Langmuir 2009, 25, 2749-2756
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Effect of Tail Architecture on Self-Assembly of Amphiphiles for Polymeric Micelles Lisheng Cheng and Dapeng Cao* DiVision of Molecular and Materials Simulation, Key Laboratory of Nanomaterials, Ministry of Education, Beijing UniVersity of Chemical Technology, Beijing 100029, People’s Republic of China ReceiVed NoVember 20, 2008. ReVised Manuscript ReceiVed December 22, 2008 Brownian dynamics simulations were carried out to explore the self-assembly of amphiphilic copolymers composed of a linear hydrophilic head and a hydrophobic tail with different architectures. In order to investigate the effect of architecture of hydrophobic tail on self-assembling behavior, these architectures of linear, branched, starlike, and dendritic tails were selected for comparison, and the branching parameter of the tail was employed to characterize the tail architectures. The critical micelle concentration (cmc), dynamics of aggregation, aggregate distribution, gyration radius distribution, density profiles of micelle, shape anisotropy, and thermal stability were examined for the four typical types of copolymers. The calculated results reveal that the self-assembly of linear tail copolymer has the lowest cmc, and the consequently formed polymeric micelles have narrow dispersion and greater aggregate size, and the micelle is closer to spherical shape. It was found that the cmc is inversely proportional to the branching parameter. Linear tail aggregates in solution to form polymeric micelles with higher physical stability, compared to other architectures of tail. The size of polymeric micelle increases with the increase of the branching parameter of the tail, and it exhibits an exponential relationship with the branching parameter. In addition, the micelles formed from copolymers with a high branching parameter of the tail were found to have higher thermal stability. This work provides useful information on designing self-assembling systems for preparing polymeric micelles applied to drug delivery.
1. Introduction On the basis of bottom-up strategies, self-assembly has been proved to be one of the most effective approaches for nanomaterials fabrication.1-5 It targets at functional materials of nanoscale with little consumption but at high efficiency. The self-assembly has made the supramolecular chemistry transfer from originally a branch of fundamental science to now an important concept in nanotechnology.2,6,7 An amphiphilic copolymer contains two moieties. One is hydrophilic and the other is hydrophobic. Due to the inter- and intramolecular interactions the copolymers can self-assemble into different morphologies, such as spherical8,9 and cylindrical10,11 micelles, vesicles,12,13 ribbons,14 lamellas,15,16 perforated * To whom correspondence should be addressed. E-mail: caodp@ mail.buct.edu.cn,
[email protected]. (1) Whitesides, G. M.; Grzybowski, B. Science 2002, 295, 2418–2421. (2) Ariga, K.; Hill, J. P.; Lee, M. V.; Vinu, A.; Charvet, R.; Acharya, S. Sci. Technol. AdV. Mater. 2008, 9, 14109–14109. (3) De Wild, M.; Berner, S.; Suzuki, H.; Ramoino, L.; Baratoff, A.; Jung, T. A. Ann. N.Y. Acad. Sci. 2003, 1006, 291–305. (4) Galisteo, J. F.; Garcia-Santamaria, F.; Golmayo, D.; Juarez, B. H.; Lopez, C.; Palacios, E. J. Opt. A: Pure Appl. Opt. 2005, 7, S244–S254. (5) Keizer, H. M.; Sijbesma, R. P. Chem. Soc. ReV. 2005, 34, 226–234. (6) Klok, H. A.; Lecommandoux, S. AdV. Mater. 2001, 13, 1217–1229. (7) Davis, J. T.; Spada, G. P. Chem. Soc. ReV. 2007, 36, 296–313. (8) Matsuzawa, Y.; Kogiso, M.; Matsumoto, M.; Shimizu, T.; Shimada, K.; Itakura, M.; Kinugasa, S. AdV. Mater. 2003, 15, 1417–1420. (9) Cho, E. J.; Kang, J. K.; Kang, S.; Lee, J. Y.; Jung, J. H. Bull. Korean Chem. Soc. 2006, 27, 1881–1884. (10) Gohy, J. F.; Lohmeijer, B. G. G.; Alexeev, A.; Wang, X. S.; Manners, I.; Winnik, M. A.; Schubert, U. S. Chem. Eur. J. 2004, 10, 4315–4323. (11) Jiang, Y.; Zhu, J. T.; Jiang, W.; Liang, H. J. J. Phys. Chem. B 2005, 109, 21549–21555. (12) Kita-Tokarczyk, K.; Grumelard, J.; Haefele, T.; Meier, W. Polymer 2005, 46, 3540–3563. (13) Zou, P.; Pan, C. Y. Macromol. Rapid Commun. 2008, 29, 763–771. (14) Lee, S. J.; Kim, E.; Seo, M. L.; Do, Y.; Lee, Y. A.; Lee, S. S.; Jung, J. H.; Kogiso, M.; Shirnizu, T. Tetrahedron 2008, 64, 1301–1308. (15) Laschewsky, A.; Mertoglu, M.; Kubowicz, S.; Thunemann, A. F. Macromolecules 2006, 39, 9337–9345. (16) Battaglia, G.; Tomas, S.; Ryan, A. J. Soft Matter 2007, 3, 470–475.
lamellas,17,18 gyroid structures,19,20 and other interesting structures.21-23 Some interesting morphologies from self-assembly of amphiphilic copolymers in nanoslits were also reported in our previous work.24,25 These structures promised a great number of applications in biomedicine,26-28 biomaterials,29-31 electronics,32-34 photonics,33,34 and catalysis,35,36 etc. Among these nanostructures, nanospheres are the earliest and most extensively studied for their various applications in nanotechnology, such as drug delivery,37,38 sensors,39 and catalysts.40 It is well-known that the structural information of a (17) Ludwigs, S.; Boker, A.; Voronov, A.; Rehse, N.; Magerle, R.; Krausch, G. Nat. Mater. 2003, 2, 744–747. (18) Valkama, S.; Ruotsalainen, T.; Nykanen, A.; Laiho, A.; Kosonen, H.; ten Brinke, G.; Ikkala, O.; Ruokolainen, J. Macromolecules 2006, 39, 9327–9336. (19) Gonzalez-Segredo, N.; Coveney, P. V. Europhys. Lett. 2004, 65, 795– 801. (20) Chin, J.; Coveney, P. V. Proc. R. Soc. London, Ser. A 2006, 462, 3575– 3600. (21) Song, B.; Wang, Z. Q.; Chen, S. L.; Zhang, X.; Fu, Y.; Smet, M.; Dehaen, W. Angew. Chem., Int. Ed. 2005, 44, 4731–4735. (22) Tian, Z.; Li, H.; Wang, M.; Zhang, A. Y.; Feng, Z. G. J. Polym. Sci., Part A: Polym. Chem. 2008, 46, 1042–1050. (23) Saksena, R. S.; Coveney, P. V. J. Phys. Chem. B 2008, 112, 2950–2957. (24) Cheng, L.; Cao, D. J. Phys. Chem. B 2007, 111, 10775–10784. (25) Cheng, L.; Cao, D. J. Chem. Phys. 2008, 128, 074902–074909. (26) Taubert, A.; Napoli, A.; Meier, W. Curr. Opin. Chem. Biol. 2004, 8, 598–603. (27) Kokkoli, E.; Mardilovich, A.; Wedekind, A.; Rexeisen, E. L.; Garg, A.; Craig, J. A. Soft Matter 2006, 2, 1015–1024. (28) Subramani, K.; Khraisat, A.; George, A. Curr. Nanosci. 2008, 4, 201– 207. (29) Zhang, S. G. Nat. Biotechnol. 2003, 21, 1171–1178. (30) Pochan, D. J.; Schneider, J. P.; Kretsinger, J.; Ozbas, B.; Rajagopal, K.; Haines, L. J. Am. Chem. Soc. 2003, 125, 11802–11803. (31) Liu, X. M.; Pramoda, K. P.; Yang, Y. Y.; Chow, S. Y.; He, C. B. Biomaterials 2004, 25, 2619–2628. (32) Clark, A. P. Z.; Cadby, A. J.; Shen, C. K. F.; Rubin, Y.; Tolbert, S. H. J. Phys. Chem. B 2006, 110, 22088–22096. (33) Ryu, J. H.; Hong, D. J.; Lee, M. Chem. Commun. 2008, 1043–1054. (34) Nie, Z. H.; Fava, D.; Kumacheva, E.; Zou, S.; Walker, G. C.; Rubinstein, M. Nat. Mater. 2007, 6, 609–614. (35) Kimizuka, N.; Nakashima, T. Langmuir 2001, 17, 6759–6761. (36) Greaves, T. L.; Weerawardena, A.; Fong, C.; Drummond, C. J. J. Phys. Chem. B 2007, 111, 4082–4088.
10.1021/la803839t CCC: $40.75 2009 American Chemical Society Published on Web 02/03/2009
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material often determines its properties. The properties of nanospheres convincingly depend on the detail of the nanoparticle or micelles fabricated from the self-assembly. For applications in drug delivery, previous reports revealed that nanoparticles with different sizes would locate at different tissues and organs,41-43 and small ones may even be eliminated by renal excretion.44 Polymeric micelles as drug delivery carrier have to be sufficiently stable to increase circulation time in blood.45,46 The shape of a micelle also influences the application of micelles for drug delivery.47 In addition, polymeric micelles are expected to have appropriate thermal stability to reduce side effects48 and to release drug molecules at tumor targets where local hyperthermia could be realized by external power.48-50 Commonly, all amphiphilic copolymers could aggregate and form micelles in appropriate media, concentration, and temperature. However, the molecular architecture of copolymers would seriously affect the micellization process and the subsequent detail conformation. Thus, molecular architecture should be involved in the design of self-assembling systems for preparation of nanoparticles. Architectures of the hydrophilic head were considered in the self-assembly of copolymers. The effect of head size on the self-assembly of amphiphiles was studied by Brownian dynamics simulation.51 Zhang et al.52 attached different architectures to a cubic attempting to control the self-assembling structures by varying the component architecture. More recently, the effect of dendritic architecture jointed to a hydrophobic linear chain was carefully examined.53 Architectures of the whole molecule also dropped researcher’s attention. Micellar behavior of triblock and diblock copolymers was compared.54-56 Star copolymers and linear diblock copolymers were also considered to understand the effect of architecture on self-assembly.57-59 Effect of architecture on self-assembly of interesting heteroarm and (37) Koneracka, M.; Muckova, M.; Zavisova, V.; Tomasovicova, N.; Kopcansky, P.; Timko, M.; Jurikova, A.; Csach, K.; Kavecansky, V.; Lancz, G. J. Phys.: Condens. Matter 2008, 204151. (38) Kang, S. W.; Lim, H. W.; Seo, S. W.; Jeon, O.; Lee, M.; Kim, B. S. Biomaterials 2008, 29, 1109–1117. (39) Gou, X. L.; Wang, G. X.; Park, J.; Liu, H.; Yang, J. Nanotechnology 2008, 19, 125606. (40) Miao, S. D.; Zhang, C. L.; Liu, Z. M.; Han, B. X.; Xie, Y.; Ding, S. F.; Yang, Z. Z. J. Phys. Chem. C 2008, 112, 774–780. (41) Moghimi, S. M. AdV. Drug DeliVery ReV. 1995, 17, 103–115. (42) Moghimi, S. M. AdV. Drug DeliVery ReV. 1995, 17, 61–73. (43) Banerjee, T.; Mitra, S.; Singh, A. K.; Sharma, R. K.; Maitra, A. Int. J. Pharm. 2002, 243, 93–105. (44) Moghimi, S. M.; Hunter, A. C.; Murray, J. C. Pharmacol. ReV. 2001, 53, 283–318. (45) Chiappetta, D. A.; Sosnik, A. Eur. J. Pharm. Biopharm. 2007, 66, 303– 317. (46) Kabanov, A. V.; Alakhov, V. Y. Crit. ReV. Ther. Drug Carrier Syst. 2002, 19, 1–72. (47) Champion, J. A.; Katare, Y. K.; Mitragotri, S. J. Controlled Release 2007, 121, 3–9. (48) Needham, D.; Dewhirst, M. W. AdV. Drug DeliVery ReV. 2001, 53, 285– 305. (49) Li, J.; Wang, B. C.; Liu, P. Med. Hypotheses 2008, 71, 249–251. (50) Li, W.; Tu, W.; Cao, D. J. Appl. Polym. Sci. 2009, 111, 701–708. (51) Bourov, G. K.; Bhattacharya, A. J. Chem. Phys. 2005, 122, 044702– 044706. (52) Zhang, X.; Zhang, Z. L.; Glotzer, S. C. Nanotechnology 2007, 18, 115706. (53) Suek, N. W.; Lamm, M. H. Langmuir 2008, 24, 3030–3036. (54) Altinok, H.; Yu, G. E.; Nixon, S. K.; Gorry, P. A.; Attwood, D.; Booth, C. Langmuir 1997, 13, 5837–5848. (55) Barbosa, S.; Cheema, M. A.; Taboada, P.; Mosquera, V. J. Phys. Chem. B 2007, 111, 10920–10928. (56) Kelarakis, A.; Havredaki, V.; Derici, L.; Yu, G. E.; Booth, C.; Hamley, I. W. J. Chem. Soc., Faraday Trans. 1998, 94, 3639–3647. (57) Pispas, S.; Hadjichristidis, N.; Potemkin, I.; Khokhlov, A. Macromolecules 2000, 33, 1741–1746. (58) Yun, J. P.; Faust, R.; Szilagyi, L. S.; Keki, S.; Zsuga, M. J. Macromol. Sci., Pure Appl. Chem. 2004, A41, 613–627. (59) Yun, J. P.; Faust, R.; Szilagyi, L. S.; Keki, S.; Zsuga, M. Macromolecules 2003, 36, 1717–1723.
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miktoarm star diblock copolymers was investigated.60 Zamurovic et al.61 systematically studied the architectural effect on micellization of copolymers with comblike architecture. Yet few interests were drawn to the architecture of the hydrophobic tail in self-assembly of amphiphilic copolymers, which would aggregate to form the core of the micelle. Actually, the architecture of the tail may affect the contribution of configurational entropy and may even bring about the coupling of entropic and enthalpic effects. Therefore, it is very necessary to explore the effect of tail architecture on micellization. Linear, dendritic, starlike, and branched copolymers represent four typical architectures. Here we studied the effect of the four tail architectures on the self-assembly of amphiphilic copolymers in a dilute solution.
2. Model and Methods In this work, we employed a coarse-grained model to represent the amphiphilic copolymers, which usually involves thousands of atoms. As we aim at qualitatively investigating the effect of the architecture of the tail on the self-assembly, such a coarse-grained model is enough. In this model, it is assumed that the hydrophilic head is a linear architecture and the hydrophobic tail has different architectures, including linear, branched, starlike, and dendritic structures. By varying the architecture of the hydrophobic tail, we could compare the self-assembling behavior of copolymers with the four typical architectures and, consequently, analyze the effect of tail architecture on the micellization. A group of atoms were modeled as a bead in this work, and different beads were connected with flexible finitely extended nonlinear elastic (FENE) bonds in a particular topology. The FENE bond potential is expressed as
UBond(rij) )
{
[ ( )]
-0.5kR02 ln 1 ∞,
r R0
2
, r < R0
(1)
r g R0
where k is the force constant and R0 is the maximum extent of the bond. Here k ) 30Tε/σ2 and R0 ) 1.5σ. These parameters are chosen to prevent chains from crossing. The miscibility of the hydrophobic tails was treated with a truncated and shifted site-site 12-6 Lennard-Jones (LJ) potential:
ULJ(r) )
{
[( σr ) - ( σr ) ] - 4ε[( rσ ) - ( rσ ) ],
4ε
12
6
12
c
6
c
0,
r e rc r > rc (2)
where rc is the cutoff radius and is set to be 2.5σ for pair interactions between hydrophobic segments. The incompatibility between hydrophobic and hydrophilic segments and the solubility of the hydrophilic head in solution were introduced with a purely repulsive Weeks-Chandler-Andersen (WCA) potential, which equals to the 12-6 LJ potential truncated at the minimum and shifted vertically by ε:
UWCA(r) )
{
[( σr ) - ( σr ) ] + ε,
4ε
12
6
0,
r e rc’ r > rc’
(3)
where rc′ ) 21/6σ. The four typical types of amphiphiles have the same number of monomers, 16 head segments and 16 tail segments (see Figure 1). This treatment ensures the possibility and convenience to study the effect of architecture on the self-assembly. All energy parameters (60) Ramzi, A.; Prager, M.; Richter, D.; Efstratiadis, V.; Hadjichristidis, N.; Young, R. N.; Allgaier, J. B. Macromolecules 1997, 30, 7171–7182. (61) Zamurovic, M.; Christodoulou, S.; Vazaios, A.; Iatrou, E.; Pitsikalis, M.; Hadjichristidis, N. Macromolecules 2007, 40, 5835–5849.
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Figure 1. Schematic of four typical types of model amphiphilic copolymers. Red represents hydrophilic and green represents hydrophobic.
of LJ potential were set to 1.0, and all segments have the same diameter and mass of unit. Because solvent occupies the major volume of a dilute copolymer solution, it is inefficient to simulate the whole system. Thus, Brownian dynamics (BD) was alternatively used to simulate the self-assembling behaviors of the typical amphiphiles. Though solvent molecules are treated implicitly, this mesoscopic simulation technology involves certain solvent effects, which allows us to study the effect of tail architecture on self-assembly in solution. Each segment in the simulating box subjects to the equation of Langevin at time t:
mir¨i ) -γiyi + FiC(r(t)) + FiR(t)
(4)
where mi and γi are the mass and friction coefficients of segment i, and r¨i, yi, FiC are the position, velocity, and conservative vectors, respectively. In this work, γi is set to 1.0. The random force vector, FiR, is assumed to be Gaussian with zero mean and satisfies the fluctuation dissipation theorem, and thus meets
〈FiR(t)〉 ) 0
(5)
〈FiR(t)FjR(t’)〉 ) 6γkBTδijδ(t - t’)
(6)
where kB is the Boltzmann constant and T is the temperature. The friction coefficient and random force couple the simulated system to a heat bath, where the random force compensates the frictional losses generating from viscous drag. The stationary solution to the Langevin equation produces a Boltzmann distribution, and therefore this simulation has canonical (NVT) ensemble thermodynamic constraints. The simulations of this work were performed in a cubic box, with open-source, massively parallel simulator, LAMMPS.62,63 LAMMPS is a highly scalable classical molecular dynamics code designed for simulating atomic, polymeric, biological, metallic, and granular systems on parallel supercomputers. In dynamics simulation, it integrates equations of motion for collections of atoms, molecules, or macroscopic particles that interact via short- or long-range forces with a variety of initial and/or boundary conditions. Our previous work has also proved the high efficiency of LAMMPS in molecular dynamics simulations of a coarse-grained polymeric system.64,65 Thus, it is applicable to study the self-assembling behaviors of amphiphilic copolymers. In this work, system sizes are varied from N ) 16 000 to N ) 64 000, where N is the number of segments in the system. The system size was chosen to ensure that the maximum distance of two hydrophobic segments belonging to the largest micelle is longer than one-fourth of the box length. System size effect was eliminated by our tests. (62) http://lammps.sandia.gov. (63) Plimpton, S. J. Comput. Phys. 1995, 117, 1–19. (64) Ni, R.; Cao, D.; Wang, W.; Jusufi, A. Macromolecules 2008, 41, 5477– 5484. (65) Hu, Y.; Ni, R.; Cao, D.; Wang, W. Langmuir 2008, 24, 10138–10144.
When generating the initial configuration, we first arranged a number of “rigid” molecules in the “large” volume lattice box applying a random rotation vector to the molecule. Then through a short time run of dynamics simulation, we “pressed” the box to make the system reach a target segmental density. In order to prevent initial-configuration dependencies, we let the system prerun at high temperature with all cutoff radius rc ) 21/6σ and then quenched the system to the target temperature. These strategies were proved to be effective in obtaining initial configurations of a system with complex-structure molecules or high concentration for molecular dynamics and Monte Carlo simulations. Amphiphiles were assumed to belong to one aggregate if any pair of whose tail segments was within the distance of 1.5σ.53 The temperature of the simulating system was set to T* ) 1.5. This choice of temperature makes the studied system have both aggregate and free molecules. If the temperature is very low, the studied system contains only aggregates and no free molecules; while the temperature is very high, the studied system contains only free molecules and no aggregates. The simulations ran 5 × 106 time steps for equilibration and 1.5 × 107 time steps for sampling with the time step ∆t ) 0.005τ ∼ 0.01τ. For some slowly relaxing systems, the number of time steps was as many as 3 × 107. Such length of simulating time was enough according to the autocorrelation function.
3. Results and Discussion 3.1. Branching Parameter. The end-to-end distance has been used to characterize the conformation of linear chains, whereas the best quantity to characterize the overall dimension of the branched chain is the radius of gyration, Rg, defined as66
Rg )
n |ri |2mi ∑ i)1 n mi ∑ i)1
(7)
where mi is the mass of segment i and ri is the vector of segment i to the mass center of the molecule or aggregate. mi is equal to one unit for every segment in this work. n is the number of segments of an aggregate or a chain of interest. Furthermore, the branching parameter, gg, representing the branching degree of a polymer, is defined as67
gg )
Rgb2 Rgl2
(8)
where Rgb2 is the mean square radius of gyration for the branched chain, and Rgl2 is the mean square radius of gyration for the linear chain. (66) Han, M.; Chen, P. Q.; Yang, X. Z. Polymer 2005, 46, 3481–3488. (67) Teraoka, I. Polymer Solutions: An Introduction to Physical Properties; Wiley: New York, 2002.
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Figure 3. Critical micelle concentration as a function of the reciprocal of the branching parameter of a hydrophobic tail. The points are the calculation data, and the line is a linear fit of the data.
Figure 2. (a) Volume fraction of free amphiphilic copolymers and (b) micellization fraction of copolymers as a function of the total volume fraction for four typical types of tails.
For further understanding the dependence of micellar behavior on tail architecture, we calculated the branching parameters. The branching parameter provides us the convenience to identify architectures quantitatively; thus, we could build up the numerical relationship of micellar properties and architecture. However, in this work our focus is placed on the architecture of the tail, so we slightly modified the formula of the branching parameter where Rg′ was updated as the mean square radius of gyration for the tail, r′i as the vector of segment i to the segment that connects with the head, n′ as the number of segments in the tail, and the ′ 2 ′ 2 branching parameter is represented by gg′ ) Rgb /Rgl for distinguishing. In the calculation of the branching parameter of the tail, the simulated system was set to infinitely dilute solution, and all segmental interactions were involved as pure repulsion. Such particular treatments avoided the interference from molecular packing and enthalpic aggregation. The values of branching parameter of the tail are 1.000 for a linear tail, 0.448 for a branched tail, 0.285 for a starlike tail, and 0.273 for a dendritic tail shown in Figure 1. 3.2. Critical Micelle Concentration. The critical micelle concentration (cmc) is a key parameter representing the molecular micellization. High cmc would make the polymeric micelles disassociate in dilute blood when they are applied as carriers for drug delivery. Thus, we first study the influence of tail architecture on cmc. Figure 2a shows the volume fraction of free amphiphilic copolymers as a function of the total volume fraction for four typical types of tails. The curves exhibit a peak, where the volume fraction of free copolymers has a maximum value, and it defines the cmc. In a polymeric solution, there exists a competition between enthalpic interaction and entropic effect, which dominates the aggregating behavior of copolymers.51,53 The competition consequently determines the cmc of the solution and what kinds of aggregates the copolymers form. Under the cmc, the entropic effects play a dominating role, whereas the enthalpic contribution dominates over the entropic effects with the increase of copolymer concentration, which makes the rapid aggregation of copolymers to form aggregates with particular properties. Figure 2a also shows that the peak becomes sharper for copolymers having smaller branching parameter tails.
The curves in Figure 2a indicate that copolymers with a linear tail have the lowest cmc, and the cmc increases with the decrease of the branching parameter of the tail. That is, micelles formed from a linear tail copolymer have low potential to disassociate in dilute blood. In order to get an insight about the dependence of cmc on tail architecture, we drew the relationship of cmc and the branching parameter of the tail in Figure 3, where the abscissa is the reciprocal value of the branching parameter of the tail. The linear fit of the calculated data nearly passes through the origin of the coordinates, which illustrates that the cmc is inversely proportional to the branching parameter of the tail (cmc ∝ 1/gg′ ). This is because cmc is the consequence of the competition of entropic and enthalpic effects. Tails with a low branching parameter have lower configurational entropy68,69 and lose less configurational entropy in the process of aggregation. However, tails with a large value of branching parameter could extend far away from the point where the tail connects to the head. These tails exist in broad space around the connecting point and have a large interface exposing to other tails. As a result, they get a high probability to interact with the tail of other molecules under the same concentration, whereas for dendritic tails, all segments are constrained near the root by bonds, and it is difficult for them to contact with other tails, to say nothing of aggregation. Therefore, copolymers with a linear tail could aggregate into the micelle at lower concentration, and the cmc increases with the decrease of the branching parameter of the tail. The micellization fraction of copolymers in solution directly determines the yield of the self-assembled micelles, which is an important factor in manufacture. We present in Figure 2b the micellization fraction of copolymers, which is defined as the number of aggregating molecules divided by the total. The results indicate that more molecules of linear tail copolymer aggregated into micelles under the same conditions. Accordingly, it has to raise the concentration of molecules in solution for dendritic tail copolymers to reach the same micellization fraction. 3.3. Aggregation Dynamics. Polymeric micelles are a dynamic system which continuously exchanges units between the micelle structure and the free units in solution.70 In order to study the effect of tail architecture on the dynamics of copolymer (68) Karplus, M.; Kushick, J. N. Macromolecules 1981, 14, 325–332. (69) Vorov, O.; Livesay, D.; Jacobs, D. Entropy 2008, 10, 285–308. (70) Florence, A. T.; Attwood, D. Physicochemical Principles of Pharmacy, 3rd ed.; Macmillan Press: Hampshire, 1998. (71) Haliloglu, T.; Mattice, W. L. Chem. Eng. Sci. 1994, 49, 2851–2857.
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Figure 4. Autocorrelation function of micelles for copolymers with a linear tail, branched tail, starlike tail, and dendritic tail at φtotal ) 0.105.
Figure 6. Aggregate size distribution in solution of φtotal ) 0.105 for four types of copolymers.
times are also plotted in Figure 5. Apparently, the linear fitting to these data is applicable for the case, which means that the characteristic relaxation time has an inversely exponential relationship with the branching parameter of the tail, and it could be described by the following equation
log tc ) 3.223gg’ + 1.104
Figure 5. Logarithm of characteristic relaxation time as a function of the branching parameter of a hydrophobic tail. The point is the calculation data, and the line is a linear fit of the data.
aggregation, we calculated the tracer autocorrelation function71 CN(t) defined as
CN(t) )
〈(N(t0 + t) - 〈N(t0)〉)(N(t0) - 〈N(t0)〉)〉 〈(N(t0) - 〈N(t0)〉)2 〉
(9)
where N(t) is the number of molecules in the aggregate which a copolymer resides at time t. We took all copolymers as tracer molecules over all simulation time for the averages. The characteristic relaxation time, tc, is defined as the required time for CN(t) to reach the value of e-1. Figure 4 shows the autocorrelation functions of the four types of copolymers at the concentration of φtotal ) 0.105. In comparison to these of micelles formed from dendritic, starlike, and branched tail copolymers, the autocorrelation function of micelles from the linear tail copolymer relaxes slowly. In order to examine the dependence of micellar dynamics on tail architecture, we calculated the logarithm of characteristic relaxation time as a function of the branching parameter of the tail in Figure 5. The characteristic relaxation time increases with the increase of branching parameter of the tail, and the characteristic relaxation time of the linear tail copolymer is greatly longer than those of another three copolymers. In order to verify the relationship of characteristic relaxation time, micelle size, and micelle shape with the branching parameter of the tail, we also simulated the self-assembly of copolymers with two additional tail architectures, whose branching parameters are 0.736 and 0.580, at the concentration of φtotal ) 0.105. Their characteristic relaxation
(10)
These obtained results reveal that the hydrophobic linear tails in the micelle core are tightly wound together to each other; thus, the formed micelles are physically stable. However, the branched, starlike, and dendritic tails are loosely bound in the core. In other word, the micelles formed from copolymers with a large branching parameter of the tail have higher stability. 3.4. Aggregate Size. Aggregate size and gyration radius distributions are useful tools for characterizing the micelles formed from the aggregation of amphiphilic copolymers. Previous work revealed that large aggregates are formed only after the solution reaches the concentration of cmc, and with the increase of concentration a Gaussian-shaped distribution appears.53 We calculated the aggregate size and gyration radius distribution of micelles in solution at the concentration of φtotal ) 0.105. Figure 6 shows the distribution of the aggregate size. The aggregate size distribution for linear tail copolymers appears as a sharp peak, which illustrates the narrow dispersity of micelles. The peak values are smaller for starlike and dendritic tail copolymers, and the micelles from them are widely dispersed. The gyration radius distribution of the micelle and micelle core are shown in Figure 7. The distributions reveal that the gyration radius of linear tail copolymer has the narrowest distribution. Therefore, a linear architecture of hydrophobic tail is a good choice for preparation of narrow dispersed micelles. The values of most probable aggregate size and radius of gyration for different architectures of tail are plotted with respect to the logarithm of the branching parameter of the tail in Figure 8. We could find that both the most probable aggregate size and radius of gyration increase with the increase of branching parameter of the tail, which agrees very well with previous experimental observations.57 This provides an approach to control the micelle size by monitoring the architecture of the tail. The linear fit of data in Figure 8 illustrates that the aggregate size and radius of gyration have an exponential relationship with the branching parameter of the tail, which could be represented by the following equations
N ) 37.383 log gg’ + 46.386
(11)
Rcore g ) 1.325 log gg’ + 5.470
(12)
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Figure 9. Density profiles of micelles at φtotal ) 0.105 for four types of copolymers. The solid lines are of micelle shells, and the dash-dot lines are of micelle cores.
Figure 7. Gyration radius distribution of (a) micelle core and (b) micelle at φtotal ) 0.105 for four types of copolymers.
Figure 8. Aggregate size and radius of gyration as functions of the logarithm of the branching parameter of a hydrophobic tail. Aggregate size links to the left axis, and radius of gyration links to the right axis. Spherical points are for the core of the micelle, and triangle points are for the whole micelle.
Rmicelle ) 2.224 log gg’ + 8.094 g
(13)
3.5. Microstructure and Asphericity. Microstructure could reflect the detail information of a polymeric micelle. The density profiles of micelles at the concentration of φtotal ) 0.105 are shown in Figure 9. The density profiles indicate that the micelles are core-shell structure with a compact core. The density of the shell formed from linear tail copolymer is larger than the left at the same radius, which illustrates the shell from the linear tail copolymer is more condensed than that of the branched tail, starlike tail, and dendritic tail copolymers. This property is particularly important for applications of polymeric micelles in controlling release of drug slowly. We could also find from Figure 9 that the micelle formed from self-assembly of linear tail copolymers has a thinner interface between the core and the shell. The density for the linear tail in the core of micelles is the highest among the four types of copolymers, which illustrates that the tails are tightly wound together. This also proved that the micelles formed from linear tail copolymers have higher stability.
Figure 10. Snapshots of micelles formed from four types of copolymers at the most probable micelle number determined by the aggregate size distribution. Red is for hydrophobic segments, and green is for hydrophilic segments.
Figure 10 shows the snapshots of micelles of different copolymers studied for the most probable aggregate size at the concentration of φtotal ) 0.105. We could see from the snapshots that with the change of tail architecture from linear to branched to starlike to dendritic, the shape of the micelle formed by selfassembly transforms from spherical to planar, and the geometric size of the micelle also decreases. The shape of micelles can also be measured numerically with the dimensionless relative shape anisotropy, κ2, which is defined as72
κ2 ) (b2 + 0.75c2)/s4
(14)
2
where s equals to the squared radius of gyration, b is asphericity, and c is acylindricity
s2 ) λ1 + λ2 + λ3 b ) λ1 - 0.5(λ2 + λ3),
(15) bg0
(16)
(72) Theodorou, D. N.; Suter, U. W. Macromolecules 1985, 18, 1206–1214.
Effect of Tail Architecture on Amphiphiles
Langmuir, Vol. 25, No. 5, 2009 2755
Figure 11. Relative shape anisotropy as a function of the branching parameter of a hydrophobic tail. Square points are for the core of the micelle, and spherical points are for the whole micelle.
c ) λ2 - λ3,
cg0
Figure 12. Normalized aggregate size as a function of temperature for four types of amphiphilic copolymers.
(17)
λ1, λ2, λ3 are the eigenvalues of the radius of gyration tensor GuV in descending order, i.e., λ1 . λ2 . λ3. GuV is given by n
GuV )
∑
1 (ru - Ru)(riV - RV) n i)1 i
(18)
where rui and Ru are the coordinates of segment i and the mass center of the micelle in the u direction. Subscripts u and V indicate two variables among x, y, and z. The quantity κ2 is between 0 and 1. Structures of tetrahedral or higher symmetry, say, spherical shape, are characterized by κ2 ) 0. For a regular planar array such as masses at the vertices of a regular polygon or a homogeneously filled polygon, κ2 ) 0.25; for the shape of cylindrical symmetry, κ2 ) 1. The values of the relative shape anisotropy of the core and micelle for each architecture are plotted with respect to the branching parameter of the tail in Figure 11. These values illustrate that the amphiphilic copolymer with linear hydrophobic tail tends to self-assemble into a spherical core, whereas those with branched, starlike, and dendritic tails tend to form a planar core. The figure shows that in the range of relatively larger branching parameter than 0.3, the symmetry of the polymeric micelle increases with the increase of the branching parameter of the tail. The shape of the whole micelle is of higher symmetry than that of the core for all types of copolymers studied in this work. 3.6. Thermal Stability. In order to investigate the architecture of the tail on the thermal stability of polymeric micelles, we simulated the self-assembly of the four types of copolymers under different temperatures. Figure 12 shows the normalized aggregate size as a function of temperature for four types of amphiphilic copolymers. It can be found that the normalized aggregate size of the branched tail, starlike tail, and dendritic tail copolymers decreases quickly with the increase of temperature, whereas it keeps little changed in a range of temperature for linear tail copolymers. Figure 13 shows the normalized radius of gyration of the micelle core as a function of temperature. The normalized radius of gyration increases first and then decreases with the increase of temperature for the linear tail, branched tail, and starlike tail copolymers. However, it decreases in the whole range of temperature studied for dendritic copolymers. The increase of radius of gyration of the micelle core results from the fact that the slight increase of temperature makes the tails stretch but not disassociate from each other. The radius of gyration decreases until some copolymers disaggregate from the micelle at high temperature. This could be verified by the little change
Figure 13. Normalized radius of gyration of the micelle core as a function of temperature for four types of amphiphilic copolymers.
Figure 14. Normalized relative shape anisotropy of the core of the micelle as a function of temperature for four types of amphiphilic copolymers.
of aggregate size with temperature increase for linear tail copolymers. In summary, we could conclude that the thermal stability of polymeric micelle increases with the increase of the branching parameter of the tail in the terms of geometry size and aggregate size. Therefore, the polymeric micelle formed from the linear tail copolymer best satisfies the requirements that the micelle keeps aggregated at the physiological temperature but disaggregates in a local heated target site where temperature is above the physiological temperature.48,49 Figure 14 shows the normalized relative shape anisotropy of the core of micelle as a function of temperature. The normalized relative shape anisotropy of the micelle core formed from tails of high branching parameter increases quickly, which indicates that the shape of the micelle core is difficult to keep unchanged for a high branching parameter of the tail, and with the increase of branching parameter, the degree of difficulty increases.
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4. Conclusions In this work, we studied the effect of tail architecture of amphiphilic copolymers on the self-assembly in solution by using Brownian dynamics simulations. A coarse-grained model was employed to represent these copolymers. Four typical types of tail architectures (linear, branched, starlike, and dendritic) were considered for comparison. The cmc, dynamics of aggregation, aggregate distribution, gyration radius distribution, density profiles of micelle, shape anisotropy, and thermal stability were examined for the polymeric micelles formed from these architectures. For convenience, we modified the formula of branching parameter and used it to represent the architecture of the tails. The obtained results indicate that the cmc is inversely proportional to the branching parameter of hydrophobic tail. We also found that the most probable aggregate size and gyration radius of the micelle increase with the decrease of the branching parameter of the tail, and they have an exponential relationship with the branching parameter. Micelles
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formed from the copolymers with tails of high branching parameter are narrowly dispersed and have spherical shape. The linear architecture makes the tails to tightly wind to each other in the micelle core. In addition, our calculated results indicate that the thermal stability of polymeric micelle increases with the increase of the branching parameter of the tail, where the thermal stability is described by the geometry size and aggregate size. Few studies were performed on the effect of tail architecture on the self-assembly of amphiphilic copolymers. Thus, this work provides important information to design the self-assembling systems to prepare polymeric micelles for applications of drug delivery. Acknowledgment. This work is supported by the National Natural Science Foundation of China (Nos. 20874005, 20776005, 20736002), NCET Program from Ministry of Education of China (NCET-06-0095), Beijing Novel Program (2006B17), “Chemical Grid Project”, and Excellent Talent Funds of BUCT. LA803839T
