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Mechanism of Hierarchical Structure Formation of Polymer/ Nanoparticle Hybrids Yu-Chiao Lin,† Hsin-Lung Chen,*,† Takeji Hashimoto,*,†,‡ and Show-An Chen† †

Department of Chemical Engineering and Frontier Research Center on Fundamental and Applied Sciences of Matters, National Tsing Hua University, Hsinchu 30013, Taiwan ‡ Kyoto University, Kyoto 606-6501, Japan S Supporting Information *

ABSTRACT: Exploiting the assembly of metallic nanoparticles (NPs) in their hybrids with polymers is an important task for creating new properties or functionalities via collective interactions or dynamics of the constituents. In this work, we present a detailed study of the process and mechanism of the reactioninduced hierarchical assembly of NPs in the hybrid wherein the metallic Pd NPs were synthesized and their numbers in the system increased with time by reducing the metal precursor, Pd(acac)2, originally dissolved uniformly in the solution of poly(2-vinylpyridine) (P2VP) with benzyl alcohol (BA) as a solvent and reduction agent. The time-resolved small-angle X-ray scattering (SAXS) experiment using synchrotron radiation revealed that the structural evolution process from the beginning of NP formation to the establishment of a fractal structure built up by the clusters of Pd NPs was constituted of four distinct stages, Stages 1−4, governed by the increasing overall NP volume fraction (ϕoverall) and hence by the increasing reduction time. At Stage 1, the NPs were uniformly distributed in the matrix of P2VP and BA with negligible interparticle interaction due to the low particle volume fraction (ϕoverall ≤ ca. 3 × 10−4). The structural evolution advanced to Stage 2 when ϕoverall was increased above 3 × 10−4. In this stage, the NPs experienced the sticky hard sphere (SHS) type attractive interaction, and as a consequence the NPs underwent a phase separation into a particle-poor phase (Phase I) and a particle-rich phase (Phase II) strikingly even in such a small ϕoverall. The net interparticle attraction created the dynamic aggregates distributed uniformly within Phase II. As the ϕoverall continued to increase to ca. 4.7 × 10−4, the structural development entered Stage 3, in which the dynamic aggregates started to form a higher-order organization, generating a larger-scale heterogeneity in Phase II. The structure developed at Stage 3 served as a precursor directing the subsequent formation of the large-scale mass-fractal network built up by the static clusters of NPs within Phase II at Stage 4. The particles within the clusters were proposed to be bridged by the P2VP chains due to pyridine−Pd coordination interaction.

1. INTRODUCTION Hybrid materials are formed by two constituents mixed at the mesoscopic and/or molecular level. One component is usually an organic polymer, and the other is an inorganic or metallic substance which may be in the form of well-defined nanoscale objects such as nanoparticles or nanorods. Hybrids are found prevalently in both technology and nature. Nature delicately integrates the toughness and flexibility of biopolymers and the mechanical strength of inorganic nanocrystals to create the biomineral hybrids with remarkable properties.1 The two building blocks often organize cooperatively into a structure with a complex hierarchy covering a broad range of length scales. The hierarchical structure thus formed renders natural hybrid materials not only the functionalities or properties of the respective constituents but also the synergetic ones arising from their collective interactions or from their supramolecular assemblies at the higher levels. Metallic nanoparticles (NPs) which have been extensively investigated for applications as optoelectronic devices2−4 and catalysts5 are one of the most important inorganic building © XXXX American Chemical Society

blocks of the hybrid materials for technological applications. Metallic NPs have been delicately combined with polymers to generate well-controlled metallic nanostructures and their assemblies through specifically designed synthetic strategies.6 In the present study, we concentrate on the morphology that the simple spherical NPs will develop in the homogeneous polymer solution as the matrix in the process of the NP formation by chemical reduction of metal salt. The morphology characterized by the spatial distribution of the NPs in the matrix and the structure of the interface between NP and polymer are the key factors controlling the properties of this type of hybrid.5−9 A uniform dispersion of NPs is often desirable for achieving effective mechanical reinforcement, while percolated structures for constructing continuous conductive paths are essential for enhancing the electrical and thermal conductivities of the hybrid materials.10−13 Received: July 15, 2016 Revised: September 10, 2016

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DOI: 10.1021/acs.macromol.6b01531 Macromolecules XXXX, XXX, XXX−XXX

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hierarchical structure from the onset of the reaction in the process of hybrid preparation is also important for clarifying the kinetic effect in structure formation. However, these critical problems remain largely unsolved thus far. In the previous work42,43 by using the small-angle X-ray scattering (SAXS) method, we have disclosed that the spherical palladium (Pd) NPs formed in the hybrid with poly(2vinylpyridine) (P2VP) self-organized into the hierarchical structure composed of three distinct levels after the completion of the reduction reaction of the metallic salt. That is about six primary NPs with the average radius of ca. 1.8 nm (level one) self-assembled to form the local clusters with the average radius of ca. 4.2 nm (level two) via the sticky hard sphere (SHS) type attractive interaction between the NPs, and these clusters further aggregated to build up the large-scale mass-fractal structure (level three) with the fractal dimension of ca. 2.3. In this work, we present a detailed study on the mechanism of the hierarchical structure formation in this polymer/NP hybrid wherein the metallic NPs were synthesized by reducing the metal precursor molecularly dissolved uniformly in a matrix of a homopolymer solution. We investigated, in situ and at a real time, the time evolution of the structure in this type of hybrid using the time-resolved SAXS method starting from a very early stage of the reduction reaction to its end. It will be shown that the hierarchical structure was established progressively with time with four characteristic stages. At the initial stage of reduction where the overall NP volume fraction, denoted as ϕoverall, was low (ϕoverall ≤ ca. 3 × 10−4), the Pd NPs were uniformly distributed in the matrix with negligible interparticle interaction as well as negligible interparticle interference of the scattered waves from the NPs; the scattering intensity profile satisfied the independent scattering criterion and is denoted hereafter as IINP(q) with q being the magnitude of the scattering vector to be defined later in section 2.2 (Stage 1). As ϕoverall, which was estimated from the experimental scattering intensity distribution, Iexp(q), satisfying the criterion of IINP(q) at the high-q range to be elaborated later in sections 4.2 and 5.1, continued to increase above the critical concentration ≅ 3 × 10−4, the interaction and interference between the NPs became obviously significant. In this case, Iexp(q) was best fitted with the theoretical scattering function based on the two-phase model where one of the phases is composed of the higher concentration of the NPs interacting through the SHS potential (denoted hereafter Phase II) and the scattering of which is expressed hereafter as ISHS/inf(q) and the other phase is composed of the lower concentration of the NPs (denoted hereafter Phase I), with scattering intensity distribution given by IINP(q) (Stage 2). The phase separation in Stage 2 was driven by the net particle−particle attraction and by the condition of ϕoverall > 3 × 10−4. As ϕoverall further increased with time, the volume fraction (ϕII) of the NPs in Phase II further increased, which brought about a further organization of the NPs in Phase II to generate an additional large length-scale heterogeneity (Stage 3). Upon a further increase of ϕoverall, ϕII further increased, and the particles in Phase II assembled to form local clusters, which subsequently built up a mass-fractal structure (Stage 4). On the other hand, Phase I kept the low NP concentration with negligible interparticle interaction throughout Stages 1−4. The present findings offer deep insight into the mechanism of the hierarchical assembly of NPs in the hybrid with polymers where the metallic NPs are generated in situ by the chemical reduction. Such a fundamental understanding is essential for

The van der Waals attraction between NPs and the depletion interaction between polymers and NPs favor the clustering and aggregation of the NPs (which will be discussed later in section 5.2). A typical approach to avoid the strong aggregation into large clusters involves the modification of the NP surface by grafting surfactants or polymers on the particle surface which serve as a compatibilizer between the NP and polymer matrix.14−16 The state of dispersion of the modified NPs is determined by several parameters, including the graft density,17 the graft/matrix chain length ratio,15,18 the molecular weight of polymers comprising matrix,19 and the chemical compositions,20 as these parameters may modulate the net interaction of the particles mediated by the polymer molecules in the matrix. Another strategy to enhance the NP dispersion deals with the in-situ chemical reactions for the synthesis of either polymers21 or NPs22 in the process of hybrid preparation.23−26 As for the former, the monomer, the initiator for the polymerization, and the NPs with proper surface modification may be uniformly mixed first, and the mixture is then brought to an appropriate condition (e.g., higher temperature) to induce the polymerization of the monomer.27−29 The formation of polymer molecules lowers the entropy of mixing with the NPs and induces the depletion interaction, which in turn leads to an aggregation of the NPs. If the polymerization is sufficiently rapid, the NPs will not have time to undergo large-length-scale aggregations before the matrix is vitrified by glass transition; in this case, the dispersion state prior to the polymerization may be effectively arrested. As for the latter, the hybrid preparation involves the mixing of (i) metallic precursors or salts, (ii) polymers capable of forming complex (e.g., coordination complex) with the salts, and (iii) the reduction agents, followed by inducing the reduction of the salts via heating the mixture. Upon the reduction, the NPs are synthesized and their volume fraction increases with the reduction time in the polymer template where the polymers may interact preferentially with the surface of the particles. The polymers associating with or adhering to the particle surfaces may limit the particle growth; eventually, NPs of a well-defined size and a narrow size distribution are formed.22,30−38 Although a considerable attempt has been made to enhance the dispersion of metallic NPs based on the above approaches, in most of the cases the NPs still exhibit certain extent of aggregation in the hybrids.39 Indeed, if the NPs are uniformly distributed over the polymer matrix, the hybrid is expected to display the properties in between those of the pure constituents. The lesson from the nature indicates that the formation of characteristic hierarchical structure in the hybrid may create new properties arising from interactions of the components; therefore, an interesting direction to explore for synthetic hybrid materials is the exploitation of the underlying driving force of NP assembly to control the hierarchical structure for creating new properties or functionalities. To accomplish a delicate control, one will need to develop a deep understanding on the interaction between the NPs mediated by the polymer molecules and on how such an interaction determines the equilibrium phase diagram of the polymer/NP mixture40,41 and the structure of the NP assembly. Moreover, since the NP diffusion in the highly viscous polymer matrix is sluggish, the structure attained within the experimental time scale may be transient and nonequilibrium in nature; consequently, understanding the time evolution of the B

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Macromolecules further developing the strategy of creating new properties or functionalities by exploiting the structure and dynamics of the hierarchical structure in the hybrids over the wide range of length and time scale.

IINP, t(q; R p) =

Ii RD2

re 2NpVp2Δρel 2 Φ 2(q; R p)

(1a)

where Ii is the incident beam intensity, RD is the distance between the sample and the detector, re is the classical radius of the electron (2.818 × 10−6 nm), Vp is the volume of the particle, Δρel is the electron density difference between the particles and their matrix, and Φ (q; Rp) is given by eq 2b. Note that Ie ≡ Iire2/RD2 is the well-known Thomson scattering from a single electron. The reduced absolute intensity of the independent scattering from the particles, denoted as IINP(q; Rp), is reduced with respect to both Ii/RD2 and Vir, viz.

2. EXPERIMENTAL SECTION 2.1. Sample Preparation. P2VP with Mw = 9.1 × 103 and Mw/Mn = 1.05 was purchased from Scientific Polymer Products Inc. (Ontario, NY). The calculated radius of gyration of the unperturbed P2VP chain is 2.6 nm. Pd(acac)2 was acquired from Alfa Aesar (Ontario, NY). Prescribed amounts of P2VP, benzyl alcohol (BA), and Pd(acac)2 were dissolved in chloroform as a good solvent to obtain a dilute homogeneous solution in which Pd(acac)2 was molecularly dissolved. The solvent (chloroform) was allowed to completely evaporate at room temperature (RT) for 1 day to prepare a uniform BA solution of P2VP and Pd(acac)2. The resultant solution, containing P2VP (15.81 wt %), BA (82.61 wt %), and Pd(acac)2 (1.58 wt %), was then heated to the reduction temperature (defined hereafter as Tred.) at 0.5 °C/s from room temperature to given Tred.’s (= 80, 100, 120, and 140 °C) to induce the reduction of Pd(acac)2 into Pd atoms, which results in formation of Pd NPs. The stoichiometric volume fraction of Pd NPs in the system was calculated to be very small, ϕPd = 4.9 × 10−4, where the density of Pd, P2VP, benzyl alcohol, and (acac)2 is 12.03, 1.17, 1.0, and 0.975 g/cm3, respectively. The time preset for collecting each scattering profile during the chemical reduction process was 5 s. The sample cell had two windows, one for the incident beam and the other one for scattered beam to pass through, which were sealed by Kapton films in order to keep the concentration of BA constant for SAXS measurements during and after the reduction. 2.2. SAXS Measurements. SAXS measurements were performed at Beamline 23A1 at the National Synchrotron Radiation Research Center (NSRRC) located at Hsin-Chu, Taiwan.44 A two-dimensional Pilatus detector with 981 × 1043 pixels was used to record the SAXS intensity. The energy of the X-ray source and the sample-to-detector distance were 14 keV and 1876 mm, respectively. The irradiated volume of the sample with the incident beam was cubic, having dimensions 0.5 mm × 0.5 mm normal to the incident beam and 0.5 mm parallel to it (sample thickness). The SAXS intensities obtained were plotted against q = (4π/λ) sin(θ/2), where λ and θ are the wavelength of X-ray (λ = 0.124 nm) and the scattering angle, respectively. The sample-to-detector distance RD was calibrated using silver behenate with the primary diffraction peak at 1.076 nm−1. The SAXS profiles were corrected for the thermal diffuse scattering (TDS) arising from the longitudinal acoustic phonons, and the scattered intensities were reduced by the volume Vir irradiated with the incident X-ray beam and the factor given by Ii/RD2, where Ii is the incident beam intensity.

IINP(q; R p) ≡

IINP, t(q; R p) IiVir /RD2

(1b)

From eqs 1a and 1b, IINP(q; Rp) is given by IINP(q; R p) = noverallΔρ2 Vp2Φ 2(q; R p)

(1c)

where noverall defined by Np/Vir is the number density of the particles in the irradiated volume Vir, and Δρ is the scattering length density (SLD) defined by

Δρ ≡ reΔρel

(1d) 2

Note that IINP,t(q; Rp)/(Ii/RD ) is the differential scattering cross section, so that IINP(q; Rp) defined by eq 1b is the differential scattering cross section reduced by Vir. In the case where the particles have a polydispersity with respect to the particle radius Rp, the reduced absolute intensity is given by IINP(q) = noverallΔρ

2

∫0

∞

⎡ 4πR 3 ⎤2 p PN(R p)⎢ Φ(q; R p)⎥ dR p ⎢⎣ 3 ⎥⎦ (2a)

where Φ(q; R p) =

3 [sin(qR p) − (qR p) cos(qR p)] (qR p)3

(2b)

The SLD of the Pd NP was evaluated to be ρPd = 8.81 × 10−3 nm−2 (corresponding to the electron density of 5.195 mol electrons/cm3 based on the mass density of the Pd crystal 12.023 g/cm3), and the SLD of the matrix composed of P2VP and PA was evaluated to be ρm = 9.79 × 10 −4 nm−2 (corresponding to the electron density of 0.577 mol electrons/cm3). We assume that PN(Rp) is given by so-called Schulz-Zimm distribution46

3. THEORETICAL BACKGROUND The hybrid system under study is composed of spherical Pd NPs with relatively narrow polydispersity in radius and the matrix phase formed by P2VP and BA. The following describes the three states of the NP distribution in the matrix phase and their corresponding scattering functions that will be used to fit the experimentally observed SAXS profiles to extract the important structural parameters. Case 1. Uniform Distribution of NPs with Negligible Interparticle Interaction and Interference. When ϕoverall is low, a given NP does not experience the interaction with other particles because they are uniformly distributed in the matrix at a large average interparticle distance. In this case, the scattered waves from the particles do not interfere with one another, so that the scattered intensity is given by the independent scattering45 or the form factor of the NPs. The absolute SAXS intensity IINP,t(q; Rp) from the volume Vir irradiated by the incident beam which contains the total number of the independently scattering particles Np is given by

⎡ exp −((Z + 1)X ) ⎤ ⎥ PN(R p) = (Z + 1)Z + 1X Z ⎢ ⎢⎣ ⟨R p⟩Γ (Z + 1) ⎥⎦

(3)

where X = Rp/⟨Rp⟩ with ⟨Rp⟩ being the number-average radius, Z = (1/PRp2) − 1 is related to the polydispersity, PRp = σp/⟨Rp⟩, with σp2 being the variance of the distribution, and Γ (Z) is the gamma function. The overall volume fraction of the NPs ϕoverall defined by eq 4 was deduced from the reduced absolute scattering intensity measurement together with the pre-evaluated Δρ = (ρPd − ρm) = 7.83 × 10−3 nm−2. ⎛ 4π ⎞ ϕoverall = noverall ⎜ ⎟R p,av 3 ⎝ 3 ⎠ C

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where ϕSHS/inf is the volume fraction of the spheres in the space. Therefore, the structure factor in eq 7 is also a function of ϕSHS/inf, which can be further denoted as SSHS/inf(q; Rp,av , ϕSHS/inf). Note that ϕSHS/inf is equal to ϕoverall if the SHSs exist in a single phase. It could be different from ϕoverall if the SHSs undergo phase separation and exist mainly in the particle-rich phase. The parameters α1 and β1 in eqs 9 and 10 are given by

where Rp,av3 is defined later by eq 6. Case 2. NPs Interacting through SHS Potential Distributed over an Infinitely Large Space. When the concentration of the NPs is sufficiently high, each NP experiences the interaction with its surrounding particles. The net particle−particle attractive interaction in the present system is described well by the SHS potential which consists of a hardcore repulsion (at the interparticle distance r < 2Rp) and a rectangular attractive well with the depth of u0 < 0 and the width of Δ (in the tail at r > 2Rp).47 The attractive potential u0 and the perturbation parameter ε ≡ Δ/(2Rp + Δ) define the dimensionless parameter called “stickiness” τ via τ=

α1 =

ISHS/inf (q) = nSHS/inf Δρ2 ⟨Vp2Φ 2(q; R p)⟩SSHS/inf (q; R p,av , ϕSHS/inf )

(5)

(14)

Case 3. NPs Assemble To Form Local Clusters Which Further Build Up a Mass-Fractal Structure. Considering the case that the hybrid forms a hierarchical structure, where several NPs experiencing SHS interaction assemble to form clusters which in turn aggregate further to construct a dendritic mass fractal structure; in this case, the SHSs are confined within the individual clusters instead of distributing over an infinitely large space described in Case 2. The corresponding scattering intensity at nonzero q values thus contains an additional contribution from the form factor of the cluster; moreover, the SHS structure factor SSHS/inf needs to be convoluted with such a cluster shape factor to account for the confinement effect.50 The reduced scattering intensity (with respect to Vir and Ii/ RD2) from the clusters composed of the particles, denoted by ISHS/cluster(q), is then expressed as42,43

(6)

Therefore, ISHS/inf(q) in eq 5 is expressed as ISHS/inf (q) = nSHS/inf Δρ2 ⟨Vp2Φ 2(q; R p)⟩SSHS/inf (q; R p,av) (7)

ISHS/cluster(q) = K {π 3n p,cluster⟨Vp 2Φ 2(q; R p)⟩

We use the following analytical formulas for SSHS/inf(q) presented by Baxter49 and Menon et al.47 SSHS/inf (q) = [A2 (q) + B2 (q)]−1

× ⟨Vc 2 Φ2(q; R c)⟩ + ⟨Vp2 Φ2(q; R p)⟩ × [SSHS/inf (q; R p,av , ϕp,cluster)*⟨Vc 2Φ 2(q; R c)⟩]}

(8)

where

λ sin ua ⎫ ⎬ 12 ua ⎭

(9)

⎧ ⎡ 1 sin ua 1 − cos ua ⎤ ⎥ B(q) = 12η⎨α1⎢ − + 2 ua ua 3 ⎦ ⎩ ⎣ 2ua ⎪

⎪

⎡1 sin ua ⎤ λ 1 − cos ua ⎫ ⎬ ⎥ + β1⎢ − − ua ua 2 ⎦ 12 ⎣ ua ⎭ ⎪

⎪

Inet(q) = SMF(q)ISHS/cluster(q)

(10)

η=

πa nSHS/inf 6

⎛ a ⎞3 ⎟⎟ ϕSHS/inf = ⎜⎜ ⎝ 2R p ⎠

(16)

where SMF(q) is the structure factor of the mass-fractal object given by51

ua = qa, a = 2Rp + Δ, and η is the effective volume fraction of the spheres having the radius a/2 (= Rp + Δ/2) as given by 3

(15)

where the symbol * denotes the convolution product, np, cluster is the number density of the particles in the cluster, K ≡ np,cluster Δρ2, ⟨Vc2Φ 2(q; Rc)⟩ is the cluster form factor averaged for the polydispersity of the radius of the cluster, Rc, and Vc is the volume of the cluster. Φ 2(q; Rc) is given by eq 2b with a replacement of Rp by Rc. It is important to note that the volume fraction to be used for SSHS/inf(q; Rp,av , ϕp,cluster) in eq 15 is the average local volume fraction of the NPs within the clusters and hence is defined as ϕ p,cluster . Note that ϕ p,cluster = np,cluster(4π⟨Rp3⟩/3). The reduced scattering intensity of the large-scale fractal object built up by the clusters, Inet, is given by42,43

⎧ sin u − u cos u 1 − cos ua a a a A(q) = 1 + 12η⎨α1 + β1 3 ua ua 2 ⎩ −

(13)

Now we rewrite eq 7 by stating explicitly that the structure factor is a function of ϕSHS/inf, viz.

where Vp is the particle volume and SSHS/inf(q) is the structure factor47 of the NPs with the SHS interaction. nSHS/inf is the number density of the particle in the space. The brackets ⟨ ⟩ denote average with respect to the distribution of Rp. We adopt the so-called “average structure factor approximation”,48 which assumes that the structure factor for the polydisperse spheres is equal to SSHS/inf(q; Rp,av), i.e., the SHS structure factor for the monodisperse sphere having the average particle size Rp,av which satisfies the demand given by PN(R p)R p3 dR p

μ ≡ λη(1 − η)

1 + (η /2) η 2 ⎛ η ⎞ λ − ⎜τ + =0 ⎟λ + 12 1 − η⎠ ⎝ (1 − η)2

ISHS/inf (q) = nSHS/inf Δρ2 ⟨Vp2Φ 2(q; R p)SSHS/inf (q; R p)⟩

∞

−3η + μ , 2(1 − η)2

and the parameter λ is a solution of the quadratic equation given by

The reduced scattering intensity (with respect to Vir and Ii/ RD2) from the SHSs distributed over an infinitely large space, denoted by ISHS/inf(q), is given by

∫0

β1 =

(12)

1 exp(u0 /kBT ) 12ε

R p,av 3 ≡

1 + 2η − μ , (1 − η)2

SMF(q) = 1 + (11)

DmΓ (Dm − 1) sin[(Dm − 1) tan−1(qξ)] (qR c,f )Dm [1 + (qξ)−2 ](Dm − 1)/2 (17)

D

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hump at ca. 0.4 nm−1 in Region II was dominated by the form factor of the cluster with the average radius ⟨Rc⟩ of ca. 4 nm. The strong intensity upturn with decreasing q observed in Region III was associated with the fractal objects with the mass fractal dimension of 2.2−2.3 built up by the clusters. 4.2. Time Evolution of SAXS Profiles during the Reaction at 80 °C. Now we proceed to present the time evolution of the SAXS profiles collected during the reduction reaction. Figure 2a displays the representative time-resolved SAXS profiles of the hybrid prepared by reducing Pd(acac)2 in the solution containing P2VP and BA at 80 °C. The SAXS profile at the room temperature as well as those obtained at 5 ~ 100 s at Tred. = 80 °C represents the scattering profile before the reduction and hence serves as the reference for examining the change of scattering intensity during the reduction process. In this case, the intensity was very weak, implying that Pd(acac)2 was molecularly dissolved uniformly in BA. The lowq scattering excess to the TDS intensity may reflect the concentration fluctuations of Pd(acac)2 and P2VP in BA. It can be seen that the SAXS intensity increased progressively with time after 200 s due to the formation of Pd NPs by the reduction reaction. The entire experimental scattering profiles Iexp(q) up to 630 s (shown by the symbols in Figure 2b) could be fitted very well by IINP(q) given by eqs 2a, 2b, and 3 as shown by the red lines, indicating that the NPs did not obviously experience interaction with other particles due to the low concentration. As a matter of fact, ϕoverall of the NPs formed upon the reaction up to 630 s, which was estimated by the best fit of Iexp(q) with IINP(q) under the fixed SLD Δρ = 7.83 × 10−3 nm−2, was only about up to 3 × 10−4 as shown in Figure 4b because of a very slow reduction rate at 80 °C and the small concentration of Pd(acac)2 added to the system. We denote this initial stage as Stage 1 hereafter. The average radius of the NPs, ⟨Rp⟩, also increased slightly from 1.26 nm at 300 s to 1.57 nm at 630 s (see Table S2 in the Supporting Information and Figure 4b), as clearly evidenced also by the low q shift of the first-order form factor peak at q ∼ 3 nm−1 (see the insets to parts a and b). The scattering profiles obtained in the time span longer than ca. 630 s after the onset of the reduction could no longer be fitted by IINP(q), as demonstrated in Figure 2c, which compares Iexp(q) (the symbols) with IINP(q) (the red curves). Although IINP(q) fitted the experimental data well at q > 1.2 nm−1 as shown in part c where the border at q = 1.2 nm−1 is indicated by the vertical broken line, an obvious deviation of IINP(q) from the experimental data was found at q < 1.2 nm−1, where the observed intensity was lower than that predicted by IINP(q) in the q range of 0.6 nm−1 < q < 1.2 nm−1, but it was higher than IINP(q) at q < 0.6 nm−1, as respectively shown by the trends indicated by the two arrows. This deviation is attributable to the effects of the structure factor SSHS/inf(q; Rp,av) being controlled by the SHS type attractive interparticle interaction.42,43 The suppression of the observed intensity relative to IINP(q) in the higher q region arises from the fact that SSHS/inf(q; Rp,av) < 1, which gives rise to the destructive interparticle interference effect, while the excess intensity relative to IINP(q) in the lower q region is due to the fact that SSHS/inf(q; Rp,av) > 1, which gives rise to the constructive interparticle interference effect. We found that the trend described above cannot be ever accounted for by the particles having the hard-core repulsion only.42,43 4.2.1. SHS Particles Dispersed in a Single Phase. The SHS potential is the simplest form of interaction potential which

where Dm is the mass-fractal dimension, Γ (x) is the gamma function of argument x, ξ is the upper cutoff length that characterizes the size of the upper bound of the mass-fractal object in real space, and Rc,f is the lower cutoff length of the mass-fractal object that should satisfy Rc,f = Rc. The scattering function given by eq 16 represents a model for a hierarchical structure with three distinct levels. The primary NPs with well-defined spherical geometry and size constitute the first level of the hierarchical structure, and the clusters assembled by a number of particles and the mass-fractal structure further built up by the clusters correspond to the second and third level, respectively.

4. RESULTS AND ANALYSES 4.1. SAXS Profiles after Completion of Reaction. Figure 1 displays the SAXS profiles of the hybrids prepared at 1800 s

Figure 1. (a) SAXS profiles of the hybrids prepared by reduction of Pd(acac)2 at Tred. = 120 and 140 °C at 1800 s after the onset of the reduction reaction. The experimentally observed profiles presented by the symbols were fitted very well by the hierarchical structure model of eq 16 presented by the solid curves superposing on the experimental data. (b) Schematic illustration of the hierarchical structure composed of three levels formed in the nanocomposites. The upper cutoff size of the fractal structure Rm exists beyond the small q-limit covered by this experiment.

after the onset of the reaction at 120 and 140 °C to demonstrate the features of the scattering from the hierarchical structure predicted by eq 16. The reactions at these two temperatures were rapid so that the reaction was found to be completed because the time-resolved SAXS profiles reached a steady profile before 1800 s. It can be seen that the experimentally observed profiles presented by the symbols were fitted very well by the hierarchical structure model, i.e., eq 16, presented by the solid curves superposing on the experimental data. The morphological parameters obtained from the model fitting are listed in Table S1 (see Supporting Information). The SAXS profiles were characterized by three regions with respect to q. The scattering intensity in Region I (at q ≳ 1 nm−1) was dominated by the structure of level one, i.e., the Pd NPs interacting through the SHS potential confined within the cluster, where the small peak (marked by “i = 1”) at q ≅ 3 nm−1 corresponds to the first-order form factor maximum of the NPs with the average radius ⟨Rp⟩ of ca. 1.7 nm, and the small but important hump (pinpointed by the thick arrow) at ca. 1.2 nm−1 arises from the interference of the scattered rays from the NPs confined within the clusters, which experienced the SHS interaction with the potential depth u0 of ca. −2.3kBT (see refs 42 and 43 for more details). The broad E

DOI: 10.1021/acs.macromol.6b01531 Macromolecules XXXX, XXX, XXX−XXX

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Figure 2. (a) Representative time-resolved SAXS profiles over the total time period of 1800 s for the hybrid prepared by reducing Pd(acac)2 at 80 °C. (b) Representative SAXS profiles illuminating Stage 1 (at t ≤ 630 s) in which the entire scattering profile Iexp(q) at the given time presented by the symbols (○) can be well fitted by the independent scattering of the NPs, IINP(q), given by eqs 2a, 2b, and 3 which is presented by the red line with the volume fraction of the NPs, ϕoverall(t), to be shown in Figure 4b and Table S2. (c) Representative SAXS profiles Iexp(q) in Stage 2 presented by the symbols (○) (630 s < t ≤ 1800 s) in which the entire scattering profile at the given time cannot be fitted by IINP(q) (i.e., the red profile with the volume fraction of the NPs ϕoverall(t) to be shown in Figure 4b and Table S2). (d) Best fit of Iexp(q) at Tred. = 80 °C and at 1800 s [profile 1 presented by the symbols (○)] with ISHS/inf on the basis of the single-phase model with ϕSHS/inf = ϕoverall = 2 × 10−2 [the blue profile (3)] and 4 × 10−4 [the red profile (2)]. The green profile (4) was obtained by multiplying the proportionality constant C = 2.36 × 10−2 to the blue profile.

discussed earlier, but it clearly failed to account for Iexp(q) at q < 1.2 nm−1. This failure of ISHS/inf(q) in the low q region at q < 1.2 nm−1 is believed to arise from the fact that SSHS/inf(q; Rp,av , ϕSHS/inf) in eq 14 at such a low NP concentration as ϕSHS/inf = 4 × 10−4 approaches unity in the low q region; thereby ISHS/inf(q) in the low q region cannot account for the interparticle interference effect controlling Iexp(q) in the low q range in the real system. In other words, ISHS/inf(q) with ϕSHS/inf = 4 × 10−4 is nothing other than the IINP(q) itself; thus, it can account for Iexp(q) in the high q region but not in the low q region. The volume fraction which controls the absolute intensity of IINP(q) = Iexp(q) in the high q region reflects ϕoverall, because IINP(q) arises from the independent scattering from individual NPs, regardless of whether the NPs exist in the single phase or in the two phases. The shape of the profile, i.e., the relative intensity distribution of Iexp(q), in the low q region could be well reproduced by ISHS/inf(q) when a larger volume fraction of

takes account of the hard-core repulsion and an attractive tail.47,49 Therefore, the SHS particles distributed over an infinitely extended single-phase space, i.e., ISHS/inf(q) given by eq 14, with ϕSHS/inf as one of the parameters influencing SSHS/inf(q; Rp,av, ϕSHS/inf) assumed to be equal to ϕoverall , was applied to fit Iexp(q)’s observed at t > 630 s. Figure 2d shows a representative Iexp(q) at a particular time 1800 s (profile 1 presented by the symbol ○) for the time span of t > 630 s, which was attempted to be fitted with ISHS/inf under the constraints of ϕSHS/inf = ϕoverall = 4 × 10−4, nSHS/inf = ϕoverall/ (4π⟨Rp3⟩/3), and the fixed SLD of Δρ = 7.83 × 10−3 nm−2. However, ISHS/inf(q) never fitted Iexp(q) satisfactorily under the imposed constraints. As a matter of fact, we found the following specific difficulties with this model based on the SHS particles in a single phase. As shown in Figure 2d by the red line (profile 2), the theoretical result of ISHS/inf(q) fitted very well with Iexp(q) only at q ≥ 1.2 nm−1 by choosing ϕSHS/inf = ϕoverall = 4 × 10−4 as F

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Macromolecules ϕSHS/inf = ϕoverall = 2 × 10−2 was chosen for SSHS/inf(q; Rp,av , ϕSHS/inf), as also shown in Figure 2d by the blue theoretical line (profile 3). However, the absolute intensity of this theoretical profile was found to be much higher than Iexp(q). It should be stressed that the blue profile (3) for ISHS/inf(q) with ϕSHS/inf = 2 × 10−2 reproduced well Iexp(q) when it was multiplied by a proportionality constant C = 2.36 × 10−2, as demonstrated by the green profile (4). This trend clearly reveals that the system was composed of the two phases in which the phase having the higher concentration of the particles (ϕSHS/inf = 2 × 10−2) had only a small phase volume fraction and the phase having the smaller concentration of the particles had a large phase volume fraction, and the former dominated the experimental intensity profile Iexp(q). 4.2.2. Evidence of SHS Particles Phase-Separated into Two Phases. The deviations described above together with the fact that the profile 4 fitted Iexp(q) reveal that (a) the NPs exhibited the SHS type attractive interactions and (b) the NPs in the time span (630 s < t < 1800 s) relevant to Figure 2c underwent phase separation into two phases as schematically illustrated in Figure 3a: one of the phases, defined hereafter as Phase I, has a

ϕI), while another phase, defined hereafter as Phase II, has a larger volume fraction of the NPs (defined as ϕII) and thereby gives rise to the reduced scattering intensity given by ISHS/inf(q; Rp,av , ϕII). If the phase volume fractions of Phase I and Phase II are denoted by Φ I and Φ II, respectively, as shown in Figure 3a, then ϕoverall = ϕIΦI + ϕIIΦII ,

ΦI + ΦII = 1

(18)

The scattering intensity from the two-phase system is given later by eq 19 to be described later in section 5.1. If the phase diagram in the parameter space of the reduced interaction potential energy −u0/kBT and the volume fraction of the NPs ϕNP is very much skewed as shown in Figure 3b, then (i) the critical volume fraction of the NPs, ϕcr, is very small and (ii) ϕII ≫ ϕI and Φ II ≪ Φ I as the theories predict.40,47 Here we consider u0 as the effective particle− particle interaction potential of the mean force which reflects the particle−particle, particle−polymer, and polymer−polymer interactions as will be discussed later in section 5.4. Under the conditions described above, it is likely that the scattering from the two-phase systems dominantly or selectively arises from Phase II, where the structure factor of which is given by SSHS/inf with the relevant volume fraction given by ϕSHS/inf = ϕII (see eq 14). If the scattering arises dominantly from Phase II, the weighting factors ϕIΦ I and ϕIIΦ II which describe the relative contribution of the scattering from Phase I and Phase II to the net scattering, respectively, satisfy ϕIIΦ II ≫ ϕIΦ I (see eqs 19 and 20 in section 5.1), which in turn gives rise to ϕoverall ≅ ϕIIΦ II from eq 18. The fact that volume fraction ϕII = 2 × 10−2 is much larger than ϕoverall = 4 × 10−4 means that Φ II ≅ ϕoverall/ ϕII ≅ 0.02. The small phase volume fraction Φ II (≅0.02) significantly lowers the absolute intensity ISHS/inf(q; Rp,ϕII) and is thereby primarily responsible for the small value of C ≅ 0.02. The full and exact analysis of C leading to C = 2.36 × 10−2 is given later in section 5.1 in conjunction with section 7 (see references and note 53). Thus, the two-phase model clarifies the dilemma illuminated by the deviations shown in Figure 2d.

Figure 3. Schematic illustrations of the two-phase model (a) and the phase diagram (b) of the hybrid composed of the NPs and the polymer solution where ϕI ≪ ϕII and Φ I ≫ Φ II.

lower volume fraction of the NPs (defined as ϕI) and thereby gives rise to the reduced scattering intensity given by IINP(q;

Figure 4. (a) Representative time-resolved SAXS profiles at Stage 2 (630 s < t ≤ 1800 s) at Tred. = 80 °C in which the entire profile was fitted by I2Phase(q) (the blue profiles obtained with the ϕII and ϕoverall shown in part b). The best fits of IINP(q)’s at q > 1.2 nm−1 which also yield ϕoverall are shown as references. (b) Average radius of the NPs ⟨Rp⟩, the average volume fraction of the NPs, ϕII, in the NP-rich phase (Phase II), and the overall volume fraction ϕoverall of the NPs obtained by the model fitting using IINP(q) for the SAXS profiles collected at t ≤ 630 s and I2Phase(q) for the SAXS profiles collected at t > 630 s. G

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Macromolecules 4.2.3. SHS Particles Undergoing Phase Separation into Two Phases at 80 °C. We calculated the theoretical scattering profiles, defined hereafter I2Phase(q), based on the two-phase model and tried to fit them with Iexp(q) obtained for the time span from 630 to 1800 s. In the two phase model, we assumed that the scattering from Phase I and that from Phase II are given by IINP(q) and ISHS/inf(q), respectively. The descriptions of the detailed method will be deferred to section 5.1. We denote the stage in which the scattering can be described by the two-phase model as Stage 2 hereafter. The best-fitted results of I2Phase(q) with Iexp(q) are displayed as the blue profiles in Figure 4a together with IINP(q) shown by the red profiles as references; it is clear that I2Phase(q) offered an excellent fit to the experimental data Iexp(q) presented by the symbols ○. The fitting yielded the following pieces of information with respect to Phase II as summarized in Table S2 of the Supporting Information: the attractive potential depth u0 of ca.−2.6kBT; the width of the attractive interaction, Δ, of ca. 0.3 nm (ε = 0.08), both of which are almost independent of time t, and the volume fraction of the NPs as a function of t, ϕII(t), as shown in Figure 4b, Table S2, and Table S6. It is noted that the good fit was found even for the SAXS profile collected at 1800 s, i.e., at the end of the time-resolved experiment; that is, the hierarchical structure characterized by the scattering pattern shown in Figure 1a was not formed even at the end of the experiment at 80 °C. Nevertheless, the values of both u0 and Δ associated with the SHS interaction of the NPs at 80 °C were close to those at 120 and 140 °C (see Table S1) in which the NPs were confined within the cluster, signaling that the strength and the width of the interparticle interaction remained virtually constant, i.e., −u0 ≅ 2.3kBT and ε = Δ/(2Rp + Δ) ≅ 0.08. Figure 4b displays the time evolution of ⟨Rp⟩, ϕoverall, and ϕII obtained by the fittings with the two-phase model described above as a function of time where ϕII is the volume fraction of the NPs in Phase II and in Stage 2 giving rise to ISHS/inf(q). The detailed discussion on Figure 4b will also be deferred to section 5.1. 4.2.4. General Features on Reaction-Induced Phase Separation of NPs and Time Evolution of the NP Assembly within Phase II. The time-resolved SAXS profiles of the hybrid prepared at 80 °C was characterized by the following two stages. At t ≤ 630 s, the Pd NPs were well dispersed in the matrix phase with negligible interparticle interaction due to such a low overall particle volume fraction as ϕoverall < 3 × 10−4, as schematically illustrated in Figure 5a. Such a time regime where the entire SAXS profile can be fitted by IINP(q) was denoted as “Stage 1”. As the particle volume fraction continued to increase above 3 × 10−4, the NPs started to experience the SHS type attractive interaction, which resulted in the phase separation into Phase I (NP-poor phase) and Phase II (NP-rich phase) as shown in Figure 5b. The scattering from Phase II is described by ISHS/inf(q),52 while that from Phase I is given by IINP(q) due to the very low NP concentration in this phase as detailed later in sections 5.1 and 5.2. The entire experimental SAXS profile was found to be well fitted by I2Phase(q) at t > 630 s (Figure 4a). This time regime was called “Stage 2”. Judging from the fact that the SAXS profiles Iexp(q) in the low-q region (q < 0.2 nm−1) at Stage 2 as well as the corresponding profiles of SSHS/inf(q; Rp,av , ϕII) which gave rise to the best-fit of the theoretical ISHS/inf(q) to Iexp(q) exhibited a Guinier-like plateau, we concluded that the NPs in Phase II are associated to form a

Figure 5. Schematic illustration for the reduction-induced time evolution of self-assembly of Pd NPs in the matrix of P2VP and BA. In Stage 1 where the NPs volume fraction is very small, ϕoverall ≤ 3 × 10−4, the NPs are uniformly distributed in a single phase, while in Stages 2−4 where ϕoverall > 3 × 10−4, the NPs are phase-separated into the NP-rich phase (Phase II) having a large particle volume fraction ϕII but a small phase volume fraction Φ II and the NP-poor phase (Phase I) having a small particle volume fraction ϕI but a large phase volume fraction Φ I (Φ I ≫ Φ II and ϕI ≪ ϕII). A further time evolution of the NPs dispersion within Phase II is schematically presented in parts c and d for Stages 2 and 3, respectively. In Stage 2, the dynamic aggregates of the NPs (schematically illustrated by the green dotted lines) are uniformly dispersed in Phase II, while in Stage 3 large length-scale higher-order structures are formed from the dynamic aggregates as schematically shown by an aggregate composed of the squares, where each square represents a dynamic aggregate. In Stage 4, a mass-fractal structure is built up by the clusters (gray cycles) which are composed of the permanent association of several NPs, as shown by the inset to part e.

type of aggregates driven by the SHS attractive interactions as shown in Figure 5c. These aggregates were postulated to be dynamic in nature, such that the association and dissociation of the particles into and from the aggregates, respectively, were in dynamic equilibrium. They were thus different from the static clusters (which were created by an essentially permanent association of the NPs) building up the mass-fractal structure. The Guinier plot using the SSHS/inf(q; Rp,av , ϕII) data at q < 0.2 nm−1 in Stage 2 yielded the average radius of gyration of the dynamic aggregates, which was found to increase from 1.7 to 2.3 nm with time. The fact that the Guinier regions of the dynamic aggregates were discernible in the SAXS profiles indicates that they were uniformly distributed without a higherlevel organization, as schematically illustrated in Figure 5c. The detailed analyses associated with the dynamic aggregates will be presented later in section 5.3. The structural evolution up to Stage 3 and Stage 4 in Phase II as shown in Figures 5d and 5e, respectively, occurred at Tred. = 100 and 120 °C, respectively. The details will be discussed in the following sections. 4.3. Time Evolution of SAXS Profiles during the Reaction at 100 °C. Figure 6a displays the representative time-resolved SAXS profiles collected during the reaction at a higher reduction temperature of 100 °C. The structural evolution at this temperature was similar to that observed at 80 °C, although the reduction rate and structural evolution of the NPs were faster than that at 80 °C. At t ≤ 90 s, Iexp(q) could H

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Figure 6. (a) Representative time-resolved SAXS profiles of the hybrid prepared by reducing Pd(acac)2 at Tred. = 100 °C over the total time period of 1800 s. (b) Representative SAXS profiles focused on Stage 1 (t ≤ 90 s) in which the entire scattering profile can be fitted by IINP(q), presented by the red profiles. (c) Representative SAXS profiles focused on Stage 2 (90 s < t ≤ 500 s) in which the entire scattering profile can be fitted by I2Phase(q) as shown by the red profiles. The SAXS profile collected at 1800 s is also displayed as a reference to show the presence of an excess intensity from that prescribed by I2Phase(q) at q < 0.3 nm−1 for the reaction time longer than 500 s. The inset (c1) to part (C) shows the intensity at q = 0.9 nm−1 as a function of time. (d) Representative SAXS profiles at Stage 3 (500 s < t ≤ 1800 s), where the SAXS profiles exhibit an excess intensity from that predicted by I2Phase(q) (i.e., the blue profiles) at q < 0.3 nm−1.

upward arrow marked at q ≈ 0.2 nm−1 in Figure 6c; however, Iexp(q) at 0.4 < q (nm−1) < 1.2 first increased with time from 100 to 300 s, but it decreased afterward, as demonstrated by the downward arrow marked at q ≈ 0.9 nm−1 and the inset which plots the intensity at q = 0.9 nm−1 as a function of time. Because ϕoverall of the NPs continued to increase over the time span of Stage 2, as shown in Figure 7, the intensity change over the region of 0.4 < q (nm−1) < 1.2 was governed by both the NP concentration (which increases with time) and the destructive interparticle interference effect arising from interparticle attraction (which tends to suppress the intensity under a given ϕΠ in this q range). At t < 300 s, the former effect dominated, such that Iexp(q) in this q region increased with time. The interference effect started to take over at t > 300 s, leading to depression of Iexp(q) with time. On the other hand, the intensity at q < 0.4 nm−1 kept increasing due to the

be well fitted by IINP(q) over the entire q range, as shown in Figure 6b, which corresponds to Stage 1 in the process of structural evolution. The form factor peak at q around 2−3 nm−1 shifted toward the lower q values with time as shown in the insets to parts a and b, implying that the average radius of the spheres increased from 1.75 to 1.98 nm as indicated in Table S3 and Figure 7. The observed intensity profile started to deviate from IINP(q) at t ≥ 100 s, and the SAXS profiles up to 500 s could be fitted satisfactorily by I2Phase(q) as displayed in Figures 6c by the red profiles, with the value of ϕII different from ϕoverall, as presented in Figure 7, and as summarized also in Table S3, signaling that the structural evolution has proceeded into Stage 2, in which the NPs underwent the phase separation driven by increasing ϕoverall of the NPs having the SHS interaction. Here, Iexp(q) at q < 0.4 nm−1 tended to increase with time, as highlighted by the I

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the permanently associated NPs which eventually constructed a fractal structure, as schematically illustrated in Figure 5e. This stage of structural formation is denoted as “Stage 4”. In this case, the higher-level association of the dynamic aggregates or the clusters at Stage 3 may serve as a precursory structure that directed the subsequent development of the mass-fractal structure composed of the static clusters. Table S3 summarizes the characteristic parameters assessed in Stages 1−3 at 100 °C. 4.4. Time Evolution of SAXS Profiles during the Reaction at 120 °C. Figure 8a shows the representative timeresolved SAXS profiles Iexp(q) of the hybrid prepared at an even higher reduction temperature of 120 °C. In contrast to the case at 80 and 100 °C, the scattering profiles Iexp(q) collected at 5 ≤ t (s) ≤ 45 (○ symbols) were well fitted by I2Phase(q), as shown by the blue profile in Figure 8b, indicating that the present time-resolved experiment could follow the structural development started from Stage 2, because the reduction of the salt at 120 °C and the structural evolution of the NPs were so rapid that Stage 1 may exist in the time span less than 5 s after the onset of the reduction. Identically to that found for the case at 80 and 100 °C, Iexp(q) at Stage 2 signifies the occurrence of the phase separation. The characteristic parameters deduced from the best-fit between Iexp(q) and I2Phase(q) in Stage 2 are summarized in Tables S4 and S6. As the reaction continued, the SAXS profiles Iexp(q) collected at t > 45 s (see Figure 8c) became closely resembling the scattering profiles associated with the hierarchical structure shown in Figure 1a. In this case, the scattering profiles were different from that observed at Stage 3 at 100 °C (Figure 6d) in that the hump at 1.2 nm−1 (marked by the arrow) was now identified at 120 °C as shown in Figure 8c. The entire SAXS profiles Iexp(q) at t > 45 s (open circles) could now be well fitted by the hierarchical structure model Inet(q) given by eq 16 with −u0 ∼ 2.3kBT as shown by the red line in Figure 8c. ISHS/cluster(q) for the clusters formed by the SHS interaction calculated by using eq 15 are also displayed by the green profile in Figure 8c for comparison. Thus, the structural development in this stage was designated as that in Stage 4. The structure in Stage 4 is schematically shown in Figure 5e. Figure 8d displays ϕoverall in Stages 2 and 4 as well as ϕII in Stage 2 and ϕp,cluster in Stage 4 as a function of time. It is noted that ϕp,cluster corresponds to the average volume fraction of the NPs within the cluster. Again, ϕp,cluster (∼0.4) was always higher than the corresponding ϕoverall, which is consistent with the local clustering of the NPs at this stage. The characteristic parameters deduced from the best-fit between Iexp(q) and I2Phase(q) in Stage 4 are summarized in Table S4.

Figure 7. Average radius of NPs ⟨Rp⟩, the average volume fraction ϕII in Phase II, and the overall volume fraction ϕoverall of the NPs obtained by the model fitting using I2Phase(q) for the SAXS profiles collected at t ≥ 90 s and IINP(q) for all SAXS profiles at q > 2 nm−1.

constructive interparticle interference effect which is enhanced by increasing NP concentration. The constructive and destructive interference effect will be clearly demonstrated later by Figure 9a. Again, the observed trends of intensity variation cannot be ever accounted for by the hard-core type repulsive interparticle interaction only. As the time progressed, a new type of scattering profile was identified at t > 500 s (see Figure 6d). In this case, an excess intensity from that predicted by I2Phase(q) was identified in Iexp(q) at q < 0.3 nm−1, and such an excess intensity grew progressively with time. More precisely, this excess intensity cannot be predicted by ISHS/inf(q) in Phase II. Although the intensity profiles at q ≤ 0.3 nm−1 now appeared like the low-q upturn in the SAXS profiles observed in Figure 1a, these scattering profiles in Figure 6d were different from those associated with the hierarchical structure identified in Figure 1a, since the hump at ca. 1.2 nm−1 as shown by the arrow arising from the cluster was not clearly observable.53 Indeed, the overall scattering profiles displayed in Figure 6d could be approximated by the supposition of I2Phase(q) (which fitted with the experimental data at q > 0.3 nm−1) and an excess intensity in the low-q region. The time regime over which such a characteristic SAXS profile was observed was denoted as “Stage 3”. The emergence of the excess scattering may imply an onset of evolution of a larger-scale heterogeneity through a higherlevel structural organization in Phase II, as schematically illustrated in Figure 5d by the green squares, which cannot be characterized either by ISHS/inf or ISHS/cluster, though the fraction of NPs incorporated in the heterogeneity is still minor. We propose that the higher-level organization was built up by the dynamic aggregates or clusters of the NPs, as schematically illustrated by the squares in Figure 5d. The values of ⟨Rp⟩, ϕII, and ϕoverall obtained by fitting the SAXS profiles at Stages 2 and 3 (t > 500 s) with I2Phase(q) are displayed in Figure 7. Stage 3 persisted from 500 s until the end of the time-resolved SAXS experiment at 100 °C (1800 s), showing that the hierarchical structure depicted in Figure 1b was still not accessed in 1800 s at 100 °C. Nevertheless, as has been shown in our previous study,42,43 when the system was stored at 100 °C for sufficiently long time (e.g., 5 h), the scattering pattern identical to that displayed in Figure 1a was observable. This implies that a further NP organization took place following Stage 3 to form the static clusters composed of

5. DISCUSSION 5.1. Phase Separation of NPs above the Critical Concentration. It is noted that the average NP radius increased slightly with time (within 0.6 nm), but from the viewpoint of phase separation and thereby the self-assembled structure of the NPs, the increase of volume fraction of NP with time is more important than the increase of the average NP size, in particular under the condition that the observed size increase was very small. When the volume fraction of the NPs having the given net attractive interaction of −u0/kBT increases with time from the point P to the point Q in Figure 3b and exceeds the critical volume fraction ϕI, the hybrids are anticipated to undergo the phase separation into two phases.40,47 The characteristic parameters (ϕI, ϕII, Φ I, and J

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Figure 8. (a) Representative time-resolved SAXS profiles of the hybrid prepared by reducing Pd(acac)2 at 120 °C over the total time period of 1800 s. (b) Representative SAXS profiles at Stage 2 (t ≤ 45 s) in which the entire scattering profile Iexp(q) (open circles) can be fitted by I2Phase(q) (i.e., the blue profiles). (c) Representative SAXS profile at Stage 4 (t > 45 s) in which the entire scattering profile Iexp(q) (open circles) can be fitted by the hierarchical structure model, Inet(q) (i.e., the red profiles), given by eq 16. The green solid curves correspond to the fits of the intensity experimental profiles Iexp(q) at q ≥ 0.5 nm−1 with ISHS/cluster(q) given by eq 15 to demonstrate that the NPs experiencing SHS interaction did form clusters. (d) Average radius of NPs ⟨Rp⟩, the average volume fraction ϕII in Phase II, ϕp,cluster, and the overall volume fraction ϕoverall of the NPs obtained by the model fittings in Stages 2 and 4. The values of the local volume fraction at Stage 4 were obtained by the model fitting using Inet(q) and were hence equal to average volume fraction of the particles within the cluster, ϕp,cluster.

Φ II) characterizing the two phases satisfy the constraints given by eq 18. The average scattering intensity distribution from the two phases, I2Phase(q), in Stage 2 is given by I2Phase(q) = ΦIIISHS/inf (q; ϕII) + ΦIIINP(q; ϕI)

I2Phase(q) = ΦIInIIΔρ2 ⟨Vp2Φ 2(q; R p)⟩SSHS/inf (q; R p,av , ϕII) + ΦInIΔρ2 ⟨Vp2Φ 2(q; R p)⟩ = ΦIInII[SSHS/inf (q; R p,av , ϕII) + α]Δρ2 ⟨Vp2Φ 2(q; R p)⟩

(19)

(20)

based on the assumption that (i) the scattering from Phase II rich in the NPs is given by ISHS/inf(q; ϕII) in eq 14 and (ii) the scattering from Phase I poor in the NPs is given by IINP(q; ϕI) in eq 2a. It is important to note that in the two-phase system the parameters nSHS/inf and ϕSHS/inf used for SSHS/inf in eq 14 should be replaced by nII and ϕII, respectively, which are related to each other by ϕII = nII(4π⟨Rp3⟩/3), and the parameter noverall in eq 2a should be replaced by nI with nI = ϕI /(4π⟨Rp3⟩/3). Here nII and nI are the number density of the particles in Phase II and Phase I, respectively. From eqs 2a, 14, and 19, one obtains

where

α ≡ ΦInI /(ΦIInII)

(21)

In the two-phase model, there are four parameters (Φ I, Φ II, ϕI, ϕII) with the two constraints given by eq 18, so that the independent parameters are two. One of the two parameters controls the absolute intensity, and the other controls the relative intensity distribution, which enable the best fitting of Iexp(q) with I2Phase(q). When q → ∞, SSHS/inf(q; Rp,av ,ϕII) → 1, so that K

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I2Phase(q), as listed in Table S6. The results in Table S6 are summarized as follows: (i) ϕI and ϕII were of the order of 10−5 and 10−2, respectively, at Tred. = 80 and 100 °C, and the disparity between ϕII and ϕI was further enhanced at the higher reduction temperature 120 °C; (ii) Φ II was less than 5 vol %, while Φ I was larger than 95 vol %. As a consequence of the results (i) and (ii), the scattering from Phase II much dominated over that from Phase I and thereby much dominantly contributed to Iexp(q) as revealed by the small value of α in Table S6 (less than 0.06 at Tred. = 80 and 100 °C and almost zero at Tred. = 120 °C) (eqs 20 and 21). Although the state of the phase separation was simplified into the macroscopic one for the sake of brevity in Figure 5b, Phase II in the real scenario were split into many phase-separated domains with characteristic sizes much larger than the length scale of our observation, i.e., 2π/qmin ≅ 60 nm with qmin being the minimum value of q covered in our experiment (0.1 nm−1). From this point of view the scattering from Phase II can be described on the basis of the SHS dispersed effectively in the infinitely large space except at the early stage of Stage 2. We did not apply the theoretical analyses based on the two-phase model to the scattering data obtained in the early stage of Stage 2 (see the lack of the data points for ϕII in the early stage of Stage 2 in Figure 4b and Figure 7). The SHS interaction led to the formation of the dynamic aggregates which distributed rather uniformly within the Phase II as schematically presented in Figure 5c, such that the Guinier region characterizing their average radius of gyration was discernible in the low-q region of the SAXS profiles Iexp(q) and the structure factor SSHS/inf(q; Rp,av , ϕII) obtained by the best fit of Iexp(q) with I2Phase(q) as will be shown in Figure 9. 4. Stage 3, where the dynamic aggregates formed at Stage 2 started to develop large length-scale heterogeneities as schematically shown in Figure 5d by the square symbols. It may be also feasible that the dynamic aggregates were first transformed into the static clusters of particles which in turn formed the large length-scale heterogeneities. In both cases, the larger-scale heterogeneities account for the excess scattering intensity in the low-q region (q < 0.3 nm−1) as shown in Figure 6d. 5. Stage 4, where the NPs in the dynamic aggregates were transformed into static clusters which further assembled to build up a larger-scale fractal structure as schematically presented in Figure 5e. It may be conceivable that the growth of the large length-scale heterogeneities in Stage 3 with time resulted in the formation of the fractal structure built up by the clusters in Stage 4. 5.3. Time Evolution of Dynamic Aggregates in Phase II at Stage 2. The time evolution of the dynamic aggregates in Phase II at Stage 2 and at Tred. = 80, 100, and 120 °C was investigated by the best-fitting between Iexp(q) and I2Phase(q) because the best-fitting yields the information on SSHS/inf(q; Rp,av , ϕII) which leads to the Guinier law in the small q limit

lim I2Phase(q) = ΦIInII(1 + α)Δρ2 ⟨Vp2Φ 2(q; R p)⟩

q →∞

= IINP(q)

(22)

Note in eq 22 that ΦIInII(1 + α) = ΦInI + ΦIInII = noverall

(23)

noverall is the average number density of the NPs in the whole two-phase system. Thus, I2Phase(q) approaches IINP(q) in the hiqh-q region of q > 1.2 nm−1, and thereby the best-fit of Iexp(q) with I2Phase(q) or IINP(q) in the high-q region justifies the evaluation of ϕoverall from Iexp(q). However, in the low-q region I2Phase(q) depends also on ϕII through SSHS/inf(q; Rp,av ,ϕII) and ϕIIΦ II through the absolute intensity. These two facts solve the dilemma brought about by the single-phase model, as encountered in Figure 2d.54 The average scattering profile from the two-phase system in Stage 4 is obtained by replacing ISHS/inf(q; ϕII) in eq 19 with ISHS/cluster(q) given by eq 15. This is based on the assumption that Phase II gives rise to ISHS/cluster(q) in Stage 4, while Phase I gives rise to IINP(q; ϕI). Thus, I2Phase(q) for Stage 4 is given by I2Phase(q) = ΦIIISHS/cluster(q) + ΦIIINP(q; ϕI)

(24)

5.2. Reduction-Induced Self-Assembly of NPs as a Function of Time and Temperature. Our time-resolved SAXS experiments have revealed that the structural evolution of P2VP/(Pd NP) hybrid in the matrix of BA, in which the NPs continued to form and thereby ϕoverall in the system also kept increasing with time by the in-situ reduction of Pd(acac)2 using BA as the reduction agent, proceeded through the four stages with the following characteristics (see Figure 5): 1. Stage 1 observed at Tred. = 80 and 100 °C, where the NPs were uniformly distributed in a single-phase state with negligible interparticle interactions due to the low particle concentration (ϕoverall < 3 × 10−4 as observed in Figures 4b and 7) as schematically presented in Figure 5a. Iexp(q) at this stage could be fitted by IINP(q) as found in Figures 2b and 6b. 2. As the particle concentration further increased with time such that ϕoverall > 3 × 10−4, the system underwent the phase separation into Phase I poor in the particles and Phase II rich in the particles, as schematically presented in Figure 5b, which is triggered by the SHS type interparticle attractive interaction and by the increased ϕoverall. The spatial organization of the NPs in Phase I may be the same as that in Stage 1 throughout the whole stages. Hence, the scattering profiles from Phase I may be given by IINP(q) irrespective of time. On the other hand, the spatial organization of the NPs in Phase II further altered with time and hence with increasing the particle concentration, which is classified into Stages 2−4 in the order of increasing time, as follows. 3. Stage 2, where the SAXS profiles in Phase II were described adequately by ISHS/inf(q), signifying that the NPs experienced the SHS interaction with the potential depth u0 ∼ −2.4kBT. Judging from the reported theoretical results shown in Figure 2 of ref 47 and Figure 5 of ref 40, (i) the particle volume fraction ϕII is anticipated to be much larger than ϕI and (ii) Φ II is anticipated to be much smaller than Φ I. These two scenarios were verified by the values of the volume fractions obtained from fitting Iexp(q) at the Stage 2 with

⎛ 1 ⎞ SSHS/inf (q) = SSHS/inf (q = 0) exp⎜ − R g,agg 2q2⎟ ⎝ 3 ⎠

(25)

where Rg,agg is the radius of gyration of the dynamic aggregates which grows with time or ϕII. SSHS/inf(q=0) gives information L

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Figure 9. (a) Representative structure factors SSHS/inf(q) in the low q region in Stage 2 labeled by (1) to (3) at Tred. = 100 °C obtained at t = 300, 400, and 500 s, respectively, which were obtained by the best fit of Iexp(q) at the respective times with I2Phase(q). The broken curves (1G), (2G), and (3G) designate the Guinier approximation given by eq 25 for the SSHS/inf(q) at the respective times. (b) Rg,agg and SSHS/inf(q=0) as a function of time at Tred. = 80, 100, and 120 °C. (c) Master curves of for Rg,agg and SSHS/inf(q=0) obtained with the time−temperature superposition principles at the reference temperature Tr,red. = 100 °C. The temperature dependence of the shift factor aT(T) is shown in the inset to part c.

on the thermal fluctuations of the number of the NPs, Np, in Phase II SSHS/inf (q = 0) = (⟨Np2⟩ − ⟨Np⟩2 )/⟨Np⟩

(26)

Figure 9 presents (a) the representative structure factors SSHS/inf(q) labeled by (1) to (3) at Tred. = 100 °C obtained at t = 300, 400, and 500 s, respectively, and their Guinier approximation shown by the broken curves marked with (1G), (2G), and (3G), respectively, and (b) the time evolution of Rg,agg(t) and SSHS/inf(q = 0; t) at Tred. = 80, 100, and 120 °C estimated from the Guinier approximation. Part b reveals that at the given Tred., Rg,agg(t) and SSHS/inf(q = 0; t) increase with time; the higher the Tred. is, the faster the growth of Rg,agg(t) and the increase of SSHS/inf(q = 0; t) are. The growth of the aggregates is anticipated to be primarily controlled by the increase of ϕII(t) with t. The time evolutions of Rg,agg(t) and SSHS/inf(q = 0; t) at these temperatures are fallen into the respective master curves only by horizontal shifts as shown in part c; the temperature-dependent shift factor aT as a reference temperature selected at Tred. = 100 °C is shown in the inset to part c. The applicability of the time−temperature superposition principle suggests that the growth of the aggregates in Stage 2 is a diffusion-controlled process. 5.4. Window of Stages 1−4. Figure 10 summarizes the window of each stage in terms of the overall volume fraction of the NPs at the various reduction temperatures. Interestingly, the windows for Stages 1 and 2 with respect to ϕoverall were almost independent of the reduction temperature. In general, the structural development at Stages 3 and 4 occurred at ϕoverall > 4.7 × 10−4. Stage 3 was accessed at 100 °C, but it could not

Figure 10. Windows of the four stages in terms of the overall volume fraction of the NPs: Stage 1 (□), Stage (2) (○), Stage 3 (◇), and Stage 4 (△) at various reduction temperatures.

be observed at 120 and 140 °C where the structural development entered Stage 4 directly from Stage 2. It is likely that the characteristic structural development at Stages 3 and 4 took place almost concurrently at this temperature due to a very rapid particle aggregation and organization. Since the hierarchical structure formed at Stage 4 gave rise to stronger scattering intensity in the low-q region relative to that associated with Stage 3, Stage 3 was not discernible clearly based on the time-resolved SAXS experiments conducted in this work. According to Figure 10, the windows of the four M

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The particle aggregation at 100 and 120 °C was able to proceed up to Stages 3 and 4, respectively, whereas that at 80 °C was limited only to Stage 2. We propose that the drastic promotion of the growth rate of the NP assembly with a small increase of temperature was due largely to the enhanced diffusivity of the particles. The classical Stokes−Einstein equation predicts the diffusion coefficient of a particle as D = kBT/(6πηmR) with ηm being the medium viscosity; because ηm varies with temperature sensitively through an exponential function, a small increase of temperature may result in a large increase of the particle diffusivity, which in turn leads to a much more rapid phase separation and aggregation of the NPs.

stages in terms of ϕoverall are as follows. Stage 1: ϕoverall < ca. 3 × 10−4; Stage 2: 3 × 10−4 < ϕoverall < 4.7 × 10−4; Stages 3/4: ϕoverall ≥ 4.7 × 10−4. The phase separation was found to take place after Stage 1, irrespective of the reduction temperature. The effective particle−particle attractive force in the matrix of the polymer solution which drives the phase separation is attributed to a balance of various interactions involving particle−particle, particle−polymer, and polymer−polymer interactions in the solvent medium (BA) such as (i) the van der Waals interaction between the NPs, (ii) the depletion interaction between the NPs and P2VP chains which rejects free P2VP chains in a narrow interstitial space between the NPs due to the conformational entropy loss of the free P2VP chains,41 (iii) the bridging and tele-bridging of the NPs by the polymer chains,40 and (iv) entropic repulsion of P2VP chains adhered or associated with the NPs. The interactions (i) to (iii) give effective attractive interactions between the NPs, while the interaction (iv) gives an effective repulsive interaction between the NPs and thereby the steric stabilization of the NPs dispersion in a single phase. The theoretical works of Schweizer and co-workers have calculated the phase diagrams of polymer/NP hybrid by considering the monomer−particle, monomer−monomer, and particle−particle interactions.40 At relatively weak monomer− particle attraction, the theory predicts an entropic depletioninduced phase separation where the NPs within the particlerich phase are in close contact with each other due to the complete rejection of the polymer chains out of the interstitial regions between the neighboring particles. When the monomer−particle attraction is sufficiently strong, the hybrid becomes a single-phase system in which the particles are sterically stabilized by the formation of a thin stable bound polymer layers around them, which in turn gives rise to a net entropic repulsion between the particles. The hybrid undergoes phase separation again at an even larger monomer−particle attraction strength, as the particles are locally bridged by the polymers.40 Because of the monomers or polymer chains intervening between the NPs, the local volume fraction of the particles in the particle-rich phase (Phase II) in this regime is lower than that associated with the closely packed spheres (e.g., 0.66 for random closely packed spheres) found in the depletion-induced aggregation. We note that in order of the increasing monomer−particle interaction strength as described above, the level (marked by “L” in Figure 3b relative to the phase diagram) of the effective particle−particle attractive potential of mean force first shifts upward to reach above the critical point where a single phase is attained and then shifts downward again below the critical point. For the hybrid system studied here, the volume fraction of the particles forming the static clusters was 0.4, which strongly suggests that the phase separation observed here was driven mainly by the enthalpic effect mediated by the strong monomer−particle attraction.40 Such a strong attraction may originate from the coordination bonding between Pd and the pyridine moiety in P2VP. The theory also predicts that the hybrid is able to undergo phase separation at a very low overall particle concentration in both the entropically (depletion) driven and enthapically driven regimes, which is consistent with our finding here. The theoretical prediction40,47 of Φ I ≫ Φ II or ϕI ≪ ϕII also is consistent with our experimental results (see Table S6).

6. CONCLUSIONS The chemical-reduction-induced temporal development of the dispersion morphology of the Pd NPs in their hybrids with P2VP and BA has been investigated by means of time-resolved SAXS at the four reduction temperatures. The quantitative analysis of the observed SAXS profiles by means of plausible scattering models revealed that the structural evolution was constituted of the four Stages 1−4 in order of increasing ϕoverall. The SAXS profiles at Stage 1 were fitted well by the independent scattering of spherical particles IINP(q), indicating that the NPs of the low concentration were uniformly distributed in the hybrid without experiencing the interparticle interaction (Figure 5a). As ϕoverall reached above 3 × 10−4, the scattering profiles showed the deviations from the prediction of IINP(q) and thereby the structural evolution proceeded to Stage 2 where the hybrid underwent the phase separation into the two phases. In this case, the overall intensity profiles could be fitted well by I2Phase(q) for the two-phase model in which the scattering from the NP-rich phase (Phase II) was given by ISHS/inf(q) with the attractive potential depth u0 ∼ −2.4kBT and the scattering from the NP-poor phase (Phase I) was given by IINP(q) (Figures 5b and 5c). The structural evolution in Phase II advanced to Stage 3 as the overall particle volume fraction reached 4.7 × 10−4; in this case, the SAXS profiles were characterized by the superposition of I2Phase(q) and an excess intensity in the low-q region. We proposed that the dynamic aggregates and/or the static clusters formed via the SHS interaction may associate to create the higher-order structures in Phase II, thereby creating the larger-scale heterogeneities (Figures 5b and 5d). The structure developed at Stage 3 may serve as the precursor for directing the transformation of the dynamic aggregates into the static clusters which further assembled to generate the large-scale mass-fractal structure at Stage 4 (Figures 5b and 5e). Considering that the volume fraction of the NPs within the static clusters was lower than that associated with closely packed sphere, we proposed that these particles were bridged by the polymers and the phase separation was driven enthapically by the net particle−particle attractive interactions mediated through the strong polymer− particle attraction. Our findings, which are fully supported by the SAXS results, shall offer deep insight into the mechanism of the hierarchical structure formation in the polymer hybrid with the metallic NPs generated by the in-situ chemical reduction. It is of interest and crucial to explore the universality of the established mechanism using the approach developed here. Moreover, it is striking to note that the phase separation observed in this work occurred at such a low concentration of the NPs as ϕoverall ≥ 3 × 10−4. This phase separation at the low concentration seems to originate from the skewed phase diagram which depends on N

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(4) Crossland, E. J. W.; Noel, N.; Sivaram, V.; Leijtens, T.; Alexander-Webber, j. A.; Snaith, H. J. Mesoporous TiO2 single crystals delivering enhanced mobility and optoelectronic device performance. Nature 2013, 495, 215−219. (5) Grossiord, N.; Loos, J.; Regev, O.; Koning, C. E. Toolbox for Dispersing Carbon Nanotubes into Polymers To Get Conductive Nanocomposites. Chem. Mater. 2006, 18, 1089−1099. (6) Li, H.; John, J. V.; Byeon, S. J.; Heo, M. S.; Sung, J. H.; Kim, K.H.; Kim, I. Controlled accommodation of metal nanostructures within the matrices of polymer architectures through solution-based synthetic strategies. Prog. Polym. Sci. 2014, 39, 1878−1907. (7) Paul, D. R.; Robeson, L. M. Polymer nanotechnology: Nanocomposites. Polymer 2008, 49, 3187−3204. (8) Chung, C.-H.; Song, T.-B.; Bob, B.; Zhu, R.; Yang, Y. Solutionprocessed flexible transparent conductors composed of silver nanowire networks embedded in indium tin oxide nanoparticle matrices. Nano Res. 2012, 5, 805−814. (9) Ginzburg, V. V. Polymer-Grafted Nanoparticles in Polymer Melts: Modeling Using the Combined SCFT−DFT Approach. Macromolecules 2013, 46, 9798−9805. (10) Sandi, G.; Kizilel, R.; Carrado, K. A.; Fernandez-Saavedra, R.; Castagnola, N. Effect of the silica precursor on the conductivity of hectorite-derived polymer nanocomposites. Electrochim. Acta 2005, 50, 3891−3896. (11) Wang, D.; Kou, R.; Choi, D.; Yang, Z.; Nie, Z.; Li, J.; Saraf, L. V.; Hu, D.; Zhang, J.; Graff, G. L.; Liu, J.; Pope, M. A.; Aksay, I. A. Ternary Self-Assembly of Ordered Metal Oxide−Graphene Nanocomposites for Electrochemical Energy Storage. ACS Nano 2010, 4, 1587−1595. (12) Huang, J.; Yin, Z.; Zheng, Q. Applications of ZnO in organic and hybrid solar cells. Energy Environ. Sci. 2011, 4, 3861−3877. (13) Kyrylyuk, A. V.; Hermant, M. C.; Schilling, T.; Klumperman, B.; Koning, C. E.; van der Schoot, P. Controlling electrical percolation in multicomponent carbon nanotube dispersions. Nat. Nanotechnol. 2011, 6, 364−369. (14) Tsutsumi, K.; Funaki, Y.; Hirokawa, Y.; Hashimoto, T. Selective Incorporation of Palladium Nanoparticles into Microphase-Separated Domains of Poly(2-vinylpyridine)-block-polyisoprene. Langmuir 1999, 15, 5200−5203. (15) Okumura, A.; Tsutsumi, K.; Hashimoto, T. Nanohybrids of Metal Nanoparticles and Block Copolymers. Control of Spatial Distribution of the Nanoparticles in Microdomain Space. Polym. J. 2000, 32, 520−523. (16) Chevigny, C.; Dalmas, F.; Di Cola, E.; Gigmes, D.; Bertin, D.; Boue, F.; Jestin, J. Polymer-Grafted-Nanoparticles Nanocomposites: Dispersion, Grafted Chain Conformation, and Rheological Behavior. Macromolecules 2011, 44, 122−133. (17) Wu, J. Density functional theory for chemical engineering: From capillarity to soft materials. AIChE J. 2006, 52, 1169−1193. (18) Hashimoto, T.; Okumura, A.; Tanabe, D. Visualization of Isolated Poly(2-vinylpyridine)-block-Polyisoprene Chains Adhered to Isolated Palladium Nanoparticles. Macromolecules 2003, 36, 7324− 7330. (19) Fornes, T. D.; Yoon, P. J.; Keskkula, H.; Paul, D. R. Nylon 6 nanocomposites: the effect of matrix molecular weight. Polymer 2001, 42, 09929−09940. (20) Jiang, J.; Oberdörster, G.; Biswas, P. Characterization of size, surface charge, and agglomeration state of nanoparticle dispersions for toxicological studies. J. Nanopart. Res. 2009, 11, 77−89. (21) Sanchez, C.; Boissière, C.; Grosso, D.; Laberty, C.; Nicole, L. Design, Synthesis, and Properties of Inorganic and Hybrid Thin Films Having Periodically Organized Nanoporosity. Chem. Mater. 2008, 20, 682−737. (22) Tanaka, H.; Koizumi, S.; Hashimoto, T.; Itoh, H.; Satoh, M.; Naka, K.; Chujo, Y. Combined in Situ and Time-Resolved SANS and SAXS Studies of Chemical Reactions at Specific Sites and SelfAssembling Processes of Reaction Products: Reduction of Palladium Ions in Self-Assembled Polyamidoamine Dendrimers as a Template. Macromolecules 2007, 40, 4327−4337.

the interactions between NPs and NPs, NPs and polymer segments, and polymer segments and polymer segments in the medium of benzyl alcohol. The asymmetry in the sizes of the interacting elements, i.e., the NPs and the segments, gives also a crucial effect on the phase diagram and thereby the phase separation. The results offer important impacts on the investigation of the phase separation mechanism and process of the colloids in aqueous or nonaqueous media or in polymer solutions and melts which have been hardly explored so far. Further studies to elucidate the underlying phase separation mechanism (e.g., spinodal decomposition or nucleation and growth) using the techniques capable of probing the structures at the larger length scales, such as small-angle light scattering (SALS), ultrasmall-angle X-ray scattering (USAXS), or ultrasmall-angle neutron scattering (USANS), are also of great significance.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b01531. Morphological parameters associated with the hierarchical structure formed at 120 and 140 °C obtained by fitting the observed SAXS profiles in Figure 1a with the scattering function in eq 16; values of the structural parameters obtained from the fits at 80 °C; values of the structural parameters obtained from the fits at 100 °C; values of the structural parameters obtained from the fits at 120 °C; values of the structural parameters obtained from the fits at 140 °C; representative values of the volume fraction of the particles in Phase I (ϕI) and Phase II (ϕII) and the phase volume fraction in Phase I (Φ I) and Phase II (Φ II) in Stage 2 (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (H.L.C.). *E-mail: [email protected] (T.H.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the support of the National Science Council, Taiwan, under Grant MOST 104-2633-M-007-001, and the Frontier Center of Fundamental and Applied Sciences of Matters of the National Tsing Hua University. We also thank to Professor U-Ser Jeng and Dr. Chun-Jen Su at the National Synchrotron Radiation Research Center for supporting the Xray scattering experiments at beamline BL23A1 and Dr. ChunYu Chen for providing the programs for model fittings.

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REFERENCES

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I2Phase(q → 0) = ΦIInIIΔρ2 ⟨Vp2⟩SSHS/inf (q → 0; R p,av , ϕII) + ΦInIΔρ2 ⟨Vp2⟩

(27)

which gives the green profile [the profile (4)] at q → 0 in Figure 2d for ϕII = 2 × 10−2. If we assume the single phase model with the same particle volume fraction, the scattered intensity at q → 0 is given by setting Φ II = 1 and Φ I = 0 in eq 27, so that

ISHS/inf (q → 0) = nIIΔρ2 ⟨Vp2⟩SSHS/inf (q → 0; R p,av , ϕII)

(28)

This intensity gives the blue profile [profile (3)] shown in Figure 2d at q → 0. If Φ IInII ≫ Φ InI, ISHS/inf(q → 0) is higher than I2Phase(q → 0) by the factor of 1/ΦII. This approximately accounts for the proportionality constant C ≅ Φ II ≅ ϕoverall/ϕII = 0.02 from eq 18 and thereby the discrepancy in the absolute intensity between the blue profile (3) and the green profile (4) shown in Figure 2d. The exact value of C is given from eqs 27 and 28 by C≡

ΦIInIISSHS/inf (q → 0; R p,av , ϕII) + ΦInI I2Phase(q → 0) = ISHS/inf (q → 0) nIISSHS/inf (q → 0; R p,av , ϕII)

(29) Equation 29 leads to C = 2.36 × 10 . This value is higher than C ≅ 2 × 10−2 due to the contribution of the second term in the numerator of eq 29. −2

P

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