Improved Methodology to Measure Surface Tension and Its

7726

Langmuir 2005, 21, 7726-7732

Improved Methodology to Measure Surface Tension and Its Application to Polystyrene or Poly(methyl methacrylate) in Styrene Solutions Martha A. Cerpa-Gallegos,† Carlos F. Jasso-Gastinel,† Vicente A. Lara-Valencia,‡ and Luis J. Gonza´lez-Ortiz*,‡ Chemical Engineering Department, University of Guadalajara, Blvd. Gral. Marcelino Garcıá Barraga´ n # 1451, Guadalajara, Jalisco. C.P. 44430, Mexico, and Chemistry Department, University of Guadalajara, Blvd. Gral. Marcelino Garcıá Barraga´ n # 1451, Guadalajara, Jalisco. C.P. 44430, Mexico Received January 12, 2005. In Final Form: June 13, 2005 Several methods to measure surface tension involve some inconveniences when applied to moderate or highly viscous polymer solutions. Therefore, an improved version of the weight drop method (WDM) is proposed here. In addition, a comparative analysis of methods is carried out, including the drop profile (DPM), the selected planes (SPM), the WDM and the one proposed here (WDSM), finding that the WDSM is as easy to apply as the SPM and the WDM, although in practical conditions it is much more accurate than either of them. Moreover, the WDSM allows to reproduce the results that can be obtained using DPM, but, in general, it is much easier to implement and apply than such method. The WDSM was used to determine surface tension in polystyrene or poly(methyl methacrylate) in styrene solutions, where the dependence of such property with polymer average molecular weight and polymer concentration was experimentally evaluated.

1. Introduction The two most common properties associated with interphases are surface and interfacial tensions. Surface tension is a controlling factor in processes involving wetting or coating operations,1 and interfacial tension is an important factor for mechanical properties,2 as well as stability and domain size in polymer-polymer dispersions.2-3 Besides, it is generally accepted that in seeded emulsion polymerizations (SEPs) the final particle morphology depends on the interplay between the three involved interfacial tensions;4-6 that is to say, those occurring between each polymer (1 or 2) and the aqueous phase (σ13 and σ23) and the one between both polymers (σ12). Moreover, such interfacial tensions are required to model the development of particle morphology in SEPs.7-8 It is well-known that σ13 and σ23 are dramatically modified by the presence of emulsifiers, which are commonly used on SEPs.5,6 Besides, some reports have shown that the end group type of polymer chains can also modify such interfacial tensions.9-11 In addition, the three * To whom correspondence should be addressed. Phone: +5233-3619-9920. Fax: +52-33-3619-4028. E-mail: ljglez@ yahoo.com.mx. † Chemical Engineering Department. ‡ Chemistry Department. (1) Koberstein, J. T. In Encyclopedia of Polymer Science and Engineering, 2nd ed.; Mark, H. F., Bikales, N. M., Overberger, C. G., Menges, G., Kroschwitz, J. I., Eds.; Wiley-Interscience: New York, 1987; Vol. 8, p 237. (2) Chen, C. C.; White, J. L. Polym. Eng. Sci. 1993, 33, 923. (3) Lepers, J.-C.; Favis, B. D. AIChE J. 1999, 45, 887. (4) Okubo, M. Makromol. Chem., Macromol. Symp. 1990, 35/36, 307. (5) Chen, Y.-C.; Dimonie, V.; El-Aasser, M. S. J. Appl. Polym. Sci. 1995, 45, 487. (6) Sundberg, D. C.; Durant, Y. G. Macromol. Symp. 1995, 92, 43. (7) Gonza´lez-Ortiz, L. J.; Asua, J. M. Macromolecules 1996, 29, 4520. (8) Mendoza-Fernańdez, S.; Jasso-Gastinel, C. F.; Gonza´lez-Ortiz, L. J. Proceedings of the Annual; Technical Conferences of Society of Plastics Engineers: Chicago, IL, 2004; Vol. L, p 2704. (9) Chen, Y.-C.; Dimonie, V.; El-Aasser, M. S. J. Appl. Polym. Sci. 1991, 42, 1049.

above-mentioned interfacial tensions depend on monomer concentration of the respective phases (at least two polymer phases and the aqueous phase),6,9,12 and in most experimental cases, such concentrations are continuously modified in the course of polymerization. In addition, surface tension of a pure polymer varies with its molecular weight,10,13-14 and interfacial tension between two immiscible polymers depends on their respective molecular weights.11,15-18 Finally, the effect of temperature on surface19,20 or interfacial tension of polymer systems has also been reported.15-18 Using an experimental system similar to the one used on SEPs, and considering the particle equilibrium swelling, interfacial tension measurements between polymer particles and aqueous phase have been proposed.21 Unfortunately, in that work, serious experimental difficulties to obtain accurate interfacial tension values were reported. On the other hand, regardless of the fact that several measurements of interfacial tension between polymer melts have been reported,2-3,16-18 there are few (10) Jalbert, C.; Koberstein, J. T.; Yilgor, I.; Gallagher, P.; Krukonis, V. Macromolecules 1993, 26, 3069. (11) Fleischer, C. A.; Koberstein, J. T.; Krukonis, V.; Wetmore, P. A. Macromolecules 1993, 26, 4172. (12) Winzor, C. L.; Sundberg, D. C. Polymer 1992, 33, 3797. (13) Sauer, B. B.; Dee, G. T. Macromolecules 1991, 24, 2124. (14) Chee, K. K. J. Appl. Polym. Sci. 1998, 70, 697. (15) Jo, W. H.; Lee, H. S.; Lee, S. C. J. Polym. Sci: Part. B: Polym. Phys. 1998, 36, 2683. (16) Demarquette, N. R.; Kamal, M. R. Polym. Eng. Sci. 1994, 34, 1823. (17) Kamal, M. R.; Lai-Fook, R.; Demarquette, N. R. Polym. Eng. Sci. 1994, 34, 1834. (18) Anastasiadis, S. H.; Gancarz, I.; Koberstein, J. T. Macromolecules 1988, 21, 2980. (19) Roe, R.-J. J. Phys. Chem. 1968, 72, 2013. (20) Wu, S. Polymer Interface and Adhesion; Marcel Dekker Inc.: New York, 1982; Chapter 3. (21) Durant, Y. G.; Sundberg, D. C.; Guillot, J. J. Appl. Polym. Sci. 1994, 53, 1469.

10.1021/la050095o CCC: $30.25 © 2005 American Chemical Society Published on Web 07/12/2005

To Measure Surface Tension

Langmuir, Vol. 21, No. 17, 2005 7727

experimental studies for interfacial tension of polymer solutions as a function of solvent concentration.22 Some correlations between interfacial and surface tensions have been developed,20 if one of them is used, the three interfacial tensions could be calculated considering the respective surface tensions of polymers 1 and 2 (as a function of monomer concentration, polymer molecular weight, polymer chain end group type and temperature) and the aqueous phase surface tension (as a function of type and effective concentration of surfactant(s)). Regardless of the wide variety of methods used to measure surface tension,23 just a few of them could be adapted to characterize highly viscous polymer solutions. One of them is called the pendant drop method (PDM), which has been widely used to determine both surface19,24-25 and interfacial tensions,16-18,24 in low viscosity systems25,26 as well as in highly viscous systems.16-19,24 Despite the numerous variations proposed for such a method, they consider that the profile shape of a drop coming out of a capillary tube is governed by the balance between surface tension (interfacial tension when the drop is immersed in another fluid) and gravity forces. The PDM considers that the pressure difference (∆P) across the interface, due to its curvature is given by the Laplace’s equation as follows:23

1 ∆P 1 + ) R1 R2 γ

(1)

where R1 and R2 are the surface orthogonal curvature radii and γ is the surface tension. Based on it, several proposals to evaluate surface or interfacial tension by comparing experimental and theoretical drop profiles have been presented,11,18,26-32 recommending different numerical methodologies to compare such profiles; henceforth, such proposals will be generically called drop profile methods (DPMs). All such methodologies require both complex numerical procedures and very accurate measurements of Cartesian coordinates of drop profile; such requirements are not easy to be fulfilled. A less demanding procedure, called the selected planes method (SPM), was originally proposed by Andreas33 in 1938; since then, it has been used by several authors.16-17,34-37 In such method, the surface tension is calculated by (22) Wagner, M.; Wolf, B. A. Macromolecules 1993, 26, 6498. (23) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; WileyInterscience: New York, 1990; Chapter II. (24) Wu, S. J. Phys. Chem. 1970, 74, 632. (25) Persson, B.; Nilsson, S.; Bergman, R. J. Colloid Interface Sci. 1999, 218, 433. (26) Lin, S.-Y.; Chen, L.-J.; Xyu, J.-W.; Wang, W.-J. Langmuir 1995, 11, 4159. (27) Anastasiadis, S. H.; Chen, J.-K.; Koberstein, J. T.; Siegel, A. F.; Sohn, J. E.; Emerson, J. A. J. Colloid Interface Sci. 1987, 119, 55. (28) Hu, W.; Koberstein, J. T.; Lingelser, J. P.; Gallot, Y. Macromolecules 1995, 28, 5209. (29) Retsos, H.; Margiolaki, I.; Messaritaki, A.; Anastasiadis, S. H. Macromolecules 2001, 34, 5295. (30) Prokop, R. M.; Hair, M. L.; Neumann, A. W. Macromolecules 1996, 29, 5902. (31) Arashiro, E. Y.; Demarquette, N. R. Mater. Res. 1999, 2, 23. (32) Semmler, A.; Ferstl, R.; Kohler, H.-H. Langmuir 1996, 12, 4165. (33) Andreas, J. M.; Hauser, E. A.; Tucker, W. B. J. Phys. Chem. 1938, 42, 1001. (34) Yoon, P. J.; White, J. L. J. Appl. Polym. Sci. 1994, 51, 1515. (35) Menke, T. J.; Funke, Z.; Maier, R.-D.; Kressler, J. Macromolecules 2000, 33, 6120. (36) Harrison, K. L.; Johnston, K. P.; Sanchez, I. C. Langmuir 1996, 12, 2637. (37) Hayami, Y. Colloid Polym. Sci. 1996, 274, 643.

γ)

g∆Fde2 H

(2)

where g is gravitational acceleration, ∆F is the density difference between the drop and its environs, de is the drop equator diameter and H is a constant tabulated as a function of ds/de,23 being ds graphically defined in Figure 1. In addition, other authors have proposed equivalent methods considering other diameters ratios.19,38-39 The relative advantage of the SPM with respect to the DPM is its application simplicity, because in the SPM it is only required to measure ds and de and to carry out some elementary calculations. However, it has been reported that considerable deviations can be expected if inaccurate diameter values are used.19,38-39 On the other hand, more than a century ago, a method requiring the drop weight was proposed (WDM).40 Such a method has been recently used to determine surface41,42 and interfacial9,42-44 tensions. In the WDM, surface or interfacial tension is calculated by Tate’s law

γ)

V∆Fg 2πrf

(3)

where V is the drop volume, r is the radius of the drop transversal section at the end of the capillary tube (at the exact moment when the drop falls from the capillary tube), and f is a factor tabulated as a function of r/RC,23,45 RC being a characteristic dimension defined as V1/3. If it is convenient, the numerator in eq 3 can be directly substituted by the drop weight experimentally determined. As in the SPM, if the involved factor is available, this method can easily be applied. To apply the above-mentioned methods, it is required that the drop reaches its equilibrium shape before it falls from the capillary tube. Therefore, to promote that the drop equilibrium shape could be quickly reached in moderate to highly viscous systems (e.g. polymer solutions, where evaporation should be reduced to avoid considerable concentration changes), it is recommended to use thin capillary tubes, which generally produce drops with r/V1/3 values lower than 0.3. Nevertheless, only one study has been focused on these systems.45 In such study, the weight drop and the surface tension was experimentally determined, and from them, in this work the corresponding f factors were calculated. Unfortunately, such factors show a noticeable dispersion (Figure 2). By the above-mentioned inconveniences, in this work, an improved methodology to measure surface tension is proposed. From now on, such a method is called the weight drop and shape method (WDSM), because it combines the measurements of weight and a shape parameter of pendant drops coming out from a capillary tube. The WDSM is used to determine the effect of molecular weight and concentration of polymer on the surface tension of polystyrene in styrene (PS/S) and poly(methyl methacrylate) in styrene (PMMA/S) solutions. (38) Roe, R.-J.; Bacchetta, V. L.; Wong, P. M. G. J. Phys. Chem. 1967, 71, 4190. (39) Winkel, D. J. Phys. Chem. 1965, 69, 348. (40) Tate, T. Philos. Mag. 1864, 27, 176. (41) Matsuki, H.; Kaneshina, S.; Yamashita, Y.; Motomura, K. Langmuir 1994, 10, 4394. (42) Fu, D.; Lu, J.-F.; Bao, T.-Z.; Li, Y.-G. Ind. Eng. Chem. Res. 2000, 39, 320. (43) Saien, J.; Salimi, A. J. Chem. Eng. Data 2004, 49, 933. (44) Chen, Y.-C.; Dimonie, V. L.; Shaffer, O. L.; El-Aasser, M. S. Polym. Int. 1993, 30, 185. (45) Wilkinson, M. C. J. Colloid Interface Sci. 1972, 40, 14.

7728

Langmuir, Vol. 21, No. 17, 2005

Cerpa-Gallegos et al. 2 dY (1 + Y ) ) - (Mz + B)(1 + Y2)1.5 dz x

Figure 1. Drop dimensions required for selected planes and drop weight and shape methods, and coordinate axes considered in eq 4.

Figure 2. f factors calculated in this work for the Y(0) values indicated in plot, and f factors available in the literature: (2) (ref 23) and (0) (ref 45).

2. Comparative Numerical Analysis of Methods 2.1. Drop Profile Calculation from the Surface Tension Value. To calculate surface Cartesian coordinates of a drop falling out from a vertical capillary tube when surface tension is available (from now on called: Laplacian profiles), in this work the following equation has been considered:

B + Mz )

(x

1

2

x 1 + (x′)

)

x′′ (1 + (x′)2)1.5

(4)

where B and M are parameters defined as

B ) 2/b - FgL/γ

(5)

M ) Fg/γ

(6)

where b is the curvature radius at drop apex (at the apex, the orthogonal curvature radii are equal), F is the drop density, and L is the drop length (defined in Figure 1). Equation 4 was obtained considering Laplace’s equation (eq 1), the mathematical expressions for the orthogonal curvature radii on Cartesians coordinates31 and the pressure due to gravitational effect ()Fg(L - z)), where coordinate axes indicated in Figure 1 were considered; in such figure, the gravitational force is exerted on positive direction of z axis. Mathematical expressions equivalent to eq 4 are the theoretical basis of the DPMs.11,18,26-32 To solve eq 4 numerically, a parameter Y was defined (eq 7), to transform it to the following differential equations system:

dx )Y dz

(7)

(8)

In this work, such a differential equations system was numerically solved applying the fourth order RungeKutta method,46 considering a large number of integration steps (more than 2.6 × 104). To predict the coordinates for a specific system, in addition to liquid properties (F and γ), the B values have to be available. Nevertheless, it was numerically determined that, for each experimental system, only B values within an extremely narrow range predict a drop shape profile, obtaining in each case practically the same profile when using B values within such range. Therefore, the B value used to calculate each system was numerically estimated. Besides, to solve the differential equations system, the border conditions (evaluated at the end of the capillary tube and denoted as x(0) and Y(0)), are required. Due to experimental evidence for tested systems, in this work, x(0) was taken as the external radius of the capillary tube. Nevertheless, different Y(0) values () tan θ; Figure 1) were considered in calculations. By the above-mentioned requirements, the coordinates for an experimental system can be calculated when F, γ, x(0), and Y(0) are available. 2.2. Equivalence Analysis of the Selected Planes and Drop Profile Methods. As a way to validate the SPM, the equivalence between such a method and the DPM was tested. For such objective, the Laplacian profiles of >103 hypothetical systems were numerically obtained with the above-mentioned program, considering each one a specific value of surface tension (γd). Then, the coordinates for each hypothetical system were used to estimated, with the SPM, the respective surface tension value (γSPM). For each tested system, a confrontation between both surface tension values (γd and γSPM) was done. Such systems were defined considering the values indicated in Table 1. In all cases, the H parameter required in eq 2 was interpolated from reported data,23 using Lagrange’s interpolation formula.47 In addition, accurate values of de and ds diameters and density were used. 2.3. Experimental Imprecision Sensitivity Analysis for the Selected Planes Method. As experimental determination of de and L values usually involve a noticeable experimental error, an experimental error sensitivity analysis was performed. For such an objective, the SPM was also used to calculate surface tension of the nine hypothetical systems described in Table 2, which were selected attending to their different drop shapes. To simulate experimental imprecision, the respective values of de and L obtained with the above-mentioned program were substituted by 10 hypothetical values that fulfill the following conditions: (a) their average is the variable value (de or L) numerically obtained, (b) their assigned deviation parameter (p; defined as the ratio between their standard deviation and their average value) is 0.005 or 0.01, and (c) the assigned values to the variables resemble a Gaussian Distribution. In Table 2, it can be noticed that for each one of the nine hypothetical systems, the two levels of imprecision (p ) 0.005 or 0.01) were considered. In addition, the SPM calculation procedure requires localization of zl coordinate (position where ds diameter must be graphically read; indicated in Figure 1) and to obtain graphically the ds value. Therefore, the experi(46) Burden, R. L.; Faires, J. D.; Reynolds, A. C. Numerical Analysis, 2nd ed.; Weber & Schmidt: Boston, 1981; Chapter 5. (47) Nakamura, S. Applied Numerical Methods with Software, 1st ed.; Prentice Hall: New York, 1991; Chapter 2.


Langmuir, Vol. 21, No. 17, 2005 7729

Table 1. Experimental Conditions Defining the Considered Hypothetical Systems experimental condition

F (kg/m3)

γ × 103 (N/m)

x(0) × 103 (m)

considered values

800 850 900 950 1000

5 15 25 35

0.1 0.2 0.26 0.3 0.4 0.5 0.6 0.7 0.8

Y(0) -0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.50 0.65 0.80

Table 2. Experimental Conditions Defining the Nine Hypothetical Systems and Their Approximated Drop Profile

mental errors that could be involved when measuring these values were also considered. To estimate the effect of such errors on the calculated surface tension value, the exact values of zl and ds (evaluated as the SPM indicates) were arbitrary multiplied by a factor that was randomly selected from a large set of values that fulfill the following conditions: (a) their average is the unit, (b) their standard

deviation is p, and (c) their values resemble a Gaussian distribution. In that way, for each experimental condition, a set of one hundred surface tension values was obtained using the SPM and the error simulation methodology before described. Such set is represented by the following surface tension ratios: (a) γ6/γd, (b) γ j /γd, and (c) γ95/γd, where the numerical subscripts denote the order number of the specific datum in the list containing the obtained surface tension values ordered in an increasing way, γ j being the average of such data set. 2.4. Equivalence Analysis of the Weight Drop and Drop Profile Methods. To evaluate the equivalence between the WDM and the DPM, the Laplacian profiles (calculated as it is indicated in section 2.1.) corresponding to >103 systems that were defined considering data shown in Table 1 were used to calculate the volume of each simulated drop. For such, at least 26 000 drop slices, whose volumes were estimated using the trapezoidal rule48 were considered. The f factor for each experimental condition was then evaluated using eq 3 and the above-mentioned volume; f factors are presented in Figure 2. From the comparison between calculated f factors and those available from the literature,23,45 the equivalence between both methods was analyzed. 2.5. Weight Drop and Shape Method. The WDSM proposed in this work is a direct consequence of the equivalence analysis just described, and it can be considered as an improved version of the WDM. There, the f factors are dependent on the characteristic Y(0) value of each system and the equivalence between its results and the ones obtained with DPM is warranted when accurate measurements are used. Therefore, when such a value is available, the application of the WDSM to estimate surface tension is similar to the WDM, with the only difference being that the factor to be used must be obtained from the corresponding curve in Figure 2, rather than from interpolating or extrapolating available data.23,45 This circumstance allows the obtention of an improved surface tension measurement, where the error involved to use an inexact f factor (independent of the characteristic Y(0) value of the system) is eliminated. 2.6. Experimental Imprecision Sensitivity Analysis for Weight Drop and Shape Method. To compare the capabilities of the SPM and the WDSM, in this section a sensitivity analysis equivalent to the one proposed for the SPM was performed, by considering the WDSM rather than the SPM when calculating surface tension. From calculations performed on sensitivity analysis for the SPM, it was numerically obtained that, for a given system, the Y(0) value is univocally related to 2L/de; therefore, such ratio can be used to estimate the Y(0) value required for the WDSM. Since the 2L/de ratio can be obtained from the same data required by the SPM, but not the Y(0) value, to compare adequately the relative imprecision involved in both methods (the WDSM and the SPM), to calculate surface tension considering inexact values of de and L, the 2L/de ratio was used to estimate the characteristic Y(0) value. A comparative analysis of surface tension ratios calculated using both methods was performed; the systems described in Table 2 were analyzed, considering in both methods an identical set of de and L pairs. 2.7. Numerical Analysis Results. 2.7.1. Capabilities Comparison between the Selected Planes and Drop Profile Methods. Results of (48) Nakamura, S. Applied Numerical Methods with Software, 1st ed.; Prentice Hall: New York, 1991; Chapter 4.

7730


Cerpa-Gallegos et al.

Table 3. Molecular Weight Averages (M h ) of Polymer Samples experimental technique membrane osmometry

gel permeation chromatography

light scattering

viscosimetry

M h Mn × 10-3 (g/mol)

sample PS19 PS44 PS110 PS220 PS550 PMMA53 PMMA100 PMMA180 PMMA300 PMMA470

Mn × 10-3 (g/mol)

Mw × 10-3 (g/mol)

Mw × 10-3 (g/mol)

Mv × 10-3 (g/mol)

18.1 41.2 106 201 540 48.9 92.5 165 292 441

19.3 44.1 111 223 561 52.7 101 178 298 467

18.7 44.0 110 212 560 53.0 102 177 307 474

21.0 45.4 102 224 573

197

Table 4. Surface Tension Ratios Calculated with the Selected Planes Method (SPM) or the Weight Drop and Shape Method (WDSM) and the Error Simulation Methodology Proposed in This Work, for Systems Described in Table 2 SPM

WDSM

system

γ6/γd

γ j /γd

γ95/γd

γ6/γd

γ j /γd

γ95/γd

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.911 0.955 0.932 0.880 0.964 0.914 0.975 0.890 0.958 0.929 0.887 0.941 0.994 0.925 0.961 0.936 0.950 0.959

0.982 1.012 0.975 0.988 1.013 0.995 1.013 0.993 1.011 1.018 1.003 1.040 1.018 1.017 1.007 1.020 1.009 1.047

1.025 1.082 1.023 1.099 1.073 1.102 1.067 1.066 1.072 1.154 1.106 1.189 1.042 1.085 1.055 1.161 1.080 1.183

0.994 0.989 0.996 0.988 0.988 0.988 0.991 0.991 0.990 0.978 0.989 0.980 0.993 0.985 0.995 0.990 0.981 0.961

1.002 0.999 1.003 1.003 0.998 1.001 0.999 1.002 0.999 1.000 1.000 0.997 0.998 0.998 0.999 0.999 0.998 0.987

1.010 1.007 1.009 1.014 1.006 1.011 1.007 1.018 1.007 1.010 1.008 1.005 1.004 1.016 1.002 1.003 1.022 1.007

the equivalence analysis of the SPM and the DPM show that, when accurate measurements are used, the SPM reproduces with negligible differences (smaller than 0.1%), the surface tension values that could be estimated by means of the DPM. Therefore, at those circumstances, the SPM can be considered a method as accurate as the DPM to determine surface tension, but experimentally much more simple. Nevertheless, if the error simulation methodology proposed here is considered, and the effect of experimental imprecision on surface tension calculated with the SPM is analyzed, the surface tension ratios shown in Table 4 are obtained. Such ratios indicate that an imprecision level as low as 0.005 (p value considered in odd systems) could produce an imprecision as high as 11% on their surface tension value calculated with the SPM (obtained for system 11, Table 4), and an imprecision level of 0.01 (p value considered in even systems) could involve errors as high as 19% (obtained for system 12, Table 4). In addition, Table 4 shows that, for tested cases, the surface tension values calculated with the SPM tend to be higher than the preassigned surface tension values; that is, the γ j /γd ratios tend to be higher than 1. Therefore, when the SPM is used, overestimated surface tension values could be expected. 2.7.2. Comparison among the Weight Drop, Weight Drop and Shape, and Drop Profile Methods. Figure 2 shows the f factors numerically calculated and the ones

available on literature23,45 as a function of r/RC. There, it can be noticed that, regardless of the different F, γ, and x(0) values used to evaluate f factors, they are located in the same master curve, provided that the same Y(0) value had been used. Therefore, in Figure 2, there is one master curve for each one of the 12 Y(0) values considered. In addition, results in Figure 2 show that the equivalence between the WDM and the DPM results is only possible for one specific Y(0) value; therefore, in this work, it was assumed that such a Y(0) value would have to be the one experimentally obtained. As a consequence of such results, the WDSM was proposed, representing an improved version of the WDM that allows to reproduce the results that can be obtained by means of the DPM. Moreover, in Figure 2, it is clearly shown that, for a specific case, the involved error as a consequence of using the WDM rather than WDSM, is noticeably dependent on the specific r/RC ratio, and on the difference between the Y(0) real value and the Y(0) value considered to evaluate the f factors reported on the literature.23,45 However, a priori, significative errors can be expected when the WDM is used. 2.7.3. Capabilities Comparison among the Weight Drop and Shape, the Selected Planes, and the Drop Profile Methods. Surface tension ratios that represent the set of surface tension values calculated by means of the WDSM for each one of systems defined in Table 2 are shown in Table 4. There, it can be observed that, in general, the odd systems (p ) 0.005) show an imprecision between 0.5 and 1.5% on surface tension measurement and, except for system 18, the rest (p ) 0.010) shows an imprecision lower than 2.5%. Besides, Table 4 shows γ j /γd values very close to 1. Comparison of results shown in Table 4 for the SPM and the WDSM allows us to establish that the WDSM is, under tested conditions, much more accurate than the SPM and, from the practical point of view, it can be considered as accurate as the DPM, but experimentally it is much easier to carry it out. 3. Experimental Measurements 3.1. Materials. As solvents, purified water from Pisa Farmaceútica Mexicana, toluene from Fermont (purity >99.7%), benzene from Productos Quı´micos Monterrey (purity >99.9%), cyclohexane analytical grade from Jalmek (purity >99.0%), and styrene from Aldrich (purity > 99%), were used as received. Polymer samples described in Table 3 (standards with narrow molecular weight distribution from Scientific Polymer Products) were used to prepare polymer solutions as they were acquired. 3.2. Experimental Procedure. First, to experimentally demonstrate the WDSM capabilities, the surface tensions of several solvents (water, toluene, benzene, cyclohexane and styrene) were measured at 20 °C, and a comparison between


Langmuir, Vol. 21, No. 17, 2005 7731

Figure 3. Schematic description of experimental equipment Table 5. Experimental Y(0) and δ Values for Different Pairs Solvent-Capillary Tube material

solvent

external diameter × 103 (m)

Y(0)

δ (%)

stainless steel stainless steel glass stainless steel stainless steel stainless steel stainless steel

styrene styrene styrene water toluene benzene cyclohexane

0.52 1.64 1.56 0.52 0.52 0.52 0.52

0.25 0.18 0.0 0.16 0.25 0.21 0.19

4 3 6 3 6 3 0

measured values and those previously reported was done.49 Afterward, separate sets of PS/S and PMMA/S solutions were prepared, including polymer concentrations as high as 50%, although only surface tension values of polymer solutions that could pass through the capillary tube and did not form a film around it (ejecting symmetrical drops) were reported. Finally, to estimate how much error can be expected when the WDM is used instead of the WDSM, the polymer solutions above-described and some solvents (water, toluene, benzene, cyclohexane and styrene) were experimentally characterized, considering both methods (that is, γWDM and γWDSM were measured); for styrene, capillary tubes with different diameters and manufactured with different materials were used. Then, the percentage difference between γWDM and γWDSM (δ) was evaluated; the δ values are shown in Table 5. Unless otherwise indicated, surface tension measurements were performed using a stainless steel capillary tube with 5.2 × 10-4 m of external diameter. Temperature was controlled to the desired value (0.5 °C. For the measurements, a digital balance (Sartorious BP211D), with 1 × 10-5 g accuracy capability, and a digital camera (Sony Handycam vision) with zoom of 330×, were adapted to a contact angle tensiometer (Kruss G10) to determine respectively the drop weight and the Y(0) value required to calculate surface tension. To favor that such experimental parameters were measured on drops at their equilibrium shape, a controlled low flow rate was used (smaller than 0.3 mL/hour). A schematic description of experimental equipment is presented in Figure 3. For each system, surface tension of more than 20 drops were measured, and with such values the respective confidence interval

(49) Dean, J. A., Ed.; Lange’s Handbook of Chemistry, 13th ed.; McGraw-Hill: New York, 1985; Chapter 10.

Figure 4. Surface tension values for polystyrene or poly(methyl methacrylate) in styrene solutions, measured at 25 °C, using the weight drop and shape method. Polymer samples: (a) PS19 (-)-), PS44 (-9-), PS110 (-4-), PS220 (-b-), PS550 (-×-); (b) PMMA53 (-)-), PMMA100 (-9-), PMMA180 (-4-), PMMA300 (-b-), PMMA470 (-×-). was estimated;50 there, percentiles of Student’s t distribution for two-tail test and confidence level of 0.95 were considered.51 3.3. Experimental Results. An excellent agreement (deviations lower than 0.6%) between this work measurements and surface tension values previously reported for tested solvents was obtained,47 demonstrating from the experimental point of view, the WDSM capabilities. Surface tension dependence of PS/S and PMMA/S solutions with molecular weight and polymer concentration can be observed in Figure 4. For these solutions, confidence intervals experimentally determined were lower than 1% of their respective surface tension average values. For both experimental systems (PS/S and PMMA/S), surface tension shows a direct relationship with both polymer concentration and polymer average molecular weight. The variation of surface tension with polymer concentration increases as the polymer average molecular weight increases. For polymer solutions experimentally tested, to compare surface tension values calculated with the WDM and the WDSM, relatively small differences between them were obtained (in general, smaller than 2%). Nevertheless, such behavior cannot be generalized a priori for all experimental systems. To evidence it, the systems shown in Table 5 were additionally characterized. There, it can be noticed that, in addition to the type of substance, the δ values are strongly dependent on the material and external diameter of the capillary tube used. For tested systems, deviations as high as 6% were obtained. Although such deviation could be accepted in cases where an additional advantage can be obtained (e.g., application simplicity or save of experimental time), it cannot be accepted when there are not foreseen important advantages. (50) Dixon, W. J.; Massey, F. J., Jr. Introduction to Statistical Analysis, 3rd ed.; McGraw-Hill: New York, 1969; Chapter 6. (51) Evans, M.; Hastings, N.; Peacock, B. Statistical Distributions, 3rd ed.; Wiley-Interscience: New York, 2000; p 218.

7732


4. Conclusions If accurate experimental measurements are performed, the SPM can be considered a method as accurate as the DPM to determine surface tension. However, when using inexact values of de and ds, considerable differences on surface tension values calculated with both methods were obtained. Regardless of the simplicity of the WDSM, as small differences on surface tension values calculated with the DPM and the WDSM were obtained, the superior accuracy of the WDSM compared to SPM at practical conditions has been shown here. Moreover, the WDSM can be considered as an improved version of the WDM, because it guarantees equivalence with DPM results and improves the accuracy of WDM. On the other hand, in tested polymer solutions, a direct relationship of surface tension with polymer average molecular weight and polymer concentration has been shown.

Cerpa-Gallegos et al.

Acknowledgment. Conacyt scholarships for M.A. C.G. (No. 117213) and V.A. L.-V. (No. 177101) and financial support by Conacyt (Projects 3338-P, 28258-U, and 39808Y) are gratefully appreciated. Abbreviations DPM: Drop profile method PDM: Pendant drop method PMMA: Poly(methyl methacrylate) PS: Polystyrene S: Styrene SEP: Seeded emulsion polymerization SPM: Selected planes method WDM: Weight drop method WDSM: Weight drop and shape method LA050095O

Improved Methodology to Measure Surface Tension and Its

Recommend Documents