How Implementation of Entropy in Driving Structural Ordering of

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Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers

How Implementation of Entropy in Driving Structural Ordering of Nanoparticles Relates to Assembly Kinetics: Insight through Reaction-Induced Interfacial Assembly of Janus Nanoparticles Ye Yang, Pengyu Chen, Yufei Cao, Zihan Huang, Guo-Long Zhu, Ziyang Xu, Xiaobin Dai, Shi Chen, Bing Miao, and Li-Tang Yan Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01378 • Publication Date (Web): 17 Jul 2018 Downloaded from http://pubs.acs.org on July 22, 2018

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How Implementation of Entropy in Driving Structural Ordering of Nanoparticles Relates to Assembly Kinetics: Insight through Reaction-Induced Interfacial Assembly of Janus Nanoparticles

Ye Yang,1,† Pengyu Chen,1,† Yufei Cao,1 Zihan Huang, 1 Guolong Zhu,1 Ziyang Xu,1 Xiaobin Dai,1 Shi Chen,1 Bing Miao2,* and Li-Tang Yan1,*

1. State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing 100084, China 2. College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China

*Corresponding author: [email protected] (L.T.Y.) [email protected] (B. M.)

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ABSTRACT: The ability to understand and exploit entropic contributions to ordering transition is of essential importance in the design of self-assembling systems with well-controlled structures. However, much less is known about the role of assembly kinetics in entropy-driven phase behaviors. Here, by combining computer simulations and theoretical analysis we report that the implementation of entropy in driving phase transition significantly depends on the kinetic process in the reaction-induced self-assembly of newly designed nanoparticle systems. In particular, such systems comprise binary Janus nanoparticles at the fluid-fluid interface and undergo phase transition driven by entropy and controlled by the polymerization reaction initiated from the surfaces of just one component of nanoparticles. Our simulations demonstrate that the competition between the reaction rate and the diffusive dynamics of nanoparticles governs the implementation of entropy in driving the phase transition from randomly-mixed phase to intercalated phase in these interfacial nanoparticle mixtures, which thereby results in diverse kinetic pathways. At slow reaction rates, the transition exhibits abrupt jump in the mixing parameter, in a similar way to first-order, equilibrium phase transition. Increasing the reaction rate diminishes the jumps until the transitions become continuous, behaving as a second-orderlike phase transition where a critical exponent, characterizing the transition, can be identified. We finally develop an analytical model of the blob theory of polymer chains to complement the simulation results and reveal essential scaling laws of the entropy-driven phase behaviors. In effect, our results allow for further opportunities to amplify the entropic contributions to the materials design via kinetic control.

KEYWORDS: entropic ordering, assembly kinetics, phase transition, interfacial nanostructure, computer simulation


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INTRODUCTION Numerous studies have demonstrated the significant impact of nanoparticle-based materials in life sciences, microelectronics, light manipulation, energy harvesting and storage.1-3 The key to regulating these materials’ macroscopic behaviors toward superior properties is to controllably manipulate the spatial arrangement of nanoparticles within a matrix material and even at interface; accomplishing this task, however, remains a considerable challenge.4-12 The seminal work on structural organization of spherical particles revealed that order can be induced by entropy alone.13,14 To be relevant, a concurrent increase of studies on the structural control of nanoscale particles show that entropy can be the sole driving force to tailor the spatial organization of nanoparticles, which is not achievable by conventional techniques.15-21 Recently, entropy-driven phase behaviors, ranging from ordering of colloidal particles to the structural organization of nanoparticles in polymer nanocomposites, have been intensively demonstrated in computer simulations and experiments.22-31 However, in contrast to the well-established equilibrium structures of entropy-driven phase behavior, the fundamental understanding on how the implementation of entropic interactions relates to the assembly kinetics remains underdeveloped, despite the fact that different self-assembly pathways can induce evidently different responses in the behavior of libraries made from the same building block.32,33 Kinetically, nanoparticle self-assembly depends on the energetic penalty of the excess interfacial area because of defect, the diffusion of nanoparticles, and the intermolecular interactions among the building blocks within the system.34,35 The assembly kinetics is particularly important for entropy-driven phase behavior as many types of entropy, e.g., translational, rotational, and vibrational entropy, contribute to the kinetic process in some sense. ACS Paragon Plus Environment

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In turn, such entropic effects provide unique opportunities to manipulate the energy landscape of the assembly process and potentially to modulate the kinetic pathway of nanoparticle assembly. Little however is known about the influence of assembly kinetics, such as diffusive behaviors and pathways, on the implementation of entropy in regulating the structural organization of nanoparticles. This might be ascribed to the complicated interplay between thermodynamics and kinetics during the process of entropic ordering, making it great challenge to identify individual contribution to the phase transition. Herein, through computational modeling we design a new nanoparticle self-assembly system that comprises binary Janus nanoparticles at the fluid-fluid interface and undergoes phase transition driven by entropy and controlled by the polymerization reaction initiated from the surfaces of just one component of nanoparticles. By tuning the reaction rate, we are able to for the first time study how the implementation of entropy in the phase transition depends on the assembly kinetics of the nanoparticles. This allows us to identify two different kinetic pathways, with dramatically different characterizations of phase transitions, from the same building block. One pathway exhibits abrupt jump and takes a similar way to first-order transition in systems that undergo an equilibrium phase transition, while a second pathway becomes continuous and behaves as a second-order-like phase transition with a characteristic critical exponent. Finally, an analytical model of the blob theory of polymer chains is developed to complement the simulation results and reveal essential scaling laws of the dependence of entropic interactions on various parameters. The good agreement between the results from computer simulation and theoretical analysis yields a fundamental insight into the entropy-driven phase behaviors as well.


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Figure 1. (a) Schematic representations of the transition from randomly-mixed phase to intercalated phase in the mixtures of grafted Janus nanoparticles where one component undergoes respectively slow and fast polymerization reactions initiated from nanoparticle surfaces. (b-d) Typical interfacial nanoparticle organizations obtained at different stages of the reaction: (b) initially random configuration at 0τ, (c) medium stage at 4000τ, and (d) late stage at 10000τ. Here D=2.76rc, [M]0=16.8%, and Pp=0.0005. The dark yellow plane represents the fluid interface whereas the fluid and monomer beads are not shown for clarity. (e) Cv, [Μ]0/[Μ], and PDI (inset) as functions of the reaction time at [M]0=16.8%, Pi=0.0025, and Pp=0.0005. (f) The temporal evolution of the mixing parameter, φ, during the reaction. The inserted images from bottom to top illustrate three typical phases of mixing states upon increasing φ, i.e., phase separated (bottom), randomly mixed (middle), and intercalated (perfect mixing at singlenanoparticle level) (top). The inserted diagram shows the time dependence of φ for the Janus nanoparticle mixtures without grafted chains (bottom) and with equally grafted chains for the both components, i.e., lA=lB=20 (top).


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RESULTS Full technical details on the simulation method and the models of different entities used in the simulations are described in Method Section. For brevity, computer simulations are carried out using dissipative particle dynamics (DPD), a method that samples the NVT ensemble.36 As illustrated by the schematic diagram in Figure 1a, two immiscible components of Janus nanoparticles are involved in the system, where the amphiphilic surface moiety of each nanoparticle anchors it at the flat interface formed by a strongly-segregating AB binary fluid mixture. While one component is unreactive, the other is reactive as the nanoparticles surfaces encompass initiator sites for the atom-transfer radical polymerization (ATRP), a form of controlled/living radical polymerization.37,38 In the present work, the radius of a bare nanoparticle, R, is fixed at 1.2rc, where rc is the truncated distance of the conservative force; chain length, l, is expressed in total number of DPD beads connected together in a chain. At the beginning of the reaction, the monomers are randomly dispersed in their preferential fluid phases, and can be initiated from the reactive nanoparticle surfaces to polymerize into grafted chains. Reaction-Induced, Entropy-Driven Phase Behaviors of Interfacial Nanostructures. Figure 1b-d show typical snapshots of the critical propagation steps, as monomers polymerize into chains on the surfaces of reactive Janus nanoparticles via the ATRP. Here the averaged surface-to-surface distance of nanoparticles, D, is 2.76rc, initial concentration of monomers at each fluid phase, [M]0, is16.8%, and propagation probability of the reactive chains, Pp, is 0.0005. One can identify from the snapshots that the polymer chains gradually grow from the nanoparticle surfaces. In Figures 1e and S1, we characterize the detailed reaction kinetics of this system by several quantities, such as the monomer conversion, Cv, and the ratio of the initial to unconverted fractions for monomers, [Μ]0/[Μ], in the log scale. It can be found that the monomers become almost fully converted into polymer chains within the time scale of the


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reaction. Importantly, the curve reveals that ln([Μ]0/[Μ]) displays a linear dependence with reaction time, as would be expected for the first-order reaction.37 This indicates that our model captures the characteristic feature of the kinetics of living radical polymerization, such as ATRP.37,38 The polydispersity index, PDI, presented in the inset of Figure 1e provides information about the molecular weight distribution. The PDI first increases dramatically before reaching a peak, and then decreases to a plateau less than 1.10. Such a peak indicates that the growth of all polymer chains does not occur simultaneously; this is a consequence of comparable rates of initiation and propagation. Through the Delaunay triangulation (as schematic representations in Figure S2), we can count the nearest neighbors of each interfacial nanoparticle, and consequently quantify the interfacial nanostructures by defining the mixing parameter, φ, as φ=Nur-Nuu-Nrr, where Nur, Nuu, and Nrr are number fractions of line contacts between unlike nanoparticle pairs and like pairs for the nearest neighboring nanoparticles.39 A higher φ value represents a better mixing for the two nanoparticle components, while the system with a lower φ value instead tends to segregation between them. Figure 1f shows the evolution of φ as a function of reaction time, where a significant increase of φ can be observed in response to the growth of grafted chains on reactive nanoparticles. Generally, there are three basic mixing states, that is, phase separation, randomly mixing, and long-ranged intercalation (see the insetting images in Figure 1f).29 The absolutely phaseseparated state (φ≈-0.95) possesses the lowest degree of mixing, where two components segregate from each other, resulting in the formation of big phase domains and obvious phase interface between two different phase domains. The randomly-mixed phase occurs at about φ≈0. In this case, although two components take a better mixing than the phase-separated state, there are still small phase domains of the same nanoparticle component, as demonstrated in the insetting image generated by Monte Carlo method. The best mixing state of nanoparticle ACS Paragon Plus Environment

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mixtures is the perfectly intercalated state (φ=0.33), where two components are uniformly dispersed at the single-nanoparticle level. Such perfect mixing is usually difficult to realize due to the intrinsic polymer limitations and a prohibited long equilibration process. However, the long-ranged intercalation with φ a little smaller than 0.33 can be achieved in practice.29 Indeed, Figure 1f shows that the growing chains on reactive nanoparticles cause a drastic increase of φ from about 0 to 0.25, being a transition from randomly-mixed phase to intercalated phase (see also Figures 2 and 3 for more details). In sharp contrast, φ keeps almost unchangeable for the system without reaction during the same period, as depicted by the bottom panel of the insetting diagram of Figure 1f. It can thereby be concluded that it is the reaction that induces this phase transition of the interfacial nanostructures. What molecular mechanism fundamentally underpins such a reaction-induced phase transition? In view of the repulsive interaction between reactive and unreactive nanoparticles, enhanced contacts between unlike nanoparticle pairs will lead to an energy increase, hindering the transition towards intercalation phase. However, such unfavorable enthalpic interactions can be overwhelmed by the entropic effect originated from the conformation of chains grafted on the reactive nanoparticles. To illustrate this, we present the schematic diagrams in Figure 1a, where the short chains grafted on reactive nanoparticles swell and extend towards their preferential fluid phases at the initial stage of the reaction. As chains grow, remarkably steric repulsion between neighboring chains is caused, which leads to the stretched chains with reduced conformational entropy. At a certain chain length, a transition from randomly-mixed phase to intercalated phase occurs because the intercalation of an unreactive nanoparticle increases the space between the chains as well as their conformational entropy. Hence, the increase in free energy associated with the direct contact between two immiscible species is compensated by the bigger increase in entropy related to the larger conformational space available to the grafted


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chains when they are sufficiently separated by the unreactive nanoparticles. Furthermore, such a reaction-induced, entropy-driven phase transition will be further quantified and be rationalized through a theoretical analysis in the last section of the work. In particular, it should be pointed out that the disparity in grafted-chain lengths of the two nanoparticle components is essentially important for the entropy-driven phase transition. To demonstrate this point, we examine a model system with two nanoparticle components of equal grafted-chain lengths, i.e., lA=lB=20, which approximates the final number average chain length of the reactive nanoparticles. The plot describing the time dependence of φ is presented in the top panel of the inset of Figure 1(f). Indeed, the value of φ only fluctuates mildly around a value close to 0, and the long-ranged intercalation phase does not occur throughout the entire period. Kinetic Pathways of Nonequilibrium First-Order or Second-Order Phase Transition. Having established the key role of entropy in providing thermodynamic driving force, we now turn to the detailed kinetic process of the reaction-induced phase behaviors, and systematically analyze how the implementation of entropy in the phase transition depends on the assembly kinetics upon various reaction rates. In the simulations, the reaction rate constants can be controlled by modifying the probabilities of initiation, Pi, and propagation, Pp, which should be sufficiently small to ensure controlled polymerization growth in the kinetically-controlled reaction regime. Therefore, we take Pi=0.0025 and various values of Pp, from 0.0002 to 0.0015, to account for the effect of reaction rate on the assembly kinetics of nanoparticles. Figures 2a and S3 display the temporal evolution of φ at four typical values of Pp. Remarkably, we observe two different types of evolution trends of φ: at high reaction rates the curves are continuous while the curves turn to be discontinuous with an abrupt φ jump at low reaction rates. This indicates that the transition from randomly-mixed phase to intercalated phase may undergo different characteristics and kinetic pathways, as schematically illustrated in Figure 1a.


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Figure 2. (a) The time dependence of φ at various propagation probabilities, Pp: (red) 0.0002, (cyan) 0.0005, (green) 0.0011, and (pink) 0.0014, where D=2.48rc and [M]0=16.3%. (b) φ as a function of the number average length of the reactive chains, LA. In (a) and (b), the thick lines are used just to guide the eye. The cyan points in the pink line represent the snapshots of (c-h). Inset: the plots of ∆φ (φ-φc) against ∆L (LA-LAc) in the log-log scale. Color scheme: Pp=0.0011 (green) and 0.0014 (pink). The points are measured along the critical φ. The lines are the power law fits. The resulting critical exponent is δ% ≈1.03. The snapshots of (c-h) are successive stages showing the representative interfacial nanoparticle organizations at various LA as marked by the points in (b). The dashed pink lines in (f) and (g) mark the short strings of the spacer nanoparticles (unreactive nanoparticles).

For the sake of thoroughly understanding the characteristics of the reaction-induced phase transitions, we present plots of φ as a function of the number average length of growing chains, LA, (Figure S4) for the fast and slow reactions respectively in Figures 2b and 3a. To the fast reactions, the φ-LA characteristics proceed in a continuous way that is reminiscent of a thermodynamic system undergoing a second-order phase transition (Figure 2b). A central feature ACS Paragon Plus Environment

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of continuous phase transition is that, approaching the critical point, physical quantities show power law dependences with characteristic critical exponents.40 To probe the idea of secondorder phase transition in our out of equilibrium systems, we look for an analogues critical behavior in the vicinity of the critical point. Here we focus on the critical exponent, δ% , defined as %

δ ϕ -ϕc ∝ ( LA − LAc ) . φc and LAc are the values of φ and LA at the critical point where φ transforms

from mild fluctuation around a small value to increase due to the reaction. The circles in the inset of Figure 2b represent data of φ-φc vs LA-LAc on a logarithmic scale. A critical exponent of p

δ% =1.03 is calculated for both P =0.0011 and 0.0014, using a power law fit―solid lines in the

inset. Although the critical exponent we extracted do not agree with any known equilibrium universality class, it is similar to the values of some nonequilibrium, second-order, phase transition systems, such as Cooper-pair insulator ( δ% =0.9)41 and 3D directed percolation ( δ% =0.8).42 This corroborates that the fast reactions result in a continuous, second-order like, phase transition with a characteristic critical exponent. In striking contrast, an abrupt φ jump can be identified in the φ-LA characteristics for the slow reaction with Pp=0.0002, as LA is increased beyond a threshold (Figure 3a). Such a jump reproduces the discontinuity for a thermodynamic system undergoing a first-order phase transition. To gain more detailed insight into the underlying physical nature of the difference between fast and slow reactions, we turn to the pathways of assembly kinetics. For this purpose, some specific points in the φ-LA plots of Pp=0.0002 and 0.0014 are marked, and the interfacial nanostructures at these points are displayed through the snapshots of Figures 2c-h and 3b-g, respectively. More detailed kinetic processes of these two cases can be found in the supporting Videos S1 and S2 and Figures S5 and S6. By comparing the snapshots in Figures 2c-h and 3b-g, it is interesting to note that the kinetic pathways of these both processes are evidently different. For the fast reaction, we observe a continuous and steady evolution of the interfacial nanostructures upon the reaction-induced transition from randomly-mixed phase to long-ranged


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intercalation (Figure 2c-h and Video S1). In this case, only short strings of unreactive nanoparticles can be identified during the transition, which break and evolve into single spacers at the late stage. However, the slow growth of grafted chains firstly leads to the formation of big phase domains of unreactive nanoparticles, as marked by the dashed circles in Figure 3b and highlighted by the locally enlarged section in Figure 3h. Such big phase domains remain until a threshold value of LA (here, LA≈6.2), beyond which the unreactive nanoparticles diffuse spontaneously out of the domain and are intercalated among the grafted nanoparticles as single spacers (Figure 3b-g and Video S2). This instant process consequently gives rise to the abrupt jump in the φ-LA plot (Figure 3a). The fact that no big domain of unreactive nanoparticles form during the fast reactions indicates that the entropic interactions from the rapidly growing chains can cause instantaneous intercalation of their neighboring nanoparticles that are unreactive, since these nanoparticles have no enough time to diffuse away from the reactive nanoparticles. These locally intercalated structures gradually connect and evolve into the intercalation phase spanning the whole interface, as shown in Figure 2c-h and Video S1. However, for the slow reactions, the reduced growth rates of grafted chains allow enough time for the unreactive nanoparticles to diffuse away and thereby escape from the entropic interactions of grafted polymer chains. This initially prompts the formation of big domains of unreactive nanoparticles, which can move through nanoparticle rearrangement to diminish the contact with the other nanoparticle component (Figure 3b, h). As the grafted chains reach a threshold length, the motion of the big domains is almost fully confined and thereby spontaneous intercalation takes place (Figure 3b-g and Video S2). These results clarify that the implementation of entropy in driving the phase transition significantly depends on the competition between the reaction rate and the diffusive dynamics of nanoparticles, which results in diverse kinetic pathways. In the last section of this work, we present quantitative analysis of such a competitive effect governing the particle intermixing.


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To further assess how entropic interaction and reaction rate relate to the pathway of assembly kinetics, we systematically compute the characteristics of reaction-induced phase transition as a function of grafting density σ and propagation rate Pp. The former variable dictates the entropic contribution of polymer’s conformations. The latter variable represents the reaction rate. In Figure 4, we show the state diagram of phase transition as a function of these variables, where the shaded regions approximately discriminate the characteristic regions of phase


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transitions. In order to ensure the states in the diagram, for each point shown in it, we performed a large scale of independent simulation runs. Figure 4 shows that the first-order phase transition is expected to occur only when Pp is less than 0.0013. Increasing σ however reduces the region of the first-order phase transition. Consequently, the second-order phase transition is more likely to occur at large values of Pp and σ. This indicates that a higher grafting density and a faster reaction rate favor the continuous phase transition. 0.08 0.07 2

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Figure 4. State diagram of phase transition as a function of the propagation probability and the grafting density, where D=2.48rc and LA=16. The green and red circles indicate the first-order and the second-order phase transitions, respectively. Boundary between different phases is represented with the dashed line for clarity only.

To quantitatively corroborate such observation, we also calculate δ% for all the propagation probabilities and the grafting densities adopted in the two-dimensional (2D) diagram of state transition, where we assume that the φ-LA characteristic is continuous at each point. Table S1 summarizes the calculation results of δ% , and demonstrates that the extreme large values of δ% (larger than 2100) are more likely to occur at small values of Pp and σ. More importantly, as indicated by the dashed line in Table S1, the boundary between the sections of large and small δ% in Table S1 coincides with the boundary between the first-order and the second-order


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state transitions in Figure 4. Such a good overlap allows a quantitative support for the discrimination between the characteristic regions of the state transitions. Moreover, these both factors suggest effective approaches that can be harnessed to tailor the pathways of assembly kinetics. For the sake of exploiting the dependence of the mixing degree on the entropy interaction and assembly kinetics, we perform systematic simulation to compute φ for a range of σ and Pp, allowing us to construct the φ landscape plotted by the colored contour map in the Pp-σ plane (Figure 5a). Figure 5a clarifies a strong dependence of the mixing degree on the grafting density and the reaction rate. It is of particular interest that, if it is observed at all, increasing σ causes a peak of φ for a certain Pp: φ increases at very small σ, but turns to slightly decrease at large σ, especially when Pp>0.0008. How can we understand the appearance of such a peak of φ? Note that here the number average length of the reactive chains in each system is controlled at LA=16. However, as shown in Figures 1e and S3, various polydispersities of grafted chains can be caused in response to different reaction rates, which may evidently modify the entropic interaction. To examine the effect of polydispersity on the entropic contribution, we quantitatively estimate the entropy change by calculating the conformation entropy of two model systems with well-defined interfacial nanostructures shown in Figure 5b. The first model system consists of one type of grafted Janus nanoparticles arranged on the interface, where the PDI of grafted chains can be tuned by randomly selecting chain lengths through Monte Carlo method (left snapshot of Figure 5b). In the other model system, the grafted nanoparticles are alternatively replaced by spacers, i.e., bare Janus nanoparticles, leading to the perfectly intercalated structure as demonstrated in the right snapshot of Figure 5b. After an equilibration stage, the conformation entropy per polymer chain, SC, that is related to unperturbed mean-square end-to-end distance, , can be estimated as SC ( N , R ) ≈ −k B R 2

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Figure 5. (a) Dependence of the mixing parameter, φ, on the propagation probability, Pp, and the grafting density, σ. Here, D=2.48rc and the number average length of the reactive chains in each system is controlled at about LA=16. (b) The model systems with (left) only grafted nanoparticles and (right) both grafted and spacer nanoparticles. (c) Change in the conformational entropy per grafted chain, ∆S, as a function of polydispersity index (PDI), for the model systems where σ=0.05 chains/rc2 and D=2.48rc. (d) The PDI of grafted chains as a function of Pp and σ. The color bars in (a) and (d) indicate the values of φ and PDI, respectively.

Figure 5c shows the plot of ∆SC against PDI of grafted polymer chains where LA=16. It can be seen clearly that increasing PDI firstly results in a dramatic increase of the entropy gain, followed by a considerable decrease at large values of PDI. A crossover occurs at PDI≈1.11. ACS Paragon Plus Environment

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This reveals that a narrow distribution of chain lengths tends to enhance the entropy gain while a wide distribution can significantly penalize the entropic contribution. We then consider the peak of φ in Figure 5a by calculate the PDI of each point in it, and summarize the data in Figure 5b. As a result, it can be observed that the polydispersity of grafted chains increases with increasing Pp and σ, accounting for the increase of φ at small σ in Figure 5a. In particular, when Pp>0.0008 and σ>0.05chains/rc2, a region with PDI>1.11 can be identified, as featured by the dark red color in the top right corner of Figure 5d. This rationalizes the slight decrease of φ at large σ in Figure 5a. Thus, it is reasonable to conclude that the appearance of the peak in Figure 5b can be fundamentally ascribed to the nonmonotonic dependence of entropy gain on the PDI of grafted chains. DISCUSSION Theoretical Analysis through Developing a Scaling Model.

h1

h0 L or D

R

D D > 2 h0

D < 2 h0

Figure 6. Schematic diagram showing the conformational transition of the grafting chains used in the theoretical analysis based on the blob concept. Here h0 and h1 denote the height of the grafting-polymer brushes without (left) and with (right) interactions between nanoparticles.

To pinpoint the mechanism of the entropy-driven phase behaviors, we develop a scaling model, based on the blob concept,44,45 to complement the simulation results and thereby reveal essential scaling laws. The detailed derivation and an extended description of the model can be found in Supporting Information. For brevity, as schematically illustrated in Figure 6, we


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consider m = ρ 0 A ≈ A / ( D + 2 R ) 2 Janus nanoparticles of radius R distributed on the interface of area A . Here, the number density of Janus nanoparticles ρ 0 ≈ 1/ ( D + 2 R )2 , with D denoting the inter-particle distance. Each nanoparticle is grafted with n ≈ R 2 / d 2 = σ R 2 polymer chains of contour length L = Na , where, d stands for the inter-chain distance on the nanoparticle surface, the grafting density σ ≈ d −2 , and N and a denote the number of statistical segments per chain and the Kuhn length. As illustrated in Figure 6, for D > 2h0 , where h0 denotes the thickness of grafted layer, Janus nanoparticles are independent of each other and the grafted chains can take their natural conformations. For D < 2h0 , the interactions between Janus nanoparticles set in, and the repulsive compression between grafted layers of adjacent Janus nanoparticles constrains chain conformational fluctuations and strongly extends chains to be perpendicular to the interface, which renders the blob size to be uniform. In this case, the average thickness of grafted Gaussian chains and the free energy per chain are obtained as h1 ≈ Na 2σ 1/ 2 and

Fc ,1 ( k BT ) = h1 d , respectively. Here, kBT stands for the thermal energy. 0.12

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14 11 8 64

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Figure 7. (a) φ as a function of LA and the nanoparticle number, m. The pink line fits the approximate transition line from randomly-mixed phase to long-ranged intercalation. (b) ∆S as a function of LA and m for the model systems. Here σ=0.05 chains/rc2. The white circle in (a) marks the critical chain length at m=72 (D=2.48rc). ACS Paragon Plus Environment

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Based on the above analysis for two scenarios of D , the phase transition line from randomly-mixed phase to long-ranged intercalation can be estimated by solving the equation D = 2h0 . Substituting h0 and adopting large R assumption lead to the transition line as N ≈ ( A1/2 m −1/ 2 ) / (2a 2σ 1/ 2 ) − R / (a 2σ 1/2 ) = k1m −1/ 2 − k2

(1)

where two parameters k1 = A1/ 2 / (2a 2σ 1/ 2 ) and k2 = R / (a 2σ 1/2 ) are defined. To validate the effectiveness of the scaling analysis regarding the phase transition line, we obtain the landscape of φ in the LA-m space by large-scale DPD simulations (Figure 7a). Recall that LA is defined based on the bead number, and thereby is equivalent to N. Given that the lower boundary of φ for the intercalation phase is about 0.15, the phase transition line can be well reproduced by eq 1 with a fitting vales of k1=430 and k2=40, as demonstrated by the pink curve in Figure 7a. Here, A=(40rc)2, a=0.5rc, R=1.2rc, and σ=0.038 chains/rc2. This yields that the analytic solutions of k1and k2 respectively are 410 and 25, in a good agreement with the fitting values for the

simulation results. We further calculate the free energy penalty due to the depression of polymer’s conformational entropy as ∆Fct mn( Fc ,1 − Fc ,0 ) h02 h1 h0 )] ≈ mR ( Na 2 ) 2 σ5/2 ≈ mn[ − ( − = k BT k BT d d 2Rd

(2)

In the intercalated state, with Janus nanoparticles of another component playing roles of spacers, the grafted chains no longer need to be stretched. Therefore, in the long-ranged intercalation phase, free energy penalty due to the depression of the polymer’s conformational entropy is saved, which provides the driving force to the phase transition from randomly-mixed phase to the intercalated phase, as indicated by eq 2. Indeed, the landscape of φ in Figure 7a presents a


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strong correlation with the landscape of entropy gain ∆S in the same parameter space (Figure 7b), which is obtained by systematic simulations based on the model systems in Figure 5b. Eq 2 also demonstrates that the entropic contribution will be enhanced through increasing the number of interfacial nanoparticles, nanoparticle size, chain length, and the grafting density, which rationalizes the simulation results in Figures 1, 5, and 7. By this token, the excellent agreement between the results from the simulation and the theoretical analysis corroborates that our scaling model captures the fundamental physics of the entropy-driven phase behaviors. 100

(a)

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d p , d c (r c )

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Figure 8. (a) Representative trajectories of a spacer nanoparticle for a slow reaction with Pp=0.0002 and (inset) a fast reaction with Pp=0.0014, where D=2.48rc and [M]0=16.3%. (b) Mean-square displacement 〈∆r 2 (t )〉 for spacer nanoparticles in the systems with (red) Pp=0.0002 and (green) 0.0014. Inset: the van Hove correlation function, GS(r, t), for spacer nanoparticles at Pp=0.0002 (slow) and 0.0014 (fast). (c) The characteristic size of the grafted-chain cage, dc, as a function of reaction time at different propagation probabilities, Pp: (red) 0.0002, (cyan) 0.0005, (green) 0.0011, and (pink) 0.0014. The orange and yellow curves represent the temporal evolution of the average displacements, dp, of spacer nanoparticles at Pp=0.0002 and 0.0014, respectively. (d) The time dependence of relative surface tension, γ , at different Pp. Color scheme is the same as that of (c). ACS Paragon Plus Environment

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Quantitative Analysis of the Intercalation Dynamics. The results in Figures 2 and 3 indicate that the competition between the reaction rate and the diffusive dynamics of nanoparticles governs the implementation of entropy in driving the intercalation and thereby leads to diverse kinetic pathways. In this section, we present a more thorough insight into the intercalation dynamics through quantifying such a competitive effect. For this purpose, the diffusion dynamics of spacer nanoparticles is examined firstly. The typical trajectories of the spacer nanoparticles diffusing at the fluid interface are illustrated in Figure 8(a) and its inset, where Pp=0.0002 and 0.0014 respectively. These stochastic trajectories demonstrate that the diffusion of the spacer nanoparticles is isotropic for both slow and fast reactions. We measure the

r r mean-square displacement (MSD) 〈∆r 2 (t )〉 = 〈| r (t ) − r (0) |2 〉 and the self-part of the van Hove r r r correlation function Gs (r , t ) = 〈δ {r − [r (t ) − r (0)]}〉 by using equilibrated configurations from r DPD simulations.46,47 Here, 〈L〉 denotes an ensemble average and r (t ) is the position vector of the monomer bead at time t. Fig. 8(b) displays the ensemble-averaged MSD of the spacer nanoparticles for the slow and fast reactions. We fit MSD to the scaling tα, yielding the diffusion exponent, α. At short time scales, the lines overlap well with one another and scale as t2, indicating that all nanoparticles diffuse via ballistic motions with few collisions with solvent. However, at long time scales, the spacer nanoparticles enter the Fickian regime with α=1. In the inset of Figure 8(b), logarithmic GS(r, t) is plotted against displacement, and the distribution is observed to follow a normal, or Gaussian, distribution. Particularly, GS(r, t) can almost overlap each other for the slow and fast reactions, indicating that the two processes obey the same statistics. Thus, all the above results demonstrate that the spacer nanoparticles for both slow and fast reactions present very similar diffusion dynamics that are isotropic. The averaged


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displacement of the spacer nanoparticles, defined as dp, can be directly obtained from the root of MSD. For a spacer nanoparticle, the chains grown from its adjacent reactive nanoparticles behave like a “cage” that can capture and thereby intercalate it when the characteristic size of the grafted-chain cage, dc, is larger than the displacement dp of the spacer nanoparticle. However, when dc is smaller than dp due to the slow growth of the grafted chains, the spacer nanoparticle will have enough time to diffuse away and thereby escape from the cage. Given that the grafted chains grow simultaneously from neighboring reactive nanoparticles consisting of the cage, the characteristic size of the grafted-chain cage can be estimated as, dc≈2Na=2LAa. As an example, Figure 8(c) shows dc and dp as a function of reaction time at different propagation probabilities Pp, where D=2.48rc (m=72). It can be identified that the curves of dp evolve quickly into the

section between the curves of dc with Pp=0.0012 and 0.0014. From the fitted line in Figure 7(a), the critical dc for the implementation of the entropy in driving the phase transition, d c , can be determined as d c ≈12rc at m=72. Thus, when the curves of dc reach d c , the value of dc is larger than dp only at Pp=0.0014, as clarified by the dashed lines and circles in Figure 8(c). These results reveal that only for the fast reaction can the spacer nanoparticles be instantaneously intercalated by the grafted-chain cage and consequently lead to the continuous phase transition, reproducing the results of Figures 2 and 3. This is also consistent with the state diagram of phase transition shown in Figure 4. Hence, the quantitative analysis corroborates the key role of the competition between the reaction rate and the diffusive dynamics of nanoparticles in the implementation of entropy and in the diverse kinetic pathways. However, we note that a full account of this concept would require a careful analysis of the presence and duration of "cage", which is a daunting task well beyond the scope of the current study. ACS Paragon Plus Environment

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Moreover, one may expect that a crossover between an abrupt and continuous intercalation transition is due to crossover from pressure-driven to diffusion-dominated particle intermixing regimes. To examine this point, we calculate the relative lateral pressure, γ , defined as γ =γ -γ 0 .48 Here γ 0 is the lateral pressure of the system without reaction. Figure 8(d) shows

γ as a function of reaction time at various Pp corresponding to Figure 2(a). What is striking is that all these curves present smooth evolution with the increasing time. This definitely clarifies that the characteristics of the reaction-induced phase transitions cannot be connected to lateral pressure.

Possibly Realizing the Simulation Systems Experimentally. The systems demonstrated by the simulations can be achievable in the experimental research. The quickly developing controlled/living radical polymerization methods, including, e.g., ATRP37,38 and RAFT (reversible addition-fragmentation chain transfer),49 give the chemists an experimental toolbox in designing critical components in such systems. For example, the flat interface of the stronglysegregating AB binary fluid mixture can be selected as one of the water-oil interfaces.50 For the living

radical

polymerizations

in

the

aqueous

phase,

2,2′-Azobis

(4-methoxy-2,4-

dimethylvaleronitrile) (V-70) and 2,2′-Azobis[(2-carboxyethyl)-2-methylpropion-amidine] (VA057) can be employed as the initiators, and acrylamide (AM) and N,N-dimethylacrylamide (DMA) can be chosen as the monomers.51,52 In the oil phase, the living polymerizations for the monomers of methyl methacrylate (MMA) and n-butyl methacrylate (BMA) can be initiated by ACHN (1,1′-azobis(1-cyclohexanenitrile)) and azobis(1-cyclohexanenitrile).53 Given the tremendous progress in diversifying synthetic strategies for the preparation of well-defined Janus nanoparticles with the aim to include diverse surface functionalities,54 the two moieties of a


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Janus nanoparticle can be selectively functionalized by initiators for the living polymerizations in aqueous and oil phases respectively, through the procedures described by, e.g., Wang et al.55 and Gupta et al.56 Furthermore, the significant advancement of surface-initiated controlled radical polymerization allows useful approaches to tailor the chemical and physical properties of the grafted nanoparticles at interfaces.57 Hence, all the components for potentially realizing the systems experimentally are available.

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4000 ∆E=1058 kBT

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∆E=660 kBT ∆E=332 kBT

1000 0

0

2

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Distance (rc) Figure 9. The interaction energy of grafted nanoparticles with solvent beads as a function of its distance from the interface at various grafted-chain lengths: (red) 0rc, (orange) 1.5rc, (blue) 5.0rc. The corresponding models of Janus nanoparticles with different grafted-chain lengths are indicated by the insetting images.

Furthermore, note that the Janus nanoparticles with amphiphilic surface moieties, which can significantly contribute to the stabilization of the nanoparticles at interfaces,50,58 are used in the systems. To evaluate the energy holding these nanoparticles at the water-oil interfaces, we performed the simulations of a process where a Janus nanoparticle is pulled from the interface to the aqueous or oil phase and the interaction energy during the process is monitored. As shown in ACS Paragon Plus Environment

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Figure 9, even a bared Janus nanoparticle can result in an energy increase of about 332kBT in this process and the presence of grafted chains evidently elevates this value. Such high energy values are comparable to the results of the colloidal Janus particles with a larger size,58 and confirm that Janus nanoparticles can be stably anchored at the water-oil interfaces. Our additional simulations also corroborate that the latent heat caused by the reactions (Figure S7) and the asymmetry propagation rates of the both sides (Figure S8) cannot force the Janus nanoparticles to deviate from the interfaces. Moreover, the phase transition from randomly-mixed phase to intercalated phase can be identified in these different systems, indicating that the difference in the polymerization rates does not alter the phase-transition behaviors, although the detailed kinetics will be systematically simulated in the future study.

CONCLUSION In summary, we have presented what is to our knowledge the first investigation to uncover the pivotal role of assembly kinetics in entropy-driven, reaction-induced phase behaviors of the nanoparticle self-assembly at interface. Our simulations demonstrate that the implementation of entropy in driving the phase transition significantly depends on the competition between the reaction rate and the diffusive dynamics of nanoparticles, which results in diverse kinetic pathways. A central finding is that, at slow reaction rates, the transition exhibits abrupt jump in the mixing parameter, in a similar way to first-order, equilibrium phase transition, whereas increasing the reaction rate diminishes the jumps until the transitions become continuous, behaving as a second-order-like phase transition where a critical exponent, characterizing the transition, can be identified. We finally develop an analytical model of the blob concept of polymer chains to complement the simulation results. The excellent agreement between the results from the simulation and the theoretical analysis corroborates that our scaling model captures the fundamental physics and thereby reveals essential scaling laws of the entropy-driven


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phase behaviors. These findings could provide useful guidelines for creating interface-reactive nanocomposites toward technologically important materials and devices, and might offer further opportunities to amplify the entropic contributions to the materials design via kinetic control.

MODEL AND METHODOLOGY Coarse-Grained Molecular Simulations. Coarse-grained molecular simulations in this work use the dissipative particle dynamics (DPD) technique, a method that samples the NVT ensemble.36 For the complicated problem considered here, DPD offers an approach that can be used for modeling physical phenomena occurring at larger time and spatial scales than molecular dynamics (MD). In DPD, each bead represents a cluster of atoms, and experiences a force with the components of conservative interaction force FC, dissipative force FD, and random force FR, i.e., fi = ∑ ( FijC + FijD + FijR ) , where the sum runs over all beads j. The interaction forces are i≠ j

treated as pairwise additive and are truncated at a certain cutoff distance rc. The conservative force is a soft, repulsive force given by FijC = aij (1 − rij )r$ ij , where aij is the maximum repulsion

between beads i and j, rij =| ri − r j | / rc , and rîj = rij / | rij | . aij has a linear relationship with FloryHuggins χij parameter: χij ≈ ( aij − aii ) 3.27 , where aii=25 is the repulsion parameter between like species (i.e., χii =0), and a larger aij corresponds to a stronger bead-bead repulsion. For interactions between unlike beads, aAP=aBQ=aAp=aBq=15, aAQ=aBP=aAq =aBp=40, aPQ=apq=30, and aAB=80, where the subscripts A, B: two fluid phases; P, Q : both sites and their grafted polymer chains of reactive Janus nanoparticles; p, q: both sites of unreactive Janus particles. Clearly, the P and p beads have an affinity to the A domain while Q and q to the B domain. The repulsion between reactive and unreactive nanoparticles anchoring in the same phase domain, i.e., aPp and aQq are chosen to 27, causing the immiscibility of these both nanoparticle components. The drag


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force is FijD = −γωD (rij )(rîj ⋅ v ij )rîj , where γ is a simulation parameter related to viscosity, ωD is a weight function that goes to zero at rc, and the relative velocity is vij=vi-vj. The random force is

% R (rij )ξij rîj , where ξij is a zero-mean Gaussian random variable of unit variance and FijR = σω σ% 2 = 2kBT γ .Here, kB is the Boltzmann constant and T is the temperature of the system. We select

weight functions to take the following form: ωD (rij ) = ωR (rij ) 2 = (1 − rij ) 2 for rij

How Implementation of Entropy in Driving Structural Ordering of

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