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Equilibrium exchange kinetics in PEP–PEO block copolymer micelles. A time resolved SANS study
ARTICLE IN PRESS
Physica B 385–386 (2006) 735–737 www.elsevier.com/locate/physb
Equilibrium exchange kinetics in PEP–PEO block copolymer micelles. A time resolved SANS study R. Lund, L. Willner, J. Stellbrink, D. Richter Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany
Abstract The chain exchange kinetics in polymeric micelles has been studied by a novel time resolved small angle neutron scattering (TR-SANS) technique. The system investigated was the amphiphilic block copolymer poly(ethylene-alt-propylene)–poly(ethylene oxide) (PEP–PEO) in water=N; N-dimethylformamide (DMF) mixtures as selective solvent for PEO. The TR-SANS technique exploits the large difference in scattering length density of deuterated and protonated species allowing for a detailed study of chain exchange kinetics with virtually no perturbation from equilibrium. The measured relaxation curves show a fast initial decay which gradually slows down at longer times. This extremely broad and heterogeneous decay drastically deviates from single exponential predicted by the scaling theory of Halperin and Alexander. Instead, the data appear to follow a logarithmic time dependence. This behavior most likely stems from a heterogeneous release of core chains caused by strong topological correlations inside the small micellar cores. r 2006 Elsevier B.V. All rights reserved. PACS: 87.15.Nm; 61.12.Ex Keywords: Time resolved SANS; Block copolymer micelles; Exchange kinetics
1. Introduction Self-assembly into micellar structures is a characteristic property of block copolymers immersed in a selective solvent. A vast number of studies have been performed mainly devoted to determine and predict equilibrium structures and their dependence on block copolymer composition and architecture [1,2]. A less studied feature is the kinetic behavior at equilibrium, i.e. the dynamic exchange of the constituting polymer chains. Theoretically, the chain exchange kinetics was studied by Halperin and Alexander [3]. They described the exchange by a simple expulsion/insertion mechanism where single chains have to overcome a defined potential barrier. From the experimental point of view the chain exchange at equilibrium is difficult to be observed since a large perturbation is generally required for most of the experimental techniques. Recently, we have developed a time resolved SANS (TR-SANS) technique which relies upon simple hydrogen/ Corresponding author.
E-mail address: [email protected] (L. Willner). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.06.045
deuterium exchange [4]. The basic idea of this technique is to mix two differently labelled but otherwise identical micelles and to follow the decay of the scattered neutron intensity as a function of time. Incorporation of bulky chemical labels or strong temperature and pressure jumps, as often required for other techniques, is not necessary, thus allowing to study exchange kinetics virtually under equilibrium conditions. Applying the TR-SANS technique we have studied the equilibrium kinetics of a very asymmetric poly(ethylene-alt-propylene)–poly(ethylene oxide) block copolymer in water=N; N-dimethylformamide (DMF) mixtures. In this report results of the kinetic study and a discussion on possible explanations for the observed peculiar relaxation behavior will be presented.
2. Experimental technique and system The principles of the TR-SANS technique are illustrated in Fig. 1. Two reservoirs consisting of deuterated (black) and protonated (white) micelles are prepared in an isotopic solvent mixture whose scattering length density (gray)
ARTICLE IN PRESS 736
R. Lund et al. / Physica B 385–386 (2006) 735–737
Fig. 2. Time evolution of the SANS intensity in the PEP1–PEO20/water/ DMF (30% DMF) system at T ¼ 50 C and f ¼ 1% polymer volume fraction. Fig. 1. Illustration of the TR-SANS experiment for the determination of the chain exchange kinetics in block copolymer micelles.
exactly matches the average scattering length density of the two constituting block copolymers. In case the micelles are just differently labelled but otherwise identical, the scattering curves of the two reservoirs are the same. After mixing at t ¼ 0 the scattered intensity is high and equal to the two reservoirs, since the contrast is maximal. As a function of time chain exchange takes place and, consequently, contrast and intensity are decreasing. The average excess fraction of labelled chains residing inside the micelles is then simply given by the square root of the excess SANS intensity, ðIðtÞ I 1 Þ1=2 , where I 1 is the scattered intensity of the completely randomized mixture at t1 . Thus, this technique gives a direct access to the kinetics in block copolymer micelles. The equilibrium kinetics of micelles prepared from the amphiphilic block copolymer PEP1–PEO20 (numbers denote nominal molecular weight in kg/mol) has been investigated. A fully deuterated and a fully protonated version of the block copolymer were synthesized by living anionic polymerization techniques. The polymers as well as the micellar characteristics are nearly identical as recently published in a separate paper [5]. Water/DMF mixtures were used to tune the exchange rate to the for SANS accessible time resolution. With 25% and 30% DMF a detailed and easy study of the relaxation behavior became possible. 3. Results and discussion A typical time evolution of SANS curves after mixing equal amounts of the two differently labelled reservoir solutions is shown in Fig. 2. The intensity gradually decreases as a function of time indicating that the conditions are appropriate for the TR-SANS method. It
Fig. 3. Relaxation kinetics of the PEP1–PEO20/water/DMF system (25% DMF) at T ¼ 47, 55, 60 and 65 C (from top to bottom). The solid line displays an example of a single exponential as predicted by Halperin and Alexander.
should be noted that a change in the shape of the curves cannot be observed at any time excluding the existence of any kinetic mechanism involving micellar clusters and that the structure is invariant. Results of the kinetic evaluation are shown for the PEP1–PEO20 micelles in water/DMF (25% DMF) in Fig. 3 for different temperatures. In this figure the square root of the excess intensity normalized to the excess intensity at t ¼ 0 is plotted versus time t. The data reveal that the exchange rate considerably increases with temperature. However, the overall shape of the curves is not changing. Independent of temperature the kinetics is fast in the beginning and significantly slows down at longer times such that a full relaxation does not occur even not at times up to 12 h. The observed heterogeneous decay is contradictory to the Halperin and Alexander model which
ARTICLE IN PRESS R. Lund et al. / Physica B 385–386 (2006) 735–737
Fig. 4. Data of Fig. 3 represented on a logarithmic time scale. Solid lines display linear fits.
predicts a single exponential decay indicated by the solid line in Fig. 3. As a first step to describe the data a distribution of relaxation rates was assumed: 1=2 Z 1 IðtÞ I 1 ¼ gðkÞ expðktÞ dk. (1) Iðt ¼ 0Þ I 1 0 For independent random processes a Gaussian distribution of activation energies is the most general approach. Using a similar strategy as Arbe et al. in Ref. [6], the data were fitted by ! kb T kb T lnðkt0 Þ hE a i 2 gðkÞ ¼ pffiffiffi exp , (2) sE a ps E a where t0 is the characteristic ‘‘attempt time’’, hE a i the mean activation energy and sE a is a smearing parameter (in energy units). This distribution could describe the data but the obtained parameters were found to be unphysical for the present system. For example an activation energy hE a i 165 kJ/mol and t0 1 1023 s are obtained which are both unreasonable. Consequently, this approach was discarded. If the data are plotted on a logarithmic time scale, as shown in Fig. 4 straight lines are observed demonstrating a logarithmic relaxation: 1=2 IðtÞ I 1 ln t. (3) Iðt ¼ 0Þ I 1
737
Such a time dependence indicates an extremely broad continuous distribution that extends over several decades. In fact a logarithmic relaxation implies that no inherent natural mean time constant exists. It is noteworthy to mention that anomalous slow relaxations have already been found in a variety of complex systems, e.g. in the relaxation in glassy systems, in the dynamics of colloids at the gel point, in friction experiments, etc. Often this behavior has been attributed to strongly coupled dynamics rather than a broad distribution of independent relaxation modes [7]. Here the path to equilibrium involves a series of correlated steps where each step is characterized with an increasing characteristic time. In fact in one general model for hierarchically constraint dynamics, Brey and Prados [8] have shown that such type of dynamics naturally gives a logarithmic relaxation. Inspired by these findings we may therefore speculate that similar processes are at play in polymeric micelles, most likely in the micellar core. Since for low molecular surfactant micelles a single expulsion rate has been observed, the broad distribution of relaxation rates in the present system can be related to its polymeric nature. For polymers strong geometrical restrictions in the core may give rise to effects where the individual chains cannot independently move out. The release of some chains may require some rearrangement processes which involves several chains. This not only slows down the release drastically but also gives rise to an interdependence or coupling that may explain the logarithmic time dependence.
References [1] I.W. Hamley, The Physics of Block Copolymers, Oxford University Press, New York, 1998. [2] P. Alexandridis, B. Lindman, Amphiphilic Block Copolymers, Elsevier, Amsterdam, 2000. [3] A. Halperin, S. Alexander, Macromolecules 22 (1989) 2403. [4] L. Willner, A. Poppe, J. Allgaier, M. Monkenbusch, D. Richter, Europhys. Lett. 55 (5) (2001) 667. [5] R. Lund, L. Willner, J. Stellbrink, A. Radulescu, D. Richter, Macromolecules 37 (2004) 9984. [6] A. Arbe, D. Richter, J. Colmenero, B. Farago, Phys. Rev. E 54 (1996) 3853. [7] R.G. Palmer, D.L. Stein, E. Abrahams, P.W. Anderson, Phys. Rev. Lett. 53 (1984) 958. [8] J.J. Brey, A. Prados, Phys. Rev. E 63 (2001) 21108.
Physica B 385–386 (2006) 735–737 www.elsevier.com/locate/physb
Equilibrium exchange kinetics in PEP–PEO block copolymer micelles. A time resolved SANS study R. Lund, L. Willner, J. Stellbrink, D. Richter Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany
Abstract The chain exchange kinetics in polymeric micelles has been studied by a novel time resolved small angle neutron scattering (TR-SANS) technique. The system investigated was the amphiphilic block copolymer poly(ethylene-alt-propylene)–poly(ethylene oxide) (PEP–PEO) in water=N; N-dimethylformamide (DMF) mixtures as selective solvent for PEO. The TR-SANS technique exploits the large difference in scattering length density of deuterated and protonated species allowing for a detailed study of chain exchange kinetics with virtually no perturbation from equilibrium. The measured relaxation curves show a fast initial decay which gradually slows down at longer times. This extremely broad and heterogeneous decay drastically deviates from single exponential predicted by the scaling theory of Halperin and Alexander. Instead, the data appear to follow a logarithmic time dependence. This behavior most likely stems from a heterogeneous release of core chains caused by strong topological correlations inside the small micellar cores. r 2006 Elsevier B.V. All rights reserved. PACS: 87.15.Nm; 61.12.Ex Keywords: Time resolved SANS; Block copolymer micelles; Exchange kinetics
1. Introduction Self-assembly into micellar structures is a characteristic property of block copolymers immersed in a selective solvent. A vast number of studies have been performed mainly devoted to determine and predict equilibrium structures and their dependence on block copolymer composition and architecture [1,2]. A less studied feature is the kinetic behavior at equilibrium, i.e. the dynamic exchange of the constituting polymer chains. Theoretically, the chain exchange kinetics was studied by Halperin and Alexander [3]. They described the exchange by a simple expulsion/insertion mechanism where single chains have to overcome a defined potential barrier. From the experimental point of view the chain exchange at equilibrium is difficult to be observed since a large perturbation is generally required for most of the experimental techniques. Recently, we have developed a time resolved SANS (TR-SANS) technique which relies upon simple hydrogen/ Corresponding author.
E-mail address: [email protected] (L. Willner). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.06.045
deuterium exchange [4]. The basic idea of this technique is to mix two differently labelled but otherwise identical micelles and to follow the decay of the scattered neutron intensity as a function of time. Incorporation of bulky chemical labels or strong temperature and pressure jumps, as often required for other techniques, is not necessary, thus allowing to study exchange kinetics virtually under equilibrium conditions. Applying the TR-SANS technique we have studied the equilibrium kinetics of a very asymmetric poly(ethylene-alt-propylene)–poly(ethylene oxide) block copolymer in water=N; N-dimethylformamide (DMF) mixtures. In this report results of the kinetic study and a discussion on possible explanations for the observed peculiar relaxation behavior will be presented.
2. Experimental technique and system The principles of the TR-SANS technique are illustrated in Fig. 1. Two reservoirs consisting of deuterated (black) and protonated (white) micelles are prepared in an isotopic solvent mixture whose scattering length density (gray)
ARTICLE IN PRESS 736
R. Lund et al. / Physica B 385–386 (2006) 735–737
Fig. 2. Time evolution of the SANS intensity in the PEP1–PEO20/water/ DMF (30% DMF) system at T ¼ 50 C and f ¼ 1% polymer volume fraction. Fig. 1. Illustration of the TR-SANS experiment for the determination of the chain exchange kinetics in block copolymer micelles.
exactly matches the average scattering length density of the two constituting block copolymers. In case the micelles are just differently labelled but otherwise identical, the scattering curves of the two reservoirs are the same. After mixing at t ¼ 0 the scattered intensity is high and equal to the two reservoirs, since the contrast is maximal. As a function of time chain exchange takes place and, consequently, contrast and intensity are decreasing. The average excess fraction of labelled chains residing inside the micelles is then simply given by the square root of the excess SANS intensity, ðIðtÞ I 1 Þ1=2 , where I 1 is the scattered intensity of the completely randomized mixture at t1 . Thus, this technique gives a direct access to the kinetics in block copolymer micelles. The equilibrium kinetics of micelles prepared from the amphiphilic block copolymer PEP1–PEO20 (numbers denote nominal molecular weight in kg/mol) has been investigated. A fully deuterated and a fully protonated version of the block copolymer were synthesized by living anionic polymerization techniques. The polymers as well as the micellar characteristics are nearly identical as recently published in a separate paper [5]. Water/DMF mixtures were used to tune the exchange rate to the for SANS accessible time resolution. With 25% and 30% DMF a detailed and easy study of the relaxation behavior became possible. 3. Results and discussion A typical time evolution of SANS curves after mixing equal amounts of the two differently labelled reservoir solutions is shown in Fig. 2. The intensity gradually decreases as a function of time indicating that the conditions are appropriate for the TR-SANS method. It
Fig. 3. Relaxation kinetics of the PEP1–PEO20/water/DMF system (25% DMF) at T ¼ 47, 55, 60 and 65 C (from top to bottom). The solid line displays an example of a single exponential as predicted by Halperin and Alexander.
should be noted that a change in the shape of the curves cannot be observed at any time excluding the existence of any kinetic mechanism involving micellar clusters and that the structure is invariant. Results of the kinetic evaluation are shown for the PEP1–PEO20 micelles in water/DMF (25% DMF) in Fig. 3 for different temperatures. In this figure the square root of the excess intensity normalized to the excess intensity at t ¼ 0 is plotted versus time t. The data reveal that the exchange rate considerably increases with temperature. However, the overall shape of the curves is not changing. Independent of temperature the kinetics is fast in the beginning and significantly slows down at longer times such that a full relaxation does not occur even not at times up to 12 h. The observed heterogeneous decay is contradictory to the Halperin and Alexander model which
ARTICLE IN PRESS R. Lund et al. / Physica B 385–386 (2006) 735–737
Fig. 4. Data of Fig. 3 represented on a logarithmic time scale. Solid lines display linear fits.
predicts a single exponential decay indicated by the solid line in Fig. 3. As a first step to describe the data a distribution of relaxation rates was assumed: 1=2 Z 1 IðtÞ I 1 ¼ gðkÞ expðktÞ dk. (1) Iðt ¼ 0Þ I 1 0 For independent random processes a Gaussian distribution of activation energies is the most general approach. Using a similar strategy as Arbe et al. in Ref. [6], the data were fitted by ! kb T kb T lnðkt0 Þ hE a i 2 gðkÞ ¼ pffiffiffi exp , (2) sE a ps E a where t0 is the characteristic ‘‘attempt time’’, hE a i the mean activation energy and sE a is a smearing parameter (in energy units). This distribution could describe the data but the obtained parameters were found to be unphysical for the present system. For example an activation energy hE a i 165 kJ/mol and t0 1 1023 s are obtained which are both unreasonable. Consequently, this approach was discarded. If the data are plotted on a logarithmic time scale, as shown in Fig. 4 straight lines are observed demonstrating a logarithmic relaxation: 1=2 IðtÞ I 1 ln t. (3) Iðt ¼ 0Þ I 1
737
Such a time dependence indicates an extremely broad continuous distribution that extends over several decades. In fact a logarithmic relaxation implies that no inherent natural mean time constant exists. It is noteworthy to mention that anomalous slow relaxations have already been found in a variety of complex systems, e.g. in the relaxation in glassy systems, in the dynamics of colloids at the gel point, in friction experiments, etc. Often this behavior has been attributed to strongly coupled dynamics rather than a broad distribution of independent relaxation modes [7]. Here the path to equilibrium involves a series of correlated steps where each step is characterized with an increasing characteristic time. In fact in one general model for hierarchically constraint dynamics, Brey and Prados [8] have shown that such type of dynamics naturally gives a logarithmic relaxation. Inspired by these findings we may therefore speculate that similar processes are at play in polymeric micelles, most likely in the micellar core. Since for low molecular surfactant micelles a single expulsion rate has been observed, the broad distribution of relaxation rates in the present system can be related to its polymeric nature. For polymers strong geometrical restrictions in the core may give rise to effects where the individual chains cannot independently move out. The release of some chains may require some rearrangement processes which involves several chains. This not only slows down the release drastically but also gives rise to an interdependence or coupling that may explain the logarithmic time dependence.
References [1] I.W. Hamley, The Physics of Block Copolymers, Oxford University Press, New York, 1998. [2] P. Alexandridis, B. Lindman, Amphiphilic Block Copolymers, Elsevier, Amsterdam, 2000. [3] A. Halperin, S. Alexander, Macromolecules 22 (1989) 2403. [4] L. Willner, A. Poppe, J. Allgaier, M. Monkenbusch, D. Richter, Europhys. Lett. 55 (5) (2001) 667. [5] R. Lund, L. Willner, J. Stellbrink, A. Radulescu, D. Richter, Macromolecules 37 (2004) 9984. [6] A. Arbe, D. Richter, J. Colmenero, B. Farago, Phys. Rev. E 54 (1996) 3853. [7] R.G. Palmer, D.L. Stein, E. Abrahams, P.W. Anderson, Phys. Rev. Lett. 53 (1984) 958. [8] J.J. Brey, A. Prados, Phys. Rev. E 63 (2001) 21108.