and Kaelble (OWRK) is chosen for the evaluation of the SFE of KDP (dispersive and polar interactions).39 Two parameters (σDcg, σPcg) are to be determined. Consequently, at least two reference liquids are required. However, it is recommended to include more reference liquids for parameter regression in order to obtain more reliable parameters.42 Considering several plausbible liquid combinations the SFE is calculated (Table 3): (1) All available reference liquids (DIM, EG, FA, GLC). (2) All available reference liquids (DIM, EG, FA, GLC), where DIM is assumed to be dispersive only.41 (3) FA as one of the liquids with highly scattering experiD P , σlg ) is disregarded. mental values (θ, σlg (4) Best-fit combination of three liquids (DIM, FA, GCN). (5) Two liquids with most reliable experimental values, that is largest contact angles and lowest experimental scattering (DIM, EG). The results are summarized in Table 3 and reveal the large impact of the choice of the reference liquid on dispersive and polar part of the SFE but also the total SFE. Hence, the determination of σcg plays certainly the most critical part when applying the Young relationship (eq 11) along with component approaches such as the OWRK-method. This is especially true for KDP or other salts with comparatively high SFE. However,

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2667 Table 3. Components of the SFT (Literature Values), the Contact Angle on the KDP Surface, and OWRK Calculations Regarding SFE of the KDP/Gas Interface (σcg) SFE according to OWRKa

SFT (literature)

b

DIM DIMc EGd FAe GCLe

D σlg [mN/m]

P σlg [mN/m]

θ [°]

D σcg [mN/m]

P σcg [mN/m]

σcg [mN/m]

R

reference liquids

49.5 ( 1 50.8 30.9 ( 1 39.5 ( 7 37.0 ( 4

1.3 ( 1 0 16.8 ( 1 18.7 ( 7 26.4 ( 4

31.4 ( 1.5

30.53 42.42 42.26 30.09 43.61

25.67 14.16 13.91 31.67 6.11

56.20 56.58 56.17 61.76 49.70

0.933 0.915 0.911 0.997 1

(1) DIMb, EG, FA, GCL (2) DIMc, EG, FA, GCL (3) DIMc, EG, GCL (4) DIMb, FA, GCL (5) DIMc, EG

15.3 ( 3 10 ( 5 8(5

a Approximation according to Stro¨m calculated from measurements according to the method of Owens, Wendt, Rabel, and Kaelble.39 Approximation according to Stro¨m calculated from measurements according to Owens.39 c Approximation according to Stro¨m.41 d Approximation according to Stro¨m calculated from measurements according to Stro¨m.41 e Approximation according to Stro¨m calculated from measurements according to Fowkes.40

b

Figure 9. Comparison between the IFT obtained from induction time experiments (σnl) and from drop shape analysis (σcl).

equation of state methods, which would normally be preferred are only applicable for cases where σcg < σlg, that is predominantly for organic surfaces.42 4. Discussion From eqs 11, 13, and 15, the IFT between the KDP prismatic surface and the saturated KDP/water/polyol solutions (σcl) can be calculated from the DSA experiments. Figure 9 compares these results with the values obtained from the induction time experiments (σnl) discussed earlier. At low fractions of polyol, negative values are obtained for σcl. For these compositions, the equilibrium contact angles were close or equal to zero. In this case, the Young relationship (eq 11) is not applicable anymore. Long-range molecular interactions must be considered when θ approaches zero.43 The courses of σcl are shown for different values of σcg according to Table 3. The σcl values calculated for σcg obtained from the OWRK calculations based on the most reliable reference liquids (49.7 mN/m, Table 3) are closest to σnl. Other combinations of reference liquids (σcg ) 56.2, 61.8 mN/m) would result in significantly higher values of σcl, which would rather correspond to the values obtained from growth rate experiments (Table 1). The impact of polyol content on σcl shows a winding course with a maximum for approximately w2 ) 0.2. Unfortunately, this somewhat questionable shape of curve of the IFT, especially for low-polyol contents, can hardly be verified with induction time measurements because experiments would be restricted to very low supersaturations (S