Simulation Analysis of the Kinetic Exchange of Copolymer Surfactants

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Simulation Analysis of the Kinetic Exchange of Copolymer Surfactants in Micelles Fabián A. García Daza, Josep Bonet Avalos, and Allan D. Mackie Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01225 • Publication Date (Web): 13 Jun 2017 Downloaded from http://pubs.acs.org on June 16, 2017

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Simulation Analysis of the Kinetic Exchange of Copolymer Surfactants in Micelles Fabián A. García Daza, Josep Bonet Avalos, and Allan D. Mackie∗

Departament d'Enginyeria Química, ETSEQ, Universitat Rovira i Virgili, Avinguda dels Països Catalans 26, 43007 Tarragona, Spain E-mail: [email protected]

Abstract The exchange of surfactants in micelles involves several processes which are dicult to characterize experimentally. Microscopic simulations have the potential to reveal some of the key aspects that take place when a surfactant spontaneously exits a micelle. We present a quantitative analysis of the kinetic exchange process over a large range of time. The study is based on a dynamic version of a single-chain mean eld theory using a coarse-grained model for poly(ethylene oxide)-poly(propylene oxide)poly(ethylene oxide) triblock copolymer systems. The kinetics described in our simulations involves three dierent regimes. After a fast initial rearrangement of the labeled chains, the system undergoes a logarithmic relaxation, which has been experimentally observed. Contrary to what has been stated in previous analyses, our simulations indicate that this regime is caused by the intrinsic physical behavior of the system, and is not only due to the polydispersity of the samples. Finally, the terminal regime is characterized by an exponential decay. The exit rates predicted by our simulations are in good agreement with the ones experimentally reported. In addition, we address the sequence of microscopic conformational changes undergone by the surfactants when leaving the micellar aggregates. We nd a subtle variation of the radius of gyration of 1

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the hydrophobic block, which challenges the vision of either a complete collapse or a full stretching commonly accepted in current theoretical and experimental literature.

Introduction

The ability of amphiphilic block copolymer systems to self-aggregate is of paramount importance in a wide range of technological applications, including: foaming 1 and emulsion processes, 2 stabilization of nanoparticles, 3 and controlled release of drugs. 4,5 This behavior emerges when the concentration of amphiphiles exceeds the critical micelle concentration (cmc). Above this concentration diverse physicochemical system properties experience a signicant change. Although the equilibrium properties of copolymer systems have been extensively studied, there is still a lack of information related to dynamic and kinetic processes. Research on the kinetic mechanisms have been carried out by way of a broad number of experimental techniques including temperature jump, 6 uorescence correlation spectroscopy 7 and time-resolved small-angle neutron scattering 813 (TR-SANS). In these experiments the system is assumed to be in thermodynamic equilibrium and near the cmc. Under these conditions expulsion and insertion of individual amphiphiles in the already formed micelles are the dominant dynamic processes for the mass exchange compared with fusion and ssion of micelles. The individual surfactant exchange has been theoretically proposed as being dominant for block copolymer surfactants by Halperin and Alexander, 14 based on the analysis of the Aniansson and Wall mechanism. 15,16 Recent experiments for triblock poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) copolymer (PEO-PPO-PEO) surfactants, also known as Pluronics, revealed that fusion and ssion rates are six orders of magnitude less than individual amphiphile insertion and expulsion rates. 17 However, detailed information of the microscopic behavior of chains involving exchange processes is still incomplete. For instance, TR-SANS data of correlation functions obtained from scattering intensities has been interpreted either as one of two contradicting processes. 2


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On the one hand, Lund and co-workers 8,9,18 interpret the neutron scattering intensity as the signature of a collapse of individual amphiphiles during the exit process. On the other hand, Choi et al. 12 suggested that a stretching of the insoluble blocks is responsible for their experimental observations. Although theoretical studies 1416 put forward explanations for the main mechanisms responsible for the exchange processes, explicit microscopic information is still missing. Within this scenario, Monte Carlo (MC) 19,20 and Brownian dynamics 21 simulations for diblock copolymers have been implemented aiming at the study of the exchange of surfactants between micelles and bulk solution. Molecular dynamics (MD) simulations 22 were used to determine the eect of aggregate size on the monomer exchange mechanism for short nonionic diblock surfactants. Moreover, the exchange of amphiphiles was also studied by dissipative particle dynamics simulations 23 for a set of diblock surfactants. Regarding simulations involving lattice models, the space discretization prevents a quantitative comparison with experimental systems. 19 Furthermore, there exist severe diculties to reach equilibrium even for short coarse-grained surfactants. 22 This overall situation calls for an alternative approach for a microscopic description of the dynamic amphiphile exchange processes. In particular, in this article we present the results of the application of a dynamic version of a mean eld method, known as the single-chain mean-eld (SCMF) theory, to explain the dynamics of the surfactant exchange between a micelle and the bulk solution. Equilibrium thermodynamic properties are obtained from a static SCMF theory. 24 The SCMF is built around the concept of an individual chain whose microscopic details are explicitly taken into consideration. Self-avoiding interactions as well as any kind of monomermonomer intramolecular potential are explicitly taken into account. The intermolecular interactions, however, are described by means of the interaction of the single chain with molecular elds obtained self-consistently. Thereby, all macroscopic properties can be formulated in terms of averages over a suciently large collection of one-chain conformations. The sampling of the chain conformation used in the SCMF can be constructed respecting the self-avoidance of the chains and keeping a sucient microscopic detail. Conceptually, the 3


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SCMF in this article diers from other popular self-consistent eld schemes, 25 where overlapping conformations are allowed, which necessarily aects the estimation of the system's free energy. Thermodynamic equilibrium properties calculated from the SCMF formalism are obtained by minimization of the free energy yielding the one-chain probability distribution function, as well as the solvent density prole. The SCMF has proven its quantitative predictive capacity in many systems, including short nonionic surfactants 2628 and block copolymer systems. 29 In particular, SCMF predictions have been successfully compared with experimental data, 29,30 standard MC 27,28 and MD simulations. 26 Although a considerable number of studies of surfactant systems in equilibrium based on SCMF calculations have been reported, only a few of them are devoted to the analysis of dynamic properties. For instance, Fang, Satulovsky and Szleifer 31,32 reported the kinetics of protein adsorption on surfaces, where the dynamics is determined by a combination of the SCMF theory and diusion equations. However, the explicit dynamics of the chains was not taken into account due to the static nature of the conformations used in the sampling to solve the SCMF equations. The dynamics was, therefore, determined by the motion of the solvent. Although this approach was adequate for proteins, it cannot be expected to apply for block copolymers whose conformational dynamics is the relevant time scale. In this article we construct a dynamic version of the SCMF in which the polymer conformations are allowed to change by means of a Metropolis process in continuously updated mean elds. An alternative self-consistent mean-eld simulation was proposed by Müller and Smith, 33 being similar in spirit to the treatment that we propose here. In the present work we have carried out a series of simulations with a dynamic SCMF scheme for a series of Pluronics. As far as the molecular description is concerned, we use a coarsegrained model for Pluronics developed in a recent study, 29 which has proven its predictive capacity in determining the cmc's of a series of dierent triblock copolymer amphiphiles. 4


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In the Computational Methodology section the theoretical aspects related with the SCMF formalism in equilibrium and its extension to the dynamic regime, as well as the details of the microscopic model, are presented. In the Results and Discussion section the predictions obtained in this work are compared to the available experimental data, in particular, with the exit rate constants. Furthermore the radius of gyration of the hydrophobic block and the angle between blocks, are also examined. Finally, a review of the most important results of this work is presented in the Conclusions section.

Computational Methodology Theoretical background The starting point is to consider a single chain that interacts with the surrounding external mean molecular elds whilst its intramolecular interactions are explicitly taken into account.

24,27,29

All thermodynamic quantities are determined from the knowledge of two

elds, namely, the solvent volume fraction prole tion function

P [α]

and the chain probability distribu-

i.e. the statistical weight of a chain conformation

of dierent congurations, Each conguration

φs (r),

{α},

α

belonging to the set

used to compute the averages over the chain conformations.

α is expressed as a set of vectors indicating the positions of all monomers

of the block copolymer. The averages, depending on the chain conformations, are performed by summing over all the conformations of the set The

P [α]

tem of

N

as well as

φs (r)

{α}, each weighted by its probability P [α].

are obtained by considering the mean-eld free energy of a sys-

physical chains in a solvent medium at a temperature

system volume

V

T,

which together with the

denes the canonical ensemble,

F = U − T S

5


(1)

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The rst term on the right hand side of this equation corresponds to the mean interaction energy given in terms of the single-chain individual contributions, namely, U ≡ N

dαP [α] (Uintra (α) + Uinter (α))

(2)

where Uintra(α) and Uinter (α) are the intramolecular and intermolecular contributions, respectively. Notice that Uinter (α) depends on the average concentrations of the mean elds of the remaining N − 1 chains and the solvent volume fraction prole. The second term on the right hand side of eq 1 includes the congurational and translational entropy of both chains and solvent, (3) S = −kB N dαP [α] log P [α] − kB drcs (r) log φs (r) Here, r is a position vector, cs(r) is the number concentration of solvent molecules at r, and φs(r) = vscs(r), where vs is the volume of a solvent molecule. Intermolecular energetic contributions are attractive since they correspond to van der Waals forces in non-electrolyte systems, whereas the inter-chain repulsive excluded volume terms, together with the excluded volume interactions with the solvent, are included by the incompressibility condition φs (r) + N φ(r) = 1

where

φ(r) =

dαP [α]φ(α, r)

(4)

(5)

In this equation we have introduced the volume fraction of monomers due to a polymer conformation α at a position r, as φ(α, r). The determination of the single-chain probabilities, P [α], and the solvent number concentration prole, cs(r), is made by minimizing the free energy functional in eq 1 with respect to these elds and subject to the volume-lling constraint given in eq 4. The constraint is introduced in the minimization of the free energy through a Lagrange multiplier eld π(r). 6


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The direct evaluation of δF/δP [α] = 0, and, δF/δcs(r) = 0 permits us to nd the P [α] and cs (r) that minimize the free energy eq 1, subject to the constraint eq 4, that is P [α] =

e−H[α]/kB T Q

(6)

where Q is a constant which ensures the normalization of the probability distribution, Q≡

dα e−H[α]/kB T

For the solvent one has cs (r) = Ns V

e−vs π(r)/kB T dr e−vs π(r)/kB T

(7)

(8)

where Ns is the total number of solvent molecules in the volume of the system V . Therefore, if one has N chains of N monomers each with a volume v, the system is constrained by the overall condition V = N N v + Nsvs. In eq 6 we have introduced the SCMF Hamiltonian, depending on the level of detail in the surfactant and solvent models. In eq 8 π(r) stands for the lateral pressure eld. Both will be described later. The solution of eqs 6 and 8, together with the condition in eq 4 gives the equilibrium solution of the SCMF problem. To implement the dynamics of the system, we have devised a method in which every conformation of the ensemble {α} is allowed to relax in the mean eld through a Metropolis algorithm. At every time step, the non-equilibrium mean elds are calculated from averages using the actual set of {α} conformations with their statistical weight P [α] at the time t. Then, the Metropolis algorithm is used to move each independent chain conformation in these non-equilibrium mean elds. The mean elds are kept constant during the motion of all the chains in the set {α} at t, to a new set of conformations {α}, corresponding to t + Δt. After this process, we update the elds to their value at t + Δt, according to the new set of chain conformations. Detailed balance in the Metropolis algorithm ensures that the nal equilibrium state is consistent with the minimum of eq 1. The 7


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solvent prole is instantaneously adjusted to the monomer prole, due to the large separation between time-scales existing in solvent motion compared with the evolution of the polymer conformations. The process is repeated until the equilibrium is reached. We have checked that this equilibrium coincides with the expected minimum in the free energy from equilibrium calculations. Furthermore, the Metropolis dynamics of the system in equilibrium permits a proper sampling of the mean-eld phase-space, needed for the study of the surfactant exit dynamic process of this work. The local displacement of chain monomers are accepted or rejected with a certain probability which depends on the change of energy between the trial and the original moves, taking the actual mean elds into account. This procedure is similar to other mean eld simulations 33 for polymer systems. For a nite set of independent chains representing the surfactant denoted by one

αn

where

o)

{α}o ,

the probability for a chain to change its congurational state

is

H[αo,n ]

p(αo → αn ) = min 1, e−(H[αn ]−H[αo ])/kB T

αo

to a new

(9)

is the SCMF Hamiltonian whose value is calculated in both, the initial (old,

and the trial state (new,

n).

Notice that, since each

α

chain is independent, all chains in

the set can be moved at once, which allows for a straightforward parallel implementation of the algorithm. Within this procedure, together with the calculation of the equilibrium state, it is also possible to follow the temporal evolution of the system and, in consequence, the dynamic properties.

Simulation Details In this work, the Pluronic copolymers are described by a coarse-grained model 29 in which the hydrophobic propylene oxide (PO), CH(CH 3 )CH2 O, and hydrophilic ethylene oxide (EO), CH2 CH2 O, groups are represented by beads of the same diameter

σ.

The Kuhn segment

of the chain is introduced by considering four and ve consecutive PO and EO beads as

8


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being rigid. The distance between consecutive beads is taken as σ . Hydrophobic and hydrophilic interactions are modeled by square well potentials centered on the beads, with inner and outer radii σ and 1.62σ , respectively. Due to the space lling constraints only unlike intermolecular interactions between EO, PO and solvent need to be considered. 34 The energetic well depth of the model was tted for each interaction from the experimental values of the cmc at a temperature of 37 ◦ C. In ref 29 we reported values EO,P O = 0.006 kB T /z , EO,s = 0.5 kB T /z and P O,s = 2.1 kB T /z . The volume of the interaction well has been set

to give a coordination number z = 26. These energetic parameters are analogous to the monomer-monomer interactions in the Flory-Huggins theory. In view of this interpretation, we may infer that z EO,s /kB T = 0.5 indicates that the EO block is close to complete solubility over the whole range of concentrations, while z P O,s /kB T = 2.1 indicates that the PO is highly insoluble. The positiveness of the coecients indicate that the interaction is repulsive, although the dierence in their value favors the contact between the EO monomer and the solvent over the PO-solvent interaction. Moreover, z EO,P O /kB T = 0.006 shows that the cross interaction between monomers is unimportant. Three fundamental concentration elds appear in the SCMF Hamiltonian, which is given by, 29 H[α] = Uintra (α) + (N − 1)EO,P O dr (ΦEO (α, r)cP O (r) + ΦP O (α, r)cEO (r)) + EO,s dr ΦEO (α, r)cs (r) + P O,s dr ΦP O (α, r)cs (r) (10) + kB T dr π(r) (φEO (α, r) + φP O (α, r))

The rst term corresponds to the self-interaction energy of the conguration α. The second, third and fourth terms relate the intermolecular interactions of the chain with the surrounding external elds created by the other N − 1 chains and solvent. This amounts to the available volume for a given monomer times the concentrations of all the other species, that is, the number of contacts times the interaction energy. Finally, the fth term is interpreted 9


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as the intermolecular repulsive interactions arising from the inclusion of the incompressibility condition in eq 4 and involves the so-called lateral pressure π(r), which can be expressed in terms of φs by solving eq 8. For more details see ref 29. Once the SCMF Hamiltonian has been identied and the incompressibility condition taken into account, we proceed to perform a series of independent MC moves over the entire congurational set of sampling chains {α}, recalculating the mean elds by solution of the SCMF equations 6 with 10 and 8 at every MC cycle. To ensure a Rouse-like motion of the surfactants, 33 the moves to displace the chain conformations are crankshaft moves over randomly selected monomers along each chain at every MC cycle. 35 As already mentioned, this simulation scheme provides an opportunity to study the dynamic behavior of Pluronic chains at a xed temperature. In this work we concentrate on the study of the kinetic exchange of monomers in micellar aggregates in thermodynamic equilibrium. Assuming that the micelles are spherical, the simulation box is divided into concentric spherical shells of xed width ranging between 1.9σ and 2.1σ in the case of the shorter and longer Pluronics, respectively. In consequence eq 10 is reduced to variations along the radial coordinate only, according to this discretization scheme.

In our analysis it is important to identify the physical time-scale related to the simulation cycles, in order to report the results in terms of real time to be directly comparable with the experiments. With this purpose, we perform a series of simulations for free chains moving in the bulk solution. In the mean eld scheme, this can be done by taking N = 1 in eqs 10 and 4 and by following the evolution of the independent set of free moving chains from eq 9. Notice that such a choice implies that the interactions of the single chain with the mean elds are exactly zero, although the intrachain interactions are still fully taken into account. Therefore, by construction, the dynamics of such a free chain is diusive and Rouse-like, 36 since no hydrodynamic interactions are considered for such short molecules. We can then determine the average displacement, Δr(tcyc )2 , of the center of mass of the chain and obtain the diusion coecient in the dynamic SCMF scheme given in terms of

10


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MC cycles, according to

(11)

Δr(tcyc )2 tcyc →∞ 6tcyc

DSCM F = lim

where the square displacement is taken as (r (t ) − r (0) ) /N being N the number of sampling chains in the set {α}. It is important to also note that all the lengths in the SCMF simulations are given in terms of the bead size diameter σ, and therefore any position vector is proportional to σ. In consequence the displacement term in eq 11 is given in units of σ . Within the simulations the diusion constant calculated from the above equation is given in units of σ /cycle. We x the correspondence between MC cycles and the experimental time by way of the dimensionless equality N{α} i=1

i

cyc

2

i

2

{α}

{α}

2

2

(12)

DSCM F D tcyc = 2 t 2 σ l

where l corresponds to the physical dimensions of σ. Since the data for the diusion coecient for our system and at the prescribed temperature are scarce and have large uncertainties, along this work we estimate the physical value through the Stokes-Einstein relation, namely k T (13) D= 6πηa B

where k T ≈ 4.28×10 kg m s at a temperature of 37 C, and the viscosity of the solvent is η = 6.91 × 10 kg m s for water. Due to the fact that our simulations reproduce the Rouse model, the hydrodynamic radius of the polymer is given by a = N σ/2. We choose the value σ = 0.2 nm for which the Stokes-Einstein estimate of the diusion coecients best ts the experimental diusion coecients available. Notice that a close value σ = 0.5 nm is reported in ref 37, which is similar to the one employed in recent MD simulations for a series of Pluronics in a coarse-grained model. For instance, we nd for Pluronic P85 (EO PO EO ) a diusion coecient of 3.6 × 10 m s which is close to the reported value of 3.7 × 10 m s in ref 39, in the −21

B

−4

2 −2

◦

−1 −1

38

26

−11

40

26

2 −1

−11

11


2 −1

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case of P104 (EO27 PO61 EO27 ) we obtained a value of 2.9 × 10−11 m2 s−1 which is not far from the interval 3.7−4.5×10−11 m2 s−1 reported in ref 40 for temperatures between 35 and 40 ◦ C, and, a value of 3.0 × 10−11 m2 s−1 was calculated for P123 (EO 20 PO69 EO20 ) being comparable to the value in the interval 2.8 − 3.3 × 10−11 m2 s−1 reported from ref 41 at a temperature of 25 ◦ C. The simulations were performed in boxes with volumes between (70σ)3 and (100σ)3 with periodic boundary conditions. A total of 500 sampling chains, N{α} , were used for every Pluronic surfactant. Simulations were performed on 12-core Intel nodes and 24-core AMD nodes with RAM memories of 64 and 32 GB, respectively.

Results and Discussion Exchange dynamics of the surfactants The exchange of chains between the bulk solution and the micellar aggregate can be explicitly studied, based on the dynamic SCMF simulations, by the formulation of the proper correlation function. In our case this correlation function allows for a direct monitoring of the evolution of the chains in the micelle. Explicitly, we dene,

F (t) =

f (t) − f (∞) f (0) − f (∞)

(14)

where f (t) is the remaining fraction of labeled chains at the simulation time t which were originally in the micelle at time t = 0. If a labeled chain returns to the core of the micelle, it will contribute once more to the overall fraction f (t). This correlation function is close to the one used in TR-SANS experiments for n-alkyl-PEO 11 and poly(ethylene- alt-propylene)poly(ethylene oxide) 10 systems, where exchange kinetics in equilibrium is investigated by monitoring the behavior of deuterated and non-deuterated copolymers in micelles on mixing. These conditions are similar to the ones in our simulation scheme where, in contrast,

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no physicochemical alteration of the system needs to be taken into consideration. An evaluation of the correlation function given in eq 14 for Pluronics, EO n POm EOn , where m and n stand for the number of monomers of the copolymer blocks, reveals in general a

relaxation behavior arising from the exit process of surfactants in equilibrium micelles. In all cases, the function F (t) is monotonously decreasing, as seen in Figure 1 for Pluronic EO13 PO30 EO13 . The conversion from the simulation cycles, tcyc to the physical time scale is made according to eq 12 in the Computational Methodology section, with σ = 0.2 nm, DSCM F (L64) = 0.0023 σ 2 /cycle and D(L64) = 5.9 × 10−11 m2 s−1 , which has been estimated

from eq 13. In Figure 1 we observe two dierent relaxation regimes separated apparently by 1 0.9 0.8 0.7 0.6

F(t)

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0.5 0.4

0.3

0

0.5

1

1.5

Time [μs] Figure 1: Correlation function in a log-linear scale for Pluronic L64 (EO 13 PO30 EO13 ) obtained from the analysis of the data from dynamic SCMF simulations. a wide crossover region. Firstly, a fast relaxation, which can be interpreted as a chain rearrangement after the labeling, which is experimentally very dicult to observe. Secondly, we nd a wide non-exponential region. Thirdly and last, we see a terminal regime characterized by a long-time single exponential decay process, ∼ exp (−kt). According to Halperin and Alexander, 14 k is the exit rate constant which is found to be independent of the micellar size N . The kinetic constants in our work are obtained by tting this terminal regime.

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With regards to the initial regime, the labeled chains do not necessarily leave the micelle. On labeling, the surfactants explore the immediate regions in the corona. Since we are considering that a chain leaves the micelle when the center of mass of its PO block crosses the interface between the core and the corona, this rearrangement of chains after labeling appears as an eective decay of F (t). The apparent crossover regime can be better analyzed from Figure 2, which corresponds to the same data as in Figure 1 but represented in a linearlog plot, as has been done in some experiments. One can observe from the gure that the 1 0.9 0.8 0.7

F(t)

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0.6 0.5 0.4 0.3 -11 -9 -10 10 10 10

-8

10

-7

10

-6

10

t [s]

Figure 2: Correlation function in a linear-log scale for Pluronic L64 (EO PO EO ). 13

30

13

crossover regime is of the form F (t) ∼ − log t over two decades in time. Such a peculiar behavior has been experimentally reported in refs 79,12,13. From a theoretical point of view, the logarithmic decay is attributed to a broad distribution of exit rates. These authors argue that the broad distribution is caused by an inherent polydispersity of the polymer samples. However, our simulations show the same behavior for a strictly monodisperse sample, which suggests that the cause of the non-exponential decay lies in a fundamental physical process characteristic of this system. We have suggested that the degeneracy of the energy of the surfactants initially in the micelle is dynamically broken on leaving, giving rise to the 42

14


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observed broad spectrum of relaxation times. The processes involved in this crossover are fast and the surfactant conformations are virtually frozen during the exit. Eectively, hydrophobic PO monomers in the core of the micelle do not energetically distinguish between inter and intra chain contacts. However, on crossing the core-corona interface, the height of the barrier is proportional to the number of inter chain contacts, which are now accessible to unfavorable tail-head or tail-solvent interactions. Therefore the degeneracy of the energy states in the core is broken on exiting the micelle. The terminal regime corresponds to a single exponential decay 14 F (t) ∼ exp(−kt). In Table 1 the physical parameters of the Pluronics studied in this work are presented. Experimental values are taken from ref 43 and references therein, unless otherwise indicated. As can be observed, our calculations reveal a signicant decrease in the values of the exit Table 1: Physical characteristics and predictions for the Pluronic systems, EO n POm EOn . M W stands for the molecular weight of the surfactant, m and n are lengths of the hydrophobic and hydrophilic blocks, respectively, T is the temperature, N is the equilibrium aggregation number, while kexp and kSCM F identify the exit rate constants, experimental and predicted, respectively.

a

Pluronic M W L44 2200 L64 2900 P84 4200 P85 4600 P104 5900 P123 5750 Taken from ref 29

m 2n 23 20 30 26 43 34 40 52 61 54 69 40 at 37 ◦ C

T (◦ C) 40 27 37.7 24.2 21.4 using

N kexp (s−1 ) 145a 40 2.0 × 105 40 8.4 × 104 40 6.7 × 104 50 3.8 × 103 40 1.5 × 103 SCMF simulations in

kSCM F (s−1 ) 1.57(0.04) × 106 5.4(2.5) × 105 3.2(1.2) × 104 9.3(1.3) × 104 4.5(1.5) × 102 5.2(1.5) × 102 equilibrium regime.

rate constants with the increase of the insoluble PO units. An increase in the hydrophobic units of all surfactants produces a growth in the energetic barrier seen by a copolymer in the interior of the micelle. The escape time of the surfactant from this energy well to the bulk solution is therefore increased. The exit process is not found to be diusion controlled, but instead it is dominated by the exit rate since R2 /D 1/k , with R the characteristic size of the micelle. In addition, changes in the number of hydrophilic units, EO, only moderately aect the values of the exit rates, as has also been experimentally reported. 43 This fact im15


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plies that an increase in the width of the soluble micellar shell does not signicantly aect the height of the energetic barrier to be overcome by the copolymers. This type of terminal regime is observed for all Pluronic systems in our calculations, and is supported by the experimental data in Table 1. In particular, the exit rate constants for the Pluronics show a progressive decrease of three orders of magnitude when the number of hydrophobic PO units is increased from 23 to 69. Although a fair quantitative agreement with the experimental data is found for L64, P84 and P85, the experimental rate constant decreases more slowly than the simulated one for the largest Pluronics, P104 and P123. The nature of this discrepancy between our calculations and experimental data can be due to several factors. Firstly, in our simulations the interaction parameters in the SCMF Hamiltonian in eq 10 were tted to predict properties at a temperature of 37 ◦ C. However, experimental data cover a range of temperature from 20−40 ◦ C. Since water as a solvent is very sensitive to the temperature, the parameters of the model can undergo signicant changes with temperature. Secondly, exit rate constants are usually obtained from the experimentally determined fast relaxation times, τ1 , and the surfactant number concentration, c, combined with the Aniansson and Wall relationship 15,16 for the exit rate constant, τ1−1 = (kexp /N )(N/δN 2 + X), where δN 2 is the variance of the aggregation number distribution and X ≡ (c − cmc)/cmc; this means that values of kexp require a precise knowledge of the aggregation numbers and the corresponding deviation in their distribution. However, when determining N , discrepancies are found in the literature. For L64, a range of aggregation numbers from 19 to 69 have been reported 44,45 for temperatures of 40 and 35 ◦ C, respectively. In addition, aggregation numbers of 30 and 53 are found for P104 in static and dynamic light scattering experiments 41 at a temperature of 25 ◦ C. In the same way, values of 99 and 120 were reported for P123 at the same temperature. Lastly, the simulations for copolymers P104 and P123 are at the limit of our computer calculation capacity and could be aected by nite size eects. However, we performed simulations with at least two dierent sizes of the simulation box for all the Pluronics. For instance, simulations for Pluronics P104 and P123 were run in boxes of sizes 16


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Page 17 of 31

(80σ)3 and (100σ)3 , to identify the impact of the box size on the results. Consequently, the values reported in this article are size-independent. 42 Therefore, in view of the dierent sources of uncertainty no clear conclusion can be drawn from the discrepancy of the theoretical predictions and the experimentally reported values of the exit rate constants. It is remarkable, however, that the trends of the variation of the constants with the composition of the surfactants are well described by the simulations reported here as can be observed in Figure 3. 6

10

5

10 -1

k [s ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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4

10

3

10

2

10

30

40

50

60

70

Number of PO units Figure 3: Log-linear plot of the exit rate constants as a function of the surfactant composition. Empty circles are the SCMF predictions and empty squares refer to experimental data given in ref 43 and shown in Table 1. The lines are guides for the eye.

Analysis of the conformational evolution of the surfactants In this section we explore the correlations between the dynamics and the chain conformations, to shed some light on the chain exit mechanisms. In this subsection we will focus our attention on the exponential long-time terminal regime, where the conformational diusion

τD ∼ R2 /D ∼ 105 cycles is much smaller than the waiting time to escape from the energy well τ ∼ 1/k ∼ 109 cycles. The central logarithmic regime corresponds to τlog τD . 17


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Changes in the conformations of the surfactants entering or leaving the micelles can be tracked from the ensemble averages corresponding to the temporal behavior of the independent chains. In Figure 4 the radius of gyration of the hydrophobic PO blocks of the surfactants, Rg (r) has been calculated as a function of the radial distance r of the center of mass of the PO blocks from the center of the micelle. Both Rg and r are given in units of σ. On average, the hydrophobic PO blocks change from a stretched conformation in the (a)

(b)

4.6

[σ]

4.4 4.2

5.6

4.4

4.0

5.4

4.3

5.2

4.2

5.0 4.8

4.1

(d)

5.6 5.4 5.2 5.0

(c)

6.0 5.8

4.5

3.8

[σ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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9

(e)

7.5 7.0

8

6.5

7

(f)

6.0

4.8

6

5.5

4.6 0

10

20

30

40

0

10

r [σ]

20

30

r [σ]

40

0

10

20

30

r [σ]

Figure 4: Average radius of gyration for surfactant PO blocks as a function of distance of the center of mass of the same PO blocks from the center of the micelle for Pluronic (a) L44, (b) L64, (c) P84, (d) P85, (e) P104 and (f) P123. Vertical dashed lines represent the approximate extension of the hydrophobic micelle core and the head groups. center of the micelle to a more collapsed average conformation as it approaches the PO-EO interface. As the PO blocks enter the corona, it stretches again to a maximum, to later attain its bulk conformation as it crosses the EO-solvent interface. As a consistency check, we nd that the chains in the solvent have the same Rg as calculated from the equilibrium solution of the SCMF calculations for homogeneous elds with N = 1. In Figure 5 we show the most 18


40

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probable conformation for four stages in the transit of the L44 chain from the micelle to the bulk. A marked dierence is observed for L44 with respect to the other studied surfactants.

Figure 5: Chain conformations for Pluronic L44 when leaving the micellar region. Dashed blue and red lines represent estimates of the inner hydrophobic PO micellar core and EO hydrophilic shell boundaries, respectively. From left to right are shown representative conformations with radius of gyration 4.8σ , 3.7σ , 3.9σ and 3.8σ , respectively. Eectively, while the former is very stretched and collapses to exit, the others stretch even more in the transit through the corona (see Figure 4 for L44 and 6 for P123 for comparison). The L44 case can be understood by considering that the diameter of the core region is 19σ , being similar to the length of the fully stretched PO blocks. The combination of these

two facts results in a high stretching of the hydrophobic block inside the micelle core. The volume of the hydrophobic core is essentially proportional to the aggregation number N . In the case of L44 N = 145, which almost triples the value of N used for the other copolymers. In contrast, the remaining Pluronics, whose aggregation numbers produce a smaller inner core in comparison with their PPO length, exhibit relatively shrunk conformations in the central region of the micelle that tend to shrink even more as they move towards the core-shell interface (see Figure 6 for P123). Subsequently, these chains are found to stretch when crossing the corona region to nally reach a relaxed conguration in the bulk solution. This overall behavior can be observed in Figure 6 for a sequence of the most probable P123 conformations as a function of the distance from the center of the micelle. The intermediate stretching of the surfactant is surprising. One would expect that the chain monotonically stretches as it travels through the corona into the bulk. Instead, Figures 5 and 6 seem to indicate that the portions lying outside the corona are slightly adsorbed on the surface, due 19


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Figure 6: Chain conformations for Pluronic P123 when leaving the micellar region. Dashed blue and red lines represent the inner hydrophobic PO micellar core and EO hydrophilic shell boundaries, respectively. From left to right are shown representative conformations with radius of gyration 6.5σ, 4.5σ, 9.4σ and 7.5σ, respectively. to the fact that the solvent is repelling both PO and EO monomers. In contrast, the interpretation of experimental results from TR-SANS and uorescence spectroscopy experiments for diblock and triblock copolymer surfactants consider either a collapsed 8,10,18 or a stretched 7,10,12,13 conformation in the process of leaving the micelle. This view is inferred from the long-time relaxation of

F (t)

in eq 14, which is tted with two

exponentials. The assumption of an Arrhenius form for the kinetic constants such as k = A exp(−Fa /kB T )

yields the energy barrier

Fa ,

as well as the prefactor A. However,

the assumptions made for both Fa and A have important consequences on the way the exit process is regarded. In our case, the simulations permit us to explicitly follow the exit process. Since the rate constants are obtained in the long-time regime, the average behavior of the surfactants is consequently the relevant information to explain the process observed experimentally. Let us rst consider the radius of gyration of a completely collapsed hydrophobic PO block. As suggested in ref 14, this radius of gyration scales as Rg = 0.9σ

NP O σ . 1/3

In the case of L44 a value of

is found assuming that Rg is the radius of gyration of a compact sphere of radius

R with the same volume as 23 spherical monomers of diameter σ , namely, Rg2 = 2R2 /5.

This

observed in Figure 4, which is close to

3.7σ .

estimate is well below the minimum of

Rg (r)

On the other hand, a calculation of the radius of gyration of PO blocks for a fully stretched 20


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L44 chains results in a value close to 6.6σ, which is clearly larger than the values reported in our calculations both inside and outside the micellar region. In general, we obtain in the simulations hydrophobic block sizes between those for completely collapsed and stretched for all the other Pluronics considered. According to Figure 4, our simulations reveal that the exit of a copolymer from the interior of the micelle involves a subtle change in the radius of gyration, which corresponds neither to a fully collapsed nor stretched insoluble block. The process of exiting the micelle indicates that the rather stretched conformations near the micellar center shrink when approaching the core-corona interface to then stretch again when crossing the corona and entering the bulk. The dynamics of the exit process in the long-time regime is therefore governed by the dynamics of these processes of conformational change, which we inferred without a priori assumptions on the conformational states of the chains. Other authors, for example, base their analysis on the fact that the energy barrier should be a function of the number of hydrophilic-hydrophobic contacts between the PO blocks and the solvent. 7,12 Hence, a fully stretched conformation would have an energy barrier dependent on the number of PO blocks NP O ,

while a fully collapsed conformation should display an energy barrier proportional to

the external surface of the globule, namely, to NP2/3 O . Hence, a plot of ln kexp (Arrhenius) in terms of both NP O and NP2/3 O should indicate the preferred mechanism as the one that ts better a linear behavior. Zana, 46 according to their observations, initially stated that the behavior is compatible with a stretched PO block leaving the micelle. However, when temperature corrections are included in the analysis, 43 the abscissa in terms of NP2/3 O seems to better t to a straight line when plotting the height of the energy barrier. Therefore, this second conclusion agrees with the fact that the PO block is rather shrunk when crossing the barrier. This particular case exemplies the sensitivity of the conclusions obtained when dierent approaches or tting parameters are considered. 18 Our simulations seem to conrm, therefore, that the hydrophobic block eectively shrinks as part of the exit process, although not as much as a complete collapse. 21


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To further evaluate the changes in conformation as the copolymers leave the micelle we also calculated the angle between the vectors connecting the center of mass of the PO block and each of the centers of mass of the two EO blocks. The results are shown in Figure 7. As [rad]

2.3

(a)

(b)

(c)

(d)

(e)

(f)

2.2 2.1 2.0 1.9 1.8 1.7 2.3

[rad]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2.2 2.1 2.0 1.9 1.8 1.7 0

10

20

30

40

0

10

r [σ]

20

30

r [σ]

40

0

10

20

30

r [σ]

Figure 7: Average angle between PEO-PPO-PEO centers of mass for surfactants inside and outside a micelle for Pluronic (a) L44, (b) L64, (c) P84, (d) P85, (e) P104 and (f) P123. Vertical dashed lines represent the approximate extension of the hydrophobic micelle core and the head groups. observed, in all cases the surfactants in the center of the micelle exhibit a decrease in the angle as they approach to the PO-EO interface, perfectly correlated with the shrinking of the PO blocks due to the geometry of the triblock copolymers. The angle then increases when crossing the EO corona and nally attains the bulk value on leaving the micellar aggregate. This situation is readily observed in the case of the shorter surfactants L44 and L64, but, as the number of monomers increases, the uncertainty in the predictions of the angles unfortunately becomes signicant. The second point to be mentioned is the observed oscillations in the average angle in Fig22


40

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ure 7, particularly gures (c), (e) and (f). These oscillations, which are compatible with the fact that the hydrophilic EO blocks are slightly attracted to each other when embedded in the solvent. This implies that during the expulsion the EO monomers linger near the corona, causing the secondary stretching of the PO blocks during the exiting process. The oscillations in the angle in Figure 4 seem to conrm this fact, although the large error bars impair drawing a clear conclusion. Finally, let us mention that the average angle of all the studied surfactants is about 110 ◦ when in the bulk solution. If we compare this average with the one corresponding to an ideal chain, namely 90◦ , it is then obvious that our Pluronic surfactants show some degree of rigidity.

Conclusions In this work we present, to the best of our knowledge, the rst systematic study through simulations of the equilibrium exit kinetics for a series of block copolymeric surfactants. From a technical point of view, we implement a dynamic SCMF theory that allows us to cover a wide range of length and time scales well beyond the ones obtainable from atomistic molecular simulations. We have identied three regimes in the dynamics. We observe an initial regime characterized by the reorganization of the labeled chains, which takes place at a very short time scale. An unexpected central regime exhibiting a R(t) ∼ − log t dependency, characterized by a non-exponential exit of almost frozen conformations, at a time-scale of the order of μs for the parameters of our model. This regime has been experimentally observed 9 although at a much larger time scale. This is achieved by controlling the surface tension between the hydrophobic moiety and the solvent using a specically chosen solvent. The energetic barrier depends on the surface tension and therefore the characteristic time scales are shifted towards very long times. Nevertheless the rather compact core reported in this reference is still compatible with our hypothesis that this regime is controlled by the energy degeneracy

23


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of the core, broken when exiting the micelle. This regime has been interpreted as the result of the polydispersity of the experimental samples, 12 due to the fact that the energy barrier has to scale with the size of the hydrophobic moiety. However, our simulations indicate that such a regime is already present when the copolymers are strictly monodisperse. We nally identify a terminal regime with F (t) ∼ e−k t dominated by the average conformational change of the chains, which therefore results in one single exponential decay, if an Arrhenius form is assumed for the exit constants k . The terminal regime occurs when conformational changes τ ∼ R2 /D are faster than the exit process, the latter characterized by τ ∼ 1/k . We have compared our simulations with the reported values in ref 43. In Figure 3 we show that our simulations qualitatively describe the observed tendency with the copolymer architecture. Even quantitatively, our results are close to the ones reported, in view of the experimental uncertainty. Moreover, we have not observed the collapse of the hydrophobic moiety reported in ref 43 since, in our simulations, the polymers in the bulk are signicantly swollen.

In this article we provide a comprehensive analysis of the exit dynamics of copolymeric surfactants from micelles. The interest of our procedure lies in the fact that our model requires only the energetic interaction parameters to set the thermodynamic properties. Moreover, the determination of the physical time scale is taken from the comparison between the simulated and the physical diusion dynamics. Once these physical parameters are determined, our results are independent of further tting and, therefore, they have to be regarded as predictions directly comparable with experiments. The main importance of the results reported here is precisely the possibility of a direct comparison with experimental data. The good agreement found therefore supports the physical view depicted along this article about the dynamics of surfactant exit from micelles.

24


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Acknowledgement F.A.G.D. acknowledges nancial support from URV through his Ph.D. scholarship. The

authors acknowledge nancial assistance from the Ministerio de Economía y Competitividad of the Spanish Government for nancial support, grant CTQ2014-52687-C3-1-P

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Simulation Analysis of the Kinetic Exchange of Copolymer Surfactants

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