pubs.acs.org/Langmuir © 2010 American Chemical Society
New Theoretical Framework for Designing Nonionic Surfactant Mixtures that Exhibit a Desired Adsorption Kinetics Behavior Srinivas Nageswaran Moorkanikkara and Daniel Blankschtein* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States Received January 11, 2010 How does one design a surfactant mixture using a set of available surfactants such that it exhibits a desired adsorption kinetics behavior? The traditional approach used to address this design problem involves conducting trial-and-error experiments with specific surfactant mixtures. This approach is typically time-consuming and resource-intensive and becomes increasingly challenging when the number of surfactants that can be mixed increases. In this article, we propose a new theoretical framework to identify a surfactant mixture that most closely meets a desired adsorption kinetics behavior. Specifically, the new theoretical framework involves (a) formulating the surfactant mixture design problem as an optimization problem using an adsorption kinetics model and (b) solving the optimization problem using a commercial optimization package. The proposed framework aims to identify the surfactant mixture that most closely satisfies the desired adsorption kinetics behavior subject to the predictive capabilities of the chosen adsorption kinetics model. Experiments can then be conducted at the identified surfactant mixture condition to validate the predictions. We demonstrate the reliability and effectiveness of the proposed theoretical framework through a realistic case study by identifying a nonionic surfactant mixture consisting of up to four alkyl poly(ethylene oxide) surfactants (C10E4, C12E5, C12E6, and C10E8) such that it most closely exhibits a desired dynamic surface tension (DST) profile. Specifically, we use the Mulqueen-Stebe-Blankschtein (MSB) adsorption kinetics model (Mulqueen, M.; Stebe, K. J.; Blankschtein, D. Langmuir 2001, 17, 5196-5207) to formulate the optimization problem as well as the SNOPT commercial optimization solver to identify a surfactant mixture consisting of these four surfactants that most closely exhibits the desired DST profile. Finally, we compare the experimental DST profile measured at the surfactant mixture condition identified by the new theoretical framework with the desired DST profile and find good agreement between the two profiles.
1. Introduction Applications such as inkjet printing,1 pesticide sprays,2 detergents,3,4 and foam and emulsion generation5-7 all involve the rapid formation of fluid/fluid interfaces. Surfactants are used in these applications in order to stabilize the freshly formed interfaces by adsorbing at a desired rate. In these applications, the kinetics of surfactant adsorption is expected to play a significant role in determining the performance and effectiveness of the *To whom correspondence should be addressed. Tel: (617) 253-4594. Fax: (617) 252-1651. E-mail:
[email protected]. (1) Howe, A. M. Some aspects of colloids in photography. Curr. Opin. Colloid Interface Sci. 2000, 288–300. (2) Chang, C.-H.; Franses, E. I. Adsorption dynamics of surfactants at the air/ water interface: a critical review of mathematical models, data, and mechanisms. Colloids Surf., A 1995, 100, 1–45. (3) Zimoch, J.; Tranthnigg, B.; Hreczuch, W.; Meissner, J.; Bialowas, E.; Szymanowski, J. Detergency and dynamic surface tension reduction of oxyethylated fatty acid methyl esters. Tenside Surfactants Detergents 2002, 39, 8–16. (4) Villis, B. The dissolving behavior of surfactants in household washing machines. Tenside Surfactants Detergents 2000, 37, 52–56. (5) Bikerman, J. J. Foams; Springer-Verlag: New York, 1973. (6) Garrett, P. R.; Gratton, P. L. Dynamic surface tensions, foam and the transition from micellar solution to lamellar phase dispersion. Colloids Surf., A 1995, 103, 127–145. (7) Schr€oder, V.; Behrend, O.; Schubert, H. Effect of dynamic interfacial tension on the emulsification process using microporous, ceramic membranes. J. Colloid Interface Sci. 1998, 202, 334–340. (8) Bergink-Martens, D. J. M.; Frens, G. Dynamic surface tension of a detergent solution in its relation to washing performance. Tenside, Surfactants, Deterg. 1997, 34, 263–266. (9) Stevens, P. J. G.; Kimberly, M. O.; Murphy, D. S.; Policello, G. A. Adhesion of spray droplets to foliage: The role of dynamic surface tension and advantages of organosilicone surfa ctants. Pestic. Sci. 1993, 38, 237–245. (10) Buzzacchi, M.; Schmiedel, P.; von Rybinski, W. Dynamic surface tension of surfactant systems and its relation to foam formation and liquid film drainage on soli d surfaces. Colloids Surf., A 2006, 273, 47–54.
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surfactant formulations.3,4,8-10 The current procedure for designing surfactant formulations in these applications involves11,12 (i) identifying constituent surfactants that could be mixed to generate the formulation, (ii) preparing several sample formulations by mixing the constituent surfactants at different concentrations, and (iii) measuring the adsorption kinetics behavior of each mixture in order to screen for the mixture that exhibits a desired adsorption kinetics behavior. Tools to help design surfactant mixtures are much less developed/unavailable, in part because of the complex adsorption kinetics behaviors exhibited by surfactant mixtures. For example, ref 13 discusses several interesting dynamic surface tension (DST) profiles that can be attained with binary surfactant mixtures depending on the molecular structures of the two surfactants and their bulk solution concentrations. Considering the large number of commercially available surfactant components (e.g., alkyl poly(ethylene oxide), CiEj, nonionic surfactants, n-alcohols, and sodium dodecyl sulfate) that can potentially be chosen to prepare the surfactant mixture, the time and effort required to design surfactant mixtures that meet a desired adsorption kinetics behavior can be substantial. With the above in mind, in this article we explore a new theoretical framework to identify a surfactant mixture that closely satisfies a desired adsorption kinetics behavior. Specifically, we (11) Miller, D. Dynamic surface tension: industrial applications and characterization of commercial surfactants. Tenside, Surfactants, Deterg. 2005, 42, 204–209. (12) Oetter, G. Dynamic properties of surfactants: influence on processes and technologies. Tenside, Surfactants, Deterg. 2005, 42, 168–174. (13) Mulqueen, M.; Stebe, K. J.; Blankschtein, D. Dynamic interfacial adsorption in aqueous surfactant mixtures: theoretical study. Langmuir 2001, 17, 5196– 5207.
Published on Web 11/16/2010
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pose the need to attain a desired adsorption kinetics behavior as an optimization problem using an adsorption kinetics model and identify the surfactant mixture that most closely exhibits the desired adsorption kinetics behavior by solving the optimization problem using a commercial optimization package. The reliability of the surfactant mixture identified using the proposed framework depends on the predictive accuracy and reliability of the adsorption kinetics model used. Accordingly, one may need to validate the predictions by experimentally measuring the adsorption kinetics behavior of the surfactant mixture identified by the proposed framework, including making any necessary refinements. To demonstrate the effectiveness of the proposed framework, we use it to identify a nonionic surfactant mixture consisting of four poly(ethylene oxide), CiEj, surfactants, namely, C10E4, C12E5, C12E6, and C10E8, that most closely satisfies a desired DST profile. We utilize the MulqueenStebe-Blankschtein (MSB) adsorption kinetics model,13 which is applicable to nonionic surfactant mixtures at premicellar surfactant concentrations, and the SNOPT commercial optimization package14 to develop the framework. The remainder of this article is organized as follows. In section 2, we present the new theoretical framework used to design a nonionic surfactant mixture that most closely meets a desired DST profile. In section 3, we demonstrate the reliability and effectiveness of the developed framework through a realistic case study by identifying a surfactant mixture consisting of up to four CiEj nonionic surfactants. In section 4, we conclude and discuss applications and possible extensions of the proposed framework. In Appendix A, we discuss the four specific inputs needed to formulate the optimization problem associated with the new theoretical framework. In Appendix B, we provide information about the numerical convergence behavior exhibited by SNOPT in the case study considered. Finally, we provide general information on SNOPT and how it interfaces with the adsorption kinetics model to iteratively solve the optimization problem associated with the proposed framework as part of the Supporting Information.
2. New Theoretical Framework Used to Design Nonionic Surfactant Mixtures 2.1. Formulation of the Optimization Problem. The need to identify a nonionic surfactant mixture that most closely satisfies a desired DST profile can be posed as an optimization problem through the following specifications: (i) Choosing an objective function that quantifies the deviation between the desired DST profile and the DST profile corresponding to a specific surfactant mixture. (ii) Defining the optimization decision variables as the bulk solution concentrations of the individual surfactants comprising the mixture. (iii) Relating the decision variables to the objective function using an adsorption kinetics model that is applicable to surfactant mixtures. (iv) Including any constraints on the decision variables (see ii above) to account for limitations on the model used (see iii above). For example, if the model is applicable only over specific ranges of concentrations of the individual surfactants comprising the formulation, then these ranges can be specified as constraints on the decision variables. For a detailed discussion of specifications i-iv above, see Appendix A. In addition, for the specific values of these specifications used in the case study presented in section 3, see Table 5. In this article, we use the MSB adsorption kinetics model to relate the decision variables to the objective function (see specification (14) Gill, P. E.; Murray, W.; Saunders, M. A. SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 2005, 47, 99–131.
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iii above), that is, to predict the DST profile at a given surfactant mixture condition.13 Note that the MSB adsorption kinetics model describes the adsorption kinetics of nonionic surfactant mixtures at premicellar surfactant concentrations.13 Specifically, the MSB adsorption kinetics model assumes that the diffusion of surfactant molecules from the bulk solution to the subsurface controls the overall rate of surfactant adsorption and (i) uses Fick’s law to model the diffusive transport of surfactant molecules from the bulk solution to the subsurface and (ii) uses the molecular equilibrium adsorption isotherm (EAI) developed by Nikas et al.15 to model the equilibrium between the subsurface and the surface. The MSB adsorption kinetics model requires as inputs the following molecular parameters for each surfactant i comprising the nonionic surfactant mixture:13 (i) ai, the head cross-sectional area of surfactant i, (ii) Bij (j = 1,...,n), the second virial coefficient between surfactant i and surfactant j, (iii) Δμh0i , the difference between the standard-state chemical potentials of surfactant i at the interface and in the bulk solution, and (iv) Di, the diffusion coefficient of surfactant i. The parameter ai for each surfactant and the parameters Bij for each surfactant pair can be estimated on the basis of the molecular structures of the nonionic surfactants comprising the mixture.15 The parameter Δμh0i for each surfactant can be estimated on the basis of a single equilibrium surface tension data point for each surfactant comprising the mixture.15 Finally, the parameter Di for each surfactant can be estimated using the Wilke-Chang correlation.13,16 2.2. Solution of the Optimization Problem. Once the optimization problem is defined through the steps outlined in section 2.1, one can use a variety of commercial optimization packages to solve the optimization problem. In this article, we have chosen the SNOPT optimization package because it has been shown to be effective in numerically solving several types of nonlinear optimization problems in a variety of areas.14 Note that whereas the formulated optimization problem seeks to find the global optimal solution (the surfactant mixture that “most closely” exhibits the desired DST profile), the optimal solution found using SNOPT (and other commercial packages such as MINOS and KNITRO) corresponds to a local optimal solution that may be globally optimal.14,17,18 A common procedure for increasing the likelihood of finding the global optimal solution is to find the local optimal solutions starting at randomly chosen initial guesses for the optimal solution.19 If one observes different local optimal solutions for different initial guesses, then the objective function values corresponding to each of the local optimal solutions need to be compared to determine the likely global optimum. Accordingly, we repeatedly applied SNOPT for different (randomly chosen) initial guesses for the surfactant mixture conditions and compared the optimal surfactant mixture conditions identified in each case in order to determine the globally optimal surfactant mixture that most closely exhibits the desired DST profile. (We provide general information on SNOPT and how it interfaces with the adsorption kinetics model to iteratively solve the optimization problem associated with the proposed framework as part of the Supporting Information.) (15) Nikas, Y. J.; Puvvada, S.; Blankschtein, D. Surface tensions of aqueous nonionic surfactant mixtures. Langmuir 1992, 8, 2680–2689. (16) Wilke, C. R.; Chang, P. Correlation of diffusion coefficients in dilute solutions. AIChE J. 1955, 1, 264–270. (17) Murtagh, B. A.; Saunders, M. A.. MINOS 5.5 User Guide; Systems Optimization Laboratory, Department of Operations Research: Stanford University, July 1998. (18) Waltz, R. A.; Nocedal, J. KNITRO User’s Manual: Technical Report OTC 2003/05. Optimization Technology Center: Northwestern University, Evanston, IL, April 2003. (19) Bertsekas, D. P. Nonlinear Programming; Athena Scientific: Belmont, MA, 1999.
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Table 1. Values of Head Cross-Sectional Areas (ai), Standard-State Chemical Potential Differences (Δμh0i ), Diffusion Coefficients (Di), and Binary Interaction Parameters (Bij) for the Four CiEj Nonionic Surfactants Considered ai surfactant i (A˚2) C10E4 C12E5 C12E6 C10E8
1 2 3 4
29.6 36.6 42.0 47.9
Δμ0i (RT) -17.35 -22.08 -23.17 -21.29
Bij (A˚2) Di 106 (cm2/s) C10E4 C12E5 C12E6 C10E8 4.3 4.2 3.4 3.7
-173
-140 -108
-145 -92 -77
-109 -69 -62 -47
As an alternative, one could consider using multiple optimization solvers to increase the confidence of the optimal solution identified. The reliability of the surfactant mixture identified by the proposed framework to exhibit the predicted DST profile is limited by the accuracy of the model used to describe the adsorption kinetics behavior (specification iii in section 2.1). Therefore, one may need to measure the DST profile experimentally at the (globally optimal) surfactant mixture condition identified by the proposed framework to validate the predictions made.
3. Demonstration of the Reliability of the New Theoretical Framework In this section, we demonstrate the reliability of the framework presented in section 2 by identifying a nonionic surfactant mixture that most closely satisfies a desired DST profile. For this purpose, we consider a realistic case study by (i) generating a DST profile corresponding to a specific surfactant mixture consisting of up to four CiEj nonionic surfactants, C10E4, C12E5, C12E6, and C10E8, using the MSB adsorption kinetics model, (ii) considering the DST profile generated in (i) as the desired DST profile and applying the framework presented in section 2 starting with completely random initial guesses for the surfactant mixture conditions, and (iii) comparing the optimal surfactant mixture condition identified in (ii) with the original surfactant mixture used to generate the DST behavior in (i). If the developed framework is able to identify the surfactant mixture that was originally used to generate the desired DST profile, then we may conclude that the new theoretical framework is reliable (and effective) in designing the nonionic surfactant mixture condition subject to the accuracy of the adsorption kinetics model chosen. Note that we have considered C10E4, C12E5, C12E6, and C10E8 nonionic surfactants because the MSB adsorption kinetics model parameters are available for these four nonionic surfactants20,21,15 (Table 1). 3.1. Specification of a Desired DST Profile. The desired DST profile was generated using the MSB adsorption kinetics model corresponding to a specific surfactant mixture condition (Table 2) at T = 295 K and is shown in Figure 1. Note that we have considered the surfactant mixture specified in Table 2 for the following reasons: (a) Actual experimental DST data is available at the chosen mixture conditions.21 This will be helpful in validating the accuracy of the model used to predict the DST profile. (b) In an actual application of the proposed framework, one would typically know only the individual surfactants that could be mixed to form the mixture but would not know which subset of those (20) Mulqueen, M.; Blankschtein, D. Prediction of equilibrium surface tension and surface adsorption of aqueous surfactant mixtures containing ionic surfactants. Langmuir 1999, 15, 8832–8848. (21) Mulqueen, M.; Datwani, S. S.; Stebe, K. J.; Blankschtein, D. Dynamic surface tension of aqueous surfactant mixtures: Experimental investigation. Langmuir 2001, 17, 5801–5812.
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Figure 1. Desired DST profile predicted using the MSB adsorption kinetics model at the surfactant mixture conditions specified in Table 2. Table 2. Nonionic Surfactant Mixture Considered to Demonstrate the Reliability of the New Theoretical Framework surfactant
i
Cb,i (mol/cm3)
C10E4 C12E5 C12E6 C10E8
1 2 3 4
0 2.67 10-9 0 1.33 10-8
surfactants is actually needed. To study the performance of the framework in designing surfactant mixtures containing only a subset of available surfactants, as an illustration we considered a case where the concentrations of two of the four surfactants (C10E4 and C12E6) are zero. Note that the DST profile in Figure 1 shows the existence of two characteristic time regimes of surface tension reduction: a relatively steep reduction in surface tension between 1 and 20 s, followed by a relatively gradual reduction in surface tension between 20 and 1000 s. In ref 21, the authors rationalized such a profile as resulting from an interesting adsorption-desorption interplay between C12E5 and C10E8. Specifically, the initial steep reduction in surface tension is primarily driven by the rapid adsorption of the C10E8 molecules as a result of their higher bulk solution concentration and higher diffusion coefficient, and the later gradual reduction in surface tension is due to the slower adsorption of the C12E5 molecules and the associated partial desorption of previously adsorbed C10E8 molecules from the surface.21 3.2. Application of the New Theoretical Framework. Given the desired DST profile in Figure 1 and a list of available surfactants that could be used to prepare the mixture (C10E4, C12E5, C12E6, and C10E8), clearly it is a nontrivial task to find which of these surfactants are needed and at what concentrations they need to be mixed to meet the desired DST profile most closely. To address this challenging design problem, we implemented the theoretical framework presented in section 2 by (a) formulating an optimization problem using the MSB adsorption kinetics model and (b) solving the formulated optimization problem using SNOPT. Specifications i-iv used to formulate the optimization problem are listed in Table 5. The formulated optimization problem was solved by repeatedly using SNOPT starting at 10 sets of random initial guesses for the surfactant mixture conditions. In each case, we found that the iterations converged to the identical (local) optimal surfactant mixture conditions, to within decimal place accuracy, as reported in Table 3. (The optimization convergence behavior observed using SNOPT is discussed in Appendix B.) Langmuir 2010, 26(24), 18728–18733
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Table 3. Nonionic Surfactant Mixture Identified by the New Theoretical Framework to Optimally Meet the Desired DST Profile Shown in Figure 1 surfactant
i
Cb,i (mol/cm3)
C10E4 C12E5 C12E6 C10E8
1 2 3 4
10-12 to 5 10-12 (2.67 ( 0.001) 10-9 10-13 to 10-12 (1.33 ( 0.0002) 10-8
Table 4. Predicted Equilibrium Surface Tension Reduction Induced by the Four Nonionic Surfactants Considered at Their Respective Optimal Concentrations surfactant C10E4 C12E5 C12E6 C10E8
i 1 2 3 4
Cb,i (mol/cm3)
surface tension reduction (mN/m)
-12
5 10 2.67 10-9 10-12 1.33 10-8
0.0011 18.6 0.08 14.6
The concentrations of C12E5 and C10E8 converged to values that are identical to their respective concentrations in the original surfactant mixture (cf. Tables 3 and 2). The concentrations of C10E4 and C12E6 converged to extremely small values (10-13 to 10-12 mol/cm3), possibly because of numerical accuracy limitations. To confirm that the concentrations of C10E4 and C12E6 identified as part of the optimal surfactant mixture are indeed “trace” quantities, we estimated the relative reduction in surface tension induced by each of the four surfactants by predicting the equilibrium surface tension reduction at their respective optimal concentrations. The equilibrium surface tension reductions induced by each of the four surfactants at their respective optimal concentrations were predicted using the molecular equilibrium adsorption isotherm (EAI) and the equation of state (EOS)15 and are reported in Table 4. Table 4 clearly shows that the surface tension reductions induced by C12E5 and C10E8 are much higher than those induced by C10E4 and C12E6. Accordingly, we conclude that the concentrations of C10E4 and C12E6 identified by the new theoretical framework are indeed trace quantities (note that according to Table 2, strictly they should be zero) and do not contribute significantly to the DST behavior relative to the contributions of C12E5 and C10E8. In addition, the DST profile corresponding to the optimal surfactant mixture (including the trace quantities of C10E4 and C12E6) was virtually identical to the original desired DST profile used. On average, the difference between the desired DST profile and the optimal DST profile identified by the new theoretical framework is on the order of 10-4 mN/m. Note that the typical accuracy of experimentally measuring the actual DST values is about (0.1 mN/m22 and the typical accuracy of the adsorption kinetics models is about 1 to 2 mN/m.21 Nevertheless, in the case study considered, we allowed the new theoretical framework to identify the optimal DST profile up to its highest level of matching with the desired DST profile in order to demonstrate its reliability and effectiveness. Overall, considering that the new theoretical framework consistently identified (to within numerical accuracy) the concentrations of C12E5 and C10E8 in the original surfactant mixture used to predict the desired DST profile, we may conclude that the new framework can be utilized reliably to facilitate the design of surfactant mixtures subject to the accuracy of the adsorption kinetics model used. In a practical implementation of the new theoretical framework, one may need to conduct experimental DST measurements (22) Lin, S.-Y.; Lee, Y.-C.; Yang, M-W; Liu, H.-S. Surface equation of state of nonionic CmEn surfactants. Langmuir 2003, 19, 3164–3171.
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Figure 2. Predicted (line) and experimentally measured (symbols) dynamic surface tension of aqueous solutions of a binary surfactant mixture of C12E5 and C10E8 at bulk solution surfactant concentrations of 2.67 10-9 and 1.33 10-8 mol/cm3, respectively.
at the optimal surfactant mixture condition identified by the new theoretical framework in order to validate its predictions. With that in mind, in Figure 2 we compare the predicted DST profile and the experimental DST profile measured under the optimal surfactant mixture condition identified by the new theoretical framework.21 Figure 2 clearly shows that the experimental DST profile exhibits two characteristic timescales for the surface tension reduction, a finding that is consistent with the desired DST profile. Recall that, as part of the design problem, we originally had the choice of mixing four CiEj nonionic surfactants to obtain the desired DST profile given in Figure 1. However, the new theoretical framework determined that it is necessary to mix only two surfactants (C12E5 and C10E8), including identifying their specific concentrations, which is a very useful practical finding. With this in mind, we believe that the agreement between the experimental DST profile at the optimal surfactant mixture condition identified by the new theoretical framework and the desired DST profile is quite remarkable. Depending on the actual application and the accuracy with which the desired DST profile needs to be achieved, one may need to refine the surfactant mixture conditions further.
4. Conclusions The central aim of this article was to propose a new theoretical framework to facilitate the design of surfactant mixtures that most closely exhibit a desired adsorption kinetics behavior. The proposed framework generally combines a predictive adsorption kinetics model with optimization techniques to develop an effective tool to design optimal surfactant mixtures, an approach that has not been explored to date in the adsorption kinetics literature. Specifically, we demonstrated the reliability and effectiveness of this framework through a realistic case study by identifying a nonionic surfactant mixture consisting of up to four surfactants (C10E4, C12E5, C12E6, and C10E8) that most closely satisfies a desired DST profile. We found good agreement between the desired DST profile and the actual DST profile measured under the surfactant mixture condition identified by the proposed framework. Clearly, several variations of the framework presented in this article are possible and may be targeted to specific applications. For example, one could include the cost of different surfactants as part of the optimization problem to identify the most costeffective surfactant mixture that optimally satisfies the desired adsorption kinetics behavior. In a different context, one could DOI: 10.1021/la103735u
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also use the theoretical framework presented here to develop “analyzer tools” to estimate unknown surfactant mixture concentrations on the basis of experimentally observed adsorption kinetics behaviors. Acknowledgment. We are grateful to Professor Paul I. Barton for several helpful discussions on the optimization and the use of optimization packages.
Appendix A: Optimization Problem Specifications In this Appendix, we discuss the four specifications needed to formulate the optimization problem associated with the new theoretical framework, including the specifications used for the case study presented in section 3. A.1. Objective Function. Let the desired (d) DST profile be γd(t), and let the time range over which the desired DST profile is specified be tL e t etU, where L denotes lower and U denotes upper. The goal of optimally satisfying the desired DST profile is expressed as the following objective function Z min M ¼
tU
tL
γd ðtÞ - γðtÞ 2 wðtÞ dt γ
ðA.1Þ
where γ* is a factor included to nondimensionalize the term (γd(t) - γ(t)) and w(t) is the weighting function. The following points are noteworthy regarding the specific choice of the objective function in eq A.1: (1) The objective function (M) measures the deviation between the desired DST profile (γd(t)) and the DST corresponding to a specific surfactant mixture (γ(t)). Therefore, by minimizing M, one determines the formulation conditions for which the deviation between γd(t) and γ(t) is minimal. (2) The weighting function (w(t)) is included in the objective function to nondimensionalize time (t). If the desired range of t varies over several orders of magnitude (for example, tL = 1 s and tU = 103s, as in the case study considered, see Figure 1), then w(t) can be chosen to be 1/t ln(10). In this case, eq A.1 becomes Z min M ¼
log tU log tL
γd ðtÞ - γðtÞ 2 d log t γ
ðA.2Þ
The objective function in eq A.2 has a clear physical interpretation: M is proportional to the square of the area between profiles γd(t) and γ(t) when these two profiles are plotted as a function of log(t). However, if the desired range of t varies over a relatively narrow range (for example, tL = 1 s and tU = 10 s), then w(t) can be chosen to be a constant value, say, 1/tU, and eq A.1 becomes Z min M ¼
tU
tL
γd ðtÞ - γðtÞ 2 dt γ tU
ðA.3Þ
(3) The factor γ* is included in eqs A.1-A.3 to nondimensionalize the term (γd(t) -γ(t)) in the objective function. Because γd(t) - γ(t) typically varies in the range of 1 to 30 mN/m (see the desired DST profile for the case study in Figure 1), γ* can be chosen to be a constant value (say γ* = 72 mN/m). A.2. Decision Variables. Let the total number of nonionic surfactants that could be used to prepare the mixture be n. Note that for the case study presented in section 3 we consider surfactant mixtures consisting of up to four CiEj nonionic surfactants (n = 4): C10E4, C12E5, C12E6, and C10E8. On the basis of the need to identify the bulk solution concentrations of the n surfactants 18732 DOI: 10.1021/la103735u
Figure 3. Sample of the evolution of the predicted DST profiles as the initial guess is refined through iterations. The initial guess and its refinement corresponding to the predicted DST profiles are listed in Table 6. The average deviation between the DST profile corresponding to iteration 10 ( 3 3 3 ) and the optimal DST profile (;) is less than 0.1 mN/m; therefore, they appear to be almost identical.
comprising the nonionic surfactant mixture, it follows that the optimization decision variables should be related to the bulk solution concentrations of the n surfactants. However, it is not convenient to choose the bulk solution concentrations as the actual decision variables because their typical values span several orders of magnitude (for example, 10-11 mol/cm3 to 10-6 mol/cm3),22 thus making the search for the optimal solution difficult. Considering this, we have chosen the logarithm to base 10 of the bulk solution concentrations of the n surfactants (denoted hereafter as log Cb = {log Cb,1, log Cb,2,...,log Cb,n}) as the optimization decision variables. Note that with this choice the values of the decision variables vary over a narrower range (for example, from -11 to -6) rather than over several orders of magnitude. A.3. Relation between the Decision Variables and the Objective Function. The evaluation of the objective function (M) for a given set of values of the decision variables (log Cb) requires knowing the dynamic surface tension (γ(t)) corresponding to the given log Cb (eq A.1). For the purpose of demonstrating the new theoretical framework, in this article we have used the molecularly based MSB adsorption kinetics model13 to predict γ(t) for a given log Cb and subsequently have evaluated M using the predicted γ(t) in eq A.1. The molecular parameters used for the four CiEj nonionic surfactants considered in the case study are listed in Table 1. A.4. Constraints on the Decision Variables. Note that the MSB adsorption kinetics model predicts the adsorption kinetics behavior of nonionic surfactant mixtures only at premicellar surfactant concentrations.13 Let Cub,i denote the upper bound value of Cb,i (i = 1,...,n) below which micelles do not form in the bulk solution. Recalling that the decision variables were chosen as log Cb, the MSB adsorption kinetics model limitation is specified through the following constraints: log Cb, i elog Cbu, i , i ¼ 1, :::, n
ðA.4Þ
For the case study discussed in section 3, we chose a constant upper bound value of Cub,i = 1 10-6 mol/cm3 (i = 1,...,4) because it corresponds to a typical value of critical micelle concentrations of CiEj nonionic surfactants.22 Table 3 summarizes the various specifications needed to formulate the optimization problem associated with the new theoretical framework. Table 5 also includes specific values chosen for the case study to demonstrate the effectiveness of the new theoretical framework. Langmuir 2010, 26(24), 18728–18733
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Table 5. List of Specifications Associated with Formulating the Optimization Problem to Identify the Surfactant Mixture That Most Closely Satisfies a Desired DST Profilea optimization problem component
description of specifications
values used in the case study
desired dynamic surface tension profile, γd(t) nondimensionalization factor, γ* weighting function, w(t) lower time limit of interest, tL upper time limit of interest, tU variables total number of nonionic surfactants, n parameters of the MSB adsorption kinetics model constraints upper bound surfactant bulk solution concentrations, Cub a Also listed are the specific values used to formulate the optimization problem presented in the case study in section 3. objective function
see Figure 3 72.0 (mN/m) 1/(t ln 10) s-1 1s 1000 s 4 see Table 1 1 10-6 mol/cm3
Table 6. Refinement of an Initial Guess (Iteration 0) of the Surfactant Mixture at the End of Specific Iterations by SNOPT and the Average Deviation between the DST Profile of the Surfactant Mixture and the Desired DST Profile surfactant concentration (mol/cm3) iteration count
C10E4
C12E5
C12E6
C10E8
average deviation (mN/m)
0 (initial guess) 2 5 10 195 (optimal)
4.70 10-10 5.04 10-10 6.72 10-10 6.33 10-10 4.4 10-12
5.80 10-11 6.06 10-11 7.23 10-11 6.71 10-11 2.67 10-9
2.90 10-8 7.32 10-9 4.12 10-9 2.35 10-9 7.72 10-13
5.30 10-9 2.77 10-9 1.21 10-8 1.33 10-8 1.33 10-8
12.9 3.4 1.0 0.06 0.0001
Appendix B: SNOPT Optimization Convergence Behavior In this Appendix, we discuss the numerical convergence behavior exhibited by SNOPT when solving the optimization problem associated with the case study discussed in section 3. The total number of iterations required by SNOPT to identify the optimal solution varied between 100 and 500 depending on the initial guess. On average, each iteration required about 1 min of computer run time (for a single CPU). Figure 3 shows a sample of the evolution of the predicted DST profiles as the initial guess is refined through iterations. The initial guess and its refinement corresponding to the predicted DST profiles are listed in Table 6.
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Table 6 also includes the average deviation (in terms of the surface tension value in mN/m) between the desired DST profile and the DST profiles corresponding to the initial guess and its refinements. Figure 3 clearly reveals that SNOPT is effectively able to handle even very poor initial guesses to solve the optimization problem associated with the new theoretical framework. Supporting Information Available: General information on how SNOPT interfaces with the adsorption kinetics model to iteratively solve the optimization problem associated with the new theoretical framework. This material is available free of charge via the Internet at http://pubs.acs.org.
DOI: 10.1021/la103735u
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