Tadpole Micelles of Diblock Copolymers

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Tadpole and Mixed Linear/Tadpole Micelles of Diblock Copolymers: Thermodynamics and Chain Exchange Kinetics Ammu Prhashanna and Elena E. Dormidontova* Polymer Program, Institute of Materials Science and Physics Department, University of Connecticut, Storrs, Connecticut 06269, United States S Supporting Information *

ABSTRACT: Chain architecture is known to control macromolecular self-assembly and furthermore affect in a more complex way nanostructure stability. Equilibrium properties and chain exchange kinetics between micelles formed by tadpole-shaped diblock copolymers containing a loop-shaped hydrophobic block and a linear hydrophilic block are investigated using dissipative particle dynamics simulations. We found that tadpoles form micelles of smaller size and aggregation number than the corresponding linear diblock copolymers and have faster chain exchange kinetics, demonstrating that chain architecture can alter its effective hydrophobicity. Similar observations are made for linear diblock copolymer with a less hydrophobic core block indicating that the more compact conformation of tadpole coreforming block makes it “less hydrophobic”. We show that tadpole and linear block copolymers form mixed micelles with tadpoles (or less hydrophobic chains) located on the periphery of the micelle core. The chain exchange kinetics between mixed micelles is found to be quicker than in linear diblock copolymer micelles and slower than in tadpole micelles. Tadpole escape or less hydrophobic chain exchange between mixed micelles occurs slower (in part due to the shielding role that these chains play) than in the corresponding pure micelles, while linear more hydrophobic chain exchange only slightly changes, suggesting that the exchange kinetics of the individual components can be affected differently by mixing.

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INTRODUCTION Self-assembly of block copolymers, which has been actively studied during the past two decades, plays an important role in understanding the fundamentals of self-organization in polymer systems and is the key for many practical applications of novel materials. With advancements in polymerization techniques, a variety of new chain architectures have become available such as cyclic,1,2 star,3,4 H-shaped,5 Y-shaped,6 comb,7 tadpole,8−12 etc. Chain architecture and micellization kinetics are among the main factors determining the outcomes of self-assembly, while the chain exchange between equilibrium micelles controls the stability of polymer nanostructures.13 Research on the selfassembly of any new chain architecture adds new insights into the general database of knowledge of the main driving forces of self-organization. The present work contributes to this effort, as we investigate by means of dissipative particle dynamics (DPD) simulations how one of the simplest cases of nonlinear diblock copolymer architecture, a tadpole-shaped block copolymer with a loop-shaped intramolecularly cross-linked hydrophobic block and a linear hydrophilic block (see Figure 1), affects both the equilibrium micelle properties and kinetics of chain exchange between micelles. The interest in tadpole-shaped copolymers originates primarily from the attractive idea of unimolecular micelle formation for encapsulation/interaction with biological mole© XXXX American Chemical Society

cules. While successful realization of the concept proved to be challenging, it made available a new chain architecture, which on one hand is quite similar to a diblock copolymer but on the other hand has principal differences in the conformation and therefore properties of the cross-linked block, which could open up new features in self-assembly processes. For instance, recent experimental data show that self-assembly of tadpoleshaped copolymer with a highly cross-linked hydrophobic block can lead to “bunchy” spherical micelles or vesicle formation.9,11,12 A tadpole copolymer represents an interesting example of an essentially identical chemical structure but a conformationally different architecture than its diblock precursor, thereby making possible a direct comparison of the equilibrium and kinetic properties of their self-assembled aggregates in order to discern the effect of chain architecture. For example, in both recently reported cases of tadpole copolymers with strongly cross-linked hydrophobic9 or hydrophilic blocks8,10 the self-assembled aggregate size is found to exceed that formed from linear diblock copolymer precursor. The molecular-level chain organization inside the aggregates Received: November 15, 2016 Revised: January 27, 2017

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Figure 1. Schematic presentation of tadpole (a) and linear (b) diblock copolymers. The hydrophobic blocks of tadpole and linear diblock copolymers are shown in green and red, respectively, and the hydrophilic blocks for both polymers are shown in blue.

thermodynamic and kinetics of pure tadpole and tadpole/linear mixed micelles. As has been shown in previous papers, the DPD technique permits us to explore large space and time scales as is required to study self-assembly and kinetics of chain exchange.26,27,34−37 To this end, we analyze the size distribution and structure of micelles formed by tadpole copolymers and compare these properties to micelles formed by the linear diblock copolymer analogue. We also investigate the difference in chain exchange kinetics between equilibrium micelles of tadpole copolymers and linear diblock copolymers. The equilibrium structure of mixed tadpole/linear micelles and chain exchange between mixed micelles are analyzed. Our predictions on the effect of chain architecture on the equilibrium properties of self-assembled structures and kinetics of chain exchange can serve as guidance for further experimental research and have important implications for the practical applications of such nanomaterials.

formed from tadpole copolymers or kinetics of chain exchange has not been investigated so far. The kinetics of chain exchange between traditional diblock copolymer micelles still remains an active area of research.14,15 The chain exchange kinetics between block copolymer micelles in solution is most commonly explained based on an unimer expulsion/insertion mechanism.16,17 However, simple firstorder kinetics (e.g., an exponential decay to equilibrium) is rarely observed experimentally with multiple reports of a logarithmic time dependence, suggesting multiple dynamic processes.18−20 Choi et al.20 have argued that this complex behavior arises from the polydispersity of the core-forming block coupled with an exponential dependence of the activation energy on hydrophobic block length. This model has been applied to successfully fit micelle TR-SANS data for a number of different polymer systems. 18,21 Only for a strictly monodisperse system is a single-exponential decay reported.22 A variety of simulation techniques have been implemented to study the chain exchange process, including Monte Carlo, stochastic dynamics, and dissipative particle dynamics (DPD) techniques.23−27 In all cases of monodisperse micellar solutions considered, the chain exchange followed first-order kinetics dominated by the chain expulsion/insertion mechanism. Mixing block copolymers of different lengths or architecture represents a simple way to create new nanostructures or alter properties of pure block copolymer aggregates.28 Thus, understanding conditions leading to comicellization and an investigation of the stability of mixed aggregates is of obvious importance for new nanomaterial development and applications.29 A few experimental studies considered chain exchange in mixed micelles containing either diblock copolymers of different lengths or a mixture of diblock and triblock copolymers.30,31 One of the main conclusions of these previous reports is the independence of chain exchange on the micelle composition; i.e., chain exchange between mixed micelles occurs in a manner similar to that in pure micelles formed by each of the constituents. What has not been considered so far is the effect of nonlinear chain architecture in pure or mixed micelles on aggregate stability and/or chain exchange. In this paper we focus on this effect for the particular case of spontaneously formed spherical micelles composed of linear and tadpole diblock polymers. This work will contribute to the general understanding of the mechanism of coassembly and stability of mixed micelles, which can be of interest for various technological applications, including drug delivery.32,33 In this study, we employ DPD simulations to analyze the effect of tadpole chain architecture (as shown in Figure 1) on

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SIMULATION METHODOLOGY AND DETAILS To model tadpole and linear diblock copolymers, we employed dissipative particle dynamics (DPD) simulations,38,39 which are well-suited to study the thermodynamics and kinetics of largescale diblock copolymer assembly.34,36,37,40,41 We model A5B6 diblock copolymers, where A and B are respectively hydrophobic and hydrophilic beads with the subscript representing the number of coarse-grained units. We note that each coarsegrained unit represents a group of several atoms and in general is larger than a polymer repeat unit.42 For the tadpole block copolymer, the first and fifth bead of A block are connected (using the same harmonic spring as for a regular chemical bond) into a ring structure, as shown in Figure 1. Chemical bonds between beads are modeled as harmonic springs, Fsij = K(rij − r0) with K = 100 being the spring constant, rij is the separation distance between two beads, and r0 = 1 is the equilibrium bond length with all the beads having an identical mass.26 In DPD simulations the interparticle force (Fi) exerted on bead i by bead j is a combination of conservative (FCij ), dissipative (FDij ), and random forces (FRij ).39,43 Fi =

∑ FijC + FijD + FijR i≠j

(1)

These forces act only when two beads are within a cutoff distance of rc = 1. The dissipative interaction coefficient in FDij is taken as 4.5,44 and the values for conservative interaction force coefficients aij, which are related to the χ-parameters,39 are chosen as follows. The interactions between alike beads and B

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Macromolecules between the hydrophilic block and solvent are aWW = aAA = aBB = aBW = 25, indicating that the hydrophilic block is compatible with solvent, taken to be water (W).26,27 The interactions between the hydrophobic block and solvent or hydrophilic block were more repulsive: aAB = aAW = 38, as summarized in Table 1. In the present study, mass, distance, energy, and time Table 1. Systems Studied: Volume Fraction of Tadpole (φT), Linear (φL), and Weakly Hydrophobic (φLw) Diblock Copolymers, Conservative Interaction Parameters aAB = aAW, and Average Micelle Aggregation Numbers (Pn)a system

φT

φL

φLw

aAB

T L Lw mixed LT20 mixed LT60 mixed LLw20

0.05 0 0 0.01 0.03 0

0 0.05 0 0.04 0.02 0.04

0 0 0.05 0 0 0.01

38.0 38.0 36.0

Pnb 29 44 31 44 41 45

± ± ± ± ± ±

3 4 3 5 4 5

Figure 2. Number-average micelle aggregation number distribution for tadpole (T, squares) and linear (L, diamonds) diblock copolymer micelles and mixed micelles: LT20 (20% of tadpole, T, and 80% linear, L, down triangles) and LT60 (60% of tadpole, T, and 20% linear, L, up triangles). Simulation data are shown as symbols and thick lines are Gaussian fit.

In all cases total polymer volume fraction φ = 0.05; other conservative interaction parameters: aWW = aAA = aBB = aBW = 25). b The aggregation numbers are obtained by fitting a Gaussian distribution to the aggregation number distribution data. a

are scaled by m, rc, kBT, and mrc 2/kBT , respectively. To obtain equilibrium micelles in all cases we used a cubic periodic simulation box of size 30rc × 30rc × 30rc and started with a homogeneous solution of diblock copolymers of total volume fraction, φ = 0.05, and allow self-assembly to progress and reach equilibrium. After t = 3.2 × 105 time units the configuration data are recorded every 4 time units (100 time steps) until t = 1.6 × 106, and the equilibrium properties of micelles are analyzed. Similar to our previous work,26,27 to identify a micelle we used a distance criterion: two chains are considered to belong to the same micelle if any of their hydrophobic beads are within the cutoff distance of 1.5. Apart from pure tadpole and pure linear micelles we also considered two mixed micelle solution: LT20 with 20% of tadpole (T) and 80% linear (L) chains; LT60 with 60% of tadpole (T) and 40% linear (L) chains. In addition, a less hydrophobic linear diblock copolymer denoted Lw (aAB = aBW = 36; the rest of the other interaction parameters remain unchanged) is studied along with a mixed micelles LLw20 (20% of Lw and 80% of L). The simulations are carried out using LAMMPS by adopting the NVT ensemble26,27 with a time step, Δt = 0.04. The details of kinetics chain exchange simulations, analogous to our previous work,26,27 are discussed below.

micelles (φunimer = (6.12 ± 2.6) × 10−4), in agreement with previous simulation results.35 We also studied two mixed micelle systems: LT20, obtained by mixing 20% of tadpole and 80% of linear diblock copolymer chains, and LT60, obtained by mixing 60% of tadpole and 40% of linear diblock copolymer chains. The micelle size distributions for the mixed micelles are also shown in Figure 2. The average aggregation numbers 44 ± 5 for LT20 and 41 ± 4 for LT60 are found to be practically the same as for micelles formed by linear diblock copolymers (Table 1) with the distribution width being somewhat more narrow for LT20 mixed micelles. The micelle structure and composition (for mixed micelles) were further characterized. In all cases we obtained spherical micelles with a well-defined core−shell structure. In both mixed tadpole/linear diblock copolymer solutions only mixed micelles are observed with the composition corresponding to the average concentration of the corresponding components in solution. This behavior is consistent with earlier computer simulation studies on linear diblock and triblock copolymer mixed micelles.44−46 The radial number density profiles for micelles close to the average size are calculated starting from the center of mass of the micelle for both LT20 and LT60 mixed micelle solutions and are shown in Figure 3. As is seen from the number density distribution and micelle snapshot, for LT20 the linear diblock copolymer forms the center of the mixed micelle core, as expected based on the composition, with the tadpole block copolymer present mainly on the periphery of the core. The LT60 mixed micelle, for which the tadpole polymer dominates, also displays the maximum of the tadpole number density distribution away from the center of the micelle. Thus, in both cases of mixed micelles the hydrophobic block of the linear diblock copolymers forms the center of the micelle with hydrophobic block of tadpole polymer present on the periphery of the core. This chain arrangement is rather logical taking into account that the hydrophobic block of the linear chain is more expanded (Rg = 0.89 ± 0.01) in the micelle core compared to the tadpole (Rg = 0.71 ± 0.01). The number

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RESULTS AND DISCUSSION Equilibrium polymer micelles are obtained by self-assembly of tadpole diblock copolymers (T) or linear diblock copolymers (L) in solution. In all cases the equilibrium micelle solution contained at least 12 micelles. The micelle size distribution was analyzed and averaged over 5.0 × 105 DPD time units at equilibrium. The obtained results are shown in Figure 2. As is seen from Table 1, tadpole block copolymers (T) form micelles with a number-average aggregation number Pn of 29 ± 3, which is smaller than that for micelles formed by linear diblock copolymers (L), Pn = 44 ± 4. Correspondingly, the concentration of unimers being in equilibrium with micelles (i.e., cmc) is noticeably higher for the tadpole polymer micelles (φunimer = (1.39 ± 0.4) × 10−3) compared to linear polymer C

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Figure 3. Radial number density profiles calculated starting from the center of mass of the micelle for LT20 (solid lines) and LT60 (dashed lines) mixed micelle systems. Number density of hydrophobic beads of tadpole and linear polymers is shown in green and red, respectively, and that for hydrophilic beads in blue in all cases. Simulation snapshot is shown for LT20 micelles with hydrophobic beads of tadpole and linear polymers shown in green and red, respectively, and hydrophilic beads as blue lines.

density distributions of the hydrophilic block in micelle coronas are nearly the same for both mixed micelles. Having characterized the static properties of the system, we performed in silico micelle hybridization simulations as illustrated in Figure 4a. At time t = 0, polymer chains in the equilibrated micelles are tagged as either blue or red such that each micelle has only one color and that there are approximately equal (less than 10% difference in) numbers of red and blue chains. The free polymer chains (unimers) are also tagged. As time progresses, chains start to exchange between micelles and red micelles acquire blue chains and vice versa. The hybridization process of the micelle solution is characterized through a contrast function (C(t)).19,20,26,27 To this end, the following time-dependent function (I(t)) is calculated: I (t ) = 4

⎡⎛ N (t )

∑ ⎢⎢⎜ N

r

⎣⎝ N (t )

−

2 ⎤ 1 ⎞ N (t ) ⎥ ⎟ 2 ⎠ Ntotal ⎥⎦

Figure 4. (a) Schematic representation of micelle hybridization process. (b) Contrast functions (eq 3) for pure tadpole (T), pure linear (L), and mixed (LT20 and LT60) micelle solutions.

We note that the contrast function is proportional to the average fraction of red (blue) chains in the ensemble of aggregates. First, we have performed hybridization simulations separately for pure tadpole block copolymer micelles and for linear diblock copolymer micelles. The resulting contrast function C(t) is shown in Figure 4b. As is seen, the chain exchange process in tadpole block copolymer micelle (T) solution is considerably faster compared to that for linear diblock copolymer micelles (L). In both cases the contrast function can be satisfactorily fit into first-order exponential decay function (C(t) = exp(−t/τ)) with characteristic time decay constants of 17.5 × 103 and 102 × 103 (in DPD time units) for tadpole and linear diblock copolymers, respectively. From the micelle size distribution in Figure 2 and Table 1 we note that the average aggregation number of tadpole micelles (∼30) is smaller than that for linear diblock copolymers (∼45). Thus, it is possible that the difference in chain exchange kinetics between pure tadpole micelles and pure linear chain micelles is merely an aggregation number effect. To understand the impact of aggregation number on kinetics of chain exchange, we calculated the chain expulsion function for micelles of different aggregation numbers (see Figure S1 in the Supporting Information). While the chain expulsion process is found to be somewhat slower in micelles with higher aggregation numbers, the difference in the time constants is only about 10−15%. The difference in chain exchange kinetics between tadpole and linear diblock copolymers can be also attributed to a different number of free unimers present in solution, which, as discussed above, is larger for tadpole polymers. However,

(2)

where Nr(t) and N(t) are the number of red chains and the total number of chains in a particular micelle at time t, and Ntotal is the total number of chains in the system. The summation is performed for all aggregates in the solution, and ⟨...⟩ denotes ensemble averaging over different initial states. At time = 0, I(t) = 1, as all chains in an aggregate have one color, as time progresses, and chains start to exchange I(t) decreases and tend to zero as t → ∞ if a perfect match in the number of red and blue chains is achieved in each aggregate. This match is unattainable for unimers, thus to eliminate the influence of statistical effects in the initial micelle coloring/tagging and final imperfection in color distribution, which are affected by the micelle size distribution, I(t) is normalized, resulting in the contrast function26 ⎡ I(t ) − I(∞) ⎤1/2 C(t ) = ⎢ ⎥ ⎣ I(0) − I(∞) ⎦

(3) D

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Macromolecules comparing chain expulsion functions26,27 (see Supporting Information), which are independent of unimer concentration, from tadpole and linear diblock copolymer micelles with similar aggregation numbers (Figure S1), we found practically the same significant difference in chain dynamics as is seen for the contrast function in Figure 4. We also studied chain exchange kinetics in two mixed micelle solutions: LT20 and LT60 with 20% and 60% of chains being tadpoles, respectively. In both cases the average aggregation numbers are close to that for linear micelles (Table 1 and Figure 2). As is seen from Figure 4, the contrast function C(t) for LT20 exceeds that for LT60, implying a slower overall chain exchange, and both mixed micelles lie in between the two pure systems, indicating that chain exchange speeds up with an increase in the fraction of tadpole chains in the micelles. An intrinsically quicker dynamics of tadpole chain escape/exchange compared to linear diblock chains is a logical explanation for this trend. Having a ring structure, the hydrophobic block of the tadpole has a more compact conformation (in the large length limit Rg2 for a ring is twice smaller than that for a chain47) inside the micelle core and more importantly in solution compared to the linear analogue: 0.63 ± 0.08 (in DPD length units) and 0.71 ± 0.08 for the tadpole (T) and linear (L) copolymers, respectively. This implies that on average the hydrophobic block of a tadpole chain has fewer contacts with the solvent and hence would have a smaller potential barrier to overcome upon escaping from a micelle with concomitant quicker escape/exchange dynamics. To test this hypothesis, we reduced the (conservative) interaction parameters aAB = aBW to 36 from initial value of 38 for the hydrophobic block (B) of the linear diblock copolymer to match the average micelle size observed for tadpole diblock copolymers. Thus, effectively we obtained a somewhat less hydrophobic linear diblock copolymer (Lw) otherwise identical to the original diblock (L) in all but hydrophobicity. The micelle size distribution for the less hydrophobic linear diblock copolymer (Lw) together with original micelle size distributions for tadpole and mixed tadpole linear LT20 copolymers is shown in Figure 5a. As is seen, the unimer population and average micelle size are very similar for the less hydrophobic linear diblock (Lw) and tadpole (T) polymers, while the micelle distribution is somewhat broader for the former. Further, we investigated the chain exchange kinetics in micelles formed by the less hydrophobic linear block copolymers (Lw) and compared it to the results obtained for tadpole micelles (T). As is seen in Figure 5b, the contrast function for micelles formed by the less hydrophobic linear block copolymers (Lw) exhibits a very similar decay profile (perhaps even slightly faster) as the tadpole polymer (T) micelle. Furthermore, we also studied the mixed micelle solution formed from 20% of the less hydrophobic linear chain (Lw) and 80% of the linear diblock copolymer chains (L). This mixed micelle system exhibits a very similar size distribution as the mixed tadpole/linear micelles of the same composition (LT20), as is seen in Figure 5a. Furthermore, the density profiles for hydrophobic and hydrophilic blocks of mixed LLw20 are analogous to that of mixed LT20 micelles (Figure 5c). As is seen, the less hydrophobic linear diblock (Lw) is also partially segregated from the more hydrophobic block (despite their compatibility) and more likely to be found near the core/corona interface. This is the result of somewhat weaker hydrophobicity of Lw chains, which are partially shielding more hydrophobic chains L from the solvent. The

Figure 5. (a) Number-average micelle aggregation number distribution for tadpole (T, squares), less hydrophobic linear (Lw, aAB = aBW = 36, up triangles), and mixed micelles: LT20 (20% of tadpole, T, and 80% linear, L, down triangles) and LLw20 (20% Lw and 80% L, circles). Simulation data are shown as symbols, and thick lines are Gaussian fits. (b) Contrast functions C(t) for the chain exchange for the same systems, as in (a). (c) Radial number density profiles calculated starting from the center of mass of the micelle for mixed LLw20 micelles with number density of Lw and L hydrophobic beads are shown in green and red, respectively, and number density for hydrophilic beads are shown in blue.

analysis of chain exchange kinetics reveals that the contrast functions for these two mixed systems are nearly identical (Figure 5b). Thus, comparison of both pure tadpole micelles with micelles of less hydrophobic linear diblocks and the corresponding mixed (with linear chains) micelles shows that the faster kinetics of chain exchange by tadpole diblock copolymers can indeed be explained by weaker interactions E

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Macromolecules with solvent. So far as we know this is the first evidence that chain architecture can influence the apparent hydrophobicity of a polymer block. Next we analyzed the details of chain exchange kinetics for mixed micelles. Obviously, we can expect that the contrast functions would follow biexponential decay, as there are two chain populations in each mixed micelle. If the tadpole and linear chain exchange occurs completely independently of each other, the contrast function for the mixed micelles (Figure 4b) should be a linear combination of the contrast functions for the pure tadpole and linear diblock copolymer micelles. However, this is not the caseno acceptable fit can be achieved, as shown in Figure S2. To understand the chain exchange kinetics in the mixed micelle system, the contrast functions for tadpole and linear chain were evaluated separately. To this end, tadpole (or linear) chains were marked in red in approximately half of micelles and blue in another half, and the contrast function was calculated using eqs 2 and 3, where Ntotal now corresponds to the total number of tadpole (or linear) chains in the system. The results for LT20 and LT60 mixed micelles are shown in Figure 6. As is seen, the chain exchange process of the tadpole

chain exchange in mixed micelle affect each other: an increase in the fraction of linear chains in mixed micelles makes tadpole exchange slower while an increasing fraction of tadpoles speed up linear chain exchange. The observed coinfluence of tadpole and linear chain exchange in mixed micelles can be explained in part by small aggregate exchange, as shown in Figure 7. Indeed, in pure tadpole micelles and in mixed LT60 and LT20 not only a single chain but also small aggregates such as dimers or trimers can

Figure 6. Contrast functions calculated separately for tadpole (LT20T and LT60-T) and linear copolymers (LT20-L and LT60-L) in mixed (LT 20 and LT60) micelles in comparison to that in pure tadpole (T) or linear (L) diblock copolymer micelles.

block copolymer slows down in LT60 and even more so in LT20 mixed micelles compared to pure tadpole copolymer micelles. Correspondingly, the exchange of linear diblock copolymer chains becomes quicker in LT20 and even quicker in LT60 compared to pure linear diblock copolymer micelles. It is worthwhile to note that all contrast functions shown in Figure 6 for tadpole or linear chain exchange in mixed micelles are well-fitted by single-exponential decay functions with the decay constants listed in Table 2. The results shown in Figure 6 and Table 2 demonstrate that the kinetics of tadpole and linear Figure 7. Simulation snapshots showing the process of small aggregate (trimer) escape/exchange between tadpole micelles: (a) gathering together chains in the original micelle core, (b) aggregate escape from the original micelle, (c) aggregate splitting in solution, (d) aggregate insertion, and (e) merger into a different micelle. Hydrophobic blocks of the small aggregate are shown in cyan for clarity. Chains in the original micelle are shown in “ balls-and-sticks” representation; chains in the receiving micelle are shown as lines. Hydrophobic blocks are colored in blue (or cyan), and hydrophilic blocks are in red.

Table 2. Decay Constant, τ, for Tadpole and Linear Chain Exchange Contrast Functions Obtained by Fitting Data in Figure 6 by Single-Exponential Functions chain micelle

tadpole T

tadpole LT60

tadpole LT20

linear LT20

linear LT60

linear L

τ (×10−3)

17.5

29.5

41.1

63.3

86.8

102 F

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micelles, but not as slow as tadpole exchange in LT20 micelles. Furthermore, the exchange of more hydrophobic chains (L) occurs only slightly quicker than in pure micelles (L). The observed differences in the kinetics of chain exchange in LLw20 and LT20 are primarily attributed to the difference in small aggregate exchange contribution (Figure S3), which plays a very minor role for the former, but have a noticeable contribution for the latter. The analysis of unimer escape from mixed LT20 micelles presented in Figure S4 further confirms this observation showing that tadpole unimer escape from mixed micelles is slower than from pure tadpole micelles, while linear chain unimer escape is nearly the same as in pure linear diblock copolymer micelles. Thus, for mixed micelles with a minor contribution of small aggregate exchange, one should expect the more hydrophobic component to exchange in a similar manner as in pure micelles, but the less hydrophobic component exchange becomes somewhat slower than in the corresponding pure micelle. This effect might be attributed to several different factors: an increase in micelle aggregation number and size for mixed micelles compared to the pure micelle of the weaker hydrophobic component, which can lead to smaller cmc and slower chain passage through the corona or a higher probability of return to the micelle; slower chain diffusion in the core and smaller core shape fluctuations, which could strongly affect the escape probability of the less hydrophobic component of the mixed micelle. To sort out the effect of these different factors for chains of different architecture would require careful analysis of a larger portfolio of chain and micelle sizes, which we hope to undertake in the future. For mixed micelles formed from chains of somewhat different hydrophobicity, such as LLw20, the main reason for slowing down of less hydrophobic chain escape/exchange is a larger penalty for the solvent contact with remaining in the mixed micelle hydrophobic chains L in comparison to Lw− solvent contacts in pure Lw micelles. In other words, since less hydrophobic chains Lw form a partial shield for the more hydrophobic (L) core of mixed micelles (Figure 5c), an escape of a less hydrophobic chain decreases the shield leaving micelle core less protected, which is thermodynamically unfavorable, and results in slowing down of less hydrophobic chain escape.

escape the micelle. The advantages of small aggregate escape are smaller contact area (per chain) with solvent (see Supporting Information) and higher efficiency of chain exchange.34 The disadvantages are that it requires collective chain rearrangement in the core (Figure 7a) and a larger overall (not per chain) potential barrier for aggregate escape, which makes it a less frequent event than unimer escape. As is seen in Figure 7a,b, small aggregate escape starts from gathering together a small group of neighboring chains which separates from the core and progresses through the corona. Outside the micelle the small aggregate can stay intact or may split as is seen in Figure 7c and after some time join a different micelle (Figure 7d). Such small aggregates can be formed either from chains of the same type or contain chains of different architecture or chemical nature in mixed systems (Figure S3). Indeed, approximately 10% of linear and tadpole chains exchange as mixed dimers and more rarely trimers in LT20 micelles with the contribution of mixed small aggregates increasing to nearly 20% in LT60 mixed micelles, as shown in the Supporting Information. Such mixed aggregate escape occurs slower than tadpole expulsion, but quicker than linear chain escape, therefore slowing down tadpole escape and increasing linear chain expulsion. It is informative to compare the chain exchange kinetics in mixed tadpole/linear chain micelles, LT20, and mixed micelles of linear chains of somewhat different hydrophobicity, LLw20. The composition and aggregation number of mixed micelles is practically the same and so is the overall chain exchange rate (Figure 5b). However, the linear chains are somewhat less prone to small aggregate exchangeonly 3.7% of chains exchanged as mixed aggregates in LLw20 (Figure S3). The contrast functions for exchange of linear chains of different hydrophobicity Lw and L in mixed LLw20 micelles are shown in Figure 8. Similar to tadpole/linear copolymer mixed micelles, the overall contrast function of linear−linear mixed micelles is quite different from that for the pure micelles of the components. The exchange of the less hydrophobic chains (Lw) between mixed LLw20 micelles is slower than in pure Lw

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CONCLUSIONS Using DPD simulations, we analyzed the equilibrium properties and kinetics of chain exchange between micelles formed by tadpole-shaped block copolymers. We found that on average tadpole polymers form smaller size micelles and exhibit faster chain exchange kinetics compared to that for the linear chain analogue (Figures 2 and 4). Both observations can be explained by the more compact conformation of the core-forming block of tadpole chains leading to effectively less contact with the solvent and therefore diminished hydrophobicity. When we reduced the hydrophobicity of the linear diblock copolymer chain to match the tadpole equilibrium aggregation number, we observed nearly identical chain exchange kinetics as for tadpole copolymers (Figure 5). Thus, a tadpole copolymer with crosslinked core-forming block acts as a less hydrophobic diblock copolymer and exhibits a higher CMC, smaller equilibrium aggregation number, and quicker chain exchange kinetics. We have also investigated mixed micelles formed by tadpole and linear diblock copolymers. We found that mixed micelles are mostly monodisperse in composition and have a similar size and aggregation number to that of pure linear diblock copolymer micelles (Figure 2). However, the distribution of

Figure 8. Contrast function for the individual components (LLw20Lw and LLw-20L) of mixed micelles of linear diblock copolymers of different hydrophobicity (LLw20) in comparison to pure micelles (Lw and L). G

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chains inside the micelle core is not homogeneousthe more compact hydrophobic blocks of tadpole copolymers reside preferentially on the micelle core surface (Figure 3). Chain exchange kinetics between mixed micelles is found to slow down with an increase in the fraction of linear copolymer (Figure 4) but does not follow a simple linear combination of the decay functions of the two pure micelle components. Our analysis of chain exchange for the tadpole and linear chain components of mixed micelles shows that tadpole exchange slows down and correspondingly linear chain exchange speeds up with an increase of the counterpart fraction (Figure 6). Small mixed aggregate (mostly dimer) exchange between mixed micelles is partially responsible for this effect. The analysis of chain exchange kinetics in the mixed micelles formed by linear diblock copolymers of different core hydrophobicity, for which small aggregate exchange plays a rather minor role, shows that less hydrophobic chain exchange slows down in mixed micelles compared to pure micelles, while more hydrophobic chain exchange is less affected by being incorporated into the mixed micelle. The obtained results provide insights into the thermodynamics of self-assembly and kinetics of chain exchange between equilibrium micelles containing tadpole copolymers that can guide further experimental research and development of practical applications of tadpole-containing micelle systems. Furthermore, the analysis of chain exchange and unimer escape from mixed micelles indicates that not only the equilibrium properties but also the kinetic stability of micelles can be altered by blending together chains of different architecture, which creates a new pathway for creating new or fine-tuning existing nanomaterials.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b02481. Chain expulsion functions for tadpole and linear diblock copolymer micelles of different aggregation numbers; contrast function for mixed tadpole/linear diblock copolymer micelles in comparison with biexponential decay function; CMC and number of contacts with solvent for different systems studied; analysis of contribution of different kinetic events in mixed micelles; unimer escape in mixed LT20 mixed micelles (PDF)

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (E.E.D.). ORCID

Ammu Prhashanna: 0000-0002-9119-7552 Elena E. Dormidontova: 0000-0002-7669-8957 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research is in part supported by the start-up funding from the Institute of Materials Science, University of Connecticut and by the National Science Foundation (DMR-1410928). Data analysis was partially conducted at the High Performance Computing Cluster at the University of Connecticut. H

DOI: 10.1021/acs.macromol.6b02481 Macromolecules XXXX, XXX, XXX−XXX

Article

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DOI: 10.1021/acs.macromol.6b02481 Macromolecules XXXX, XXX, XXX−XXX