Self-assembly is ubiquitous in various fields, and self-assembled architectures are crucial for the functionality of materials. Thus, controlling the self-assembled architecture emerges as an important research subject in the chemistry and physics of materials. Fascinatingly ordered and well-defined structures from anisotropic particles have been created by optimizing the anisotropic interactions among particles, thereby paving the way for potential applications, including metamaterials and catalysts. Conversely, the self-assembly of isotropic particles (e.g., spherical silica nanoparticles) normally just results in disordered or isotropic aggregations. Constructing ordered and well-defined structures from isotropic particles is remains highly challenging [1,2,3,4,5].

Here, we focus on anisotropic self-assembly from isotropic particles, which was a phenomenon discovered by Wang et al. in 2010 [6]. The addition of alcohol to aqueous dispersions of silica nanoparticles (SNPs) under controlled pH conditions leads to the self-assembly of SNPs into a unidimensional polymer chain-like structure, although isotropic interactions prevail amongst SNPs. The reason behind the linear connection of SNPs can be interpreted as follows. Initially, two SNPs coalesce into a dimer, as the electrostatic repulsion between the SNPs becomes slightly weakened by the alcohol. When another SNP approaches the dimer around the joint point, electrostatic interactions work from the two SNPs of the dimer onto the approaching SNP. Meanwhile, when a SNP approaches the edge of the dimer, electrostatic interactions predominantly work between the SNP of the edge (i.e., a single SNP) and the approaching SNP. Consequently, SNPs are likely to adhere to the edge of the dimer, thereby preferentially forming a chain-like assembly. However, the kinetic process underlying this formation has remained unclear.

In this study, we used time-resolved small-angle X-ray scattering (SAXS), a technique that facilitates in situ monitoring of kinetic processes, obviating the need for drying, freezing, and staining of the samples. With this methodology, we address the formation mechanisms underpinning the chain-like self-assembly of the SNPs.

First, we analyzed the structure in both the initial state and a state subsequent to a sufficient duration. Fig. 1 shows the SAXS profiles at time point (t) of 0 (SNP dispersion; alcohol was not added). Fig. 1 also shows the SAXS profile at t = 730 min after the addition of ethanol and L-arginine (a buffer agent) to trigger the self-assembly of the SNPs. The silica concentration was 0.1 wt% in both profiles. Details of the experimental procedures are provided in the Supplementary Material. The SAXS profile at the initial state exhibited a plateau in the low-q regime and a steep slope and oscillations in the higher-q regime, which are typically observed in spherical nanoparticles [7]. This profile is modeled by a form factor of sphere expressed as follows:

$${P}_{{{{{{\rm{sph}}}}}}}\left(q\right)={9\left[\frac{\sin \left({qR}\right)-{qR}\cos ({qR})}{{({qR})}^{3}}\right]}^{2}$$
(1)

where R stands for the radius of the sphere. By considering the dispersity of R, assuming a Gaussian distribution with a standard deviation of 0.8 nm, the experimentally obtained data can be fitted with R = 7.5 ± 0.1 nm, represented as the solid black curve in Fig. 1.

Fig. 1
figure 1

SAXS profiles at the initial state (t = 0) and a state with a sufficient duration (t = 730 min), where the silica concentration is 0.1 wt%. The blue and gray symbols represent the data at t = 0 and 730 min, respectively. The black curve indicates the sphere model fit the initial state. The red curve shows the cylindrical wormlike chain model fitted to the low-q region in the final state. The green curve shows the model in which spheres are discretely aligned on a straight line fitted to the high-q region in the final state

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The SAXS profile at t = 730 min exhibits I(q) ~ q–1.5 in the low-q region. Gaussian chains in an unperturbed state are known to adhere to I(q) ~ q–2 while straight cylinders obey I(q) ~ q–1. Thus, the intermediate exponent (1.5) suggests that the morphology of the self-assembled SNPs may be regarded as a semiflexible chain. Indeed, the experimental data in the low-q region can be fitted by the model of a wormlike cylinder [8]. The Kuhn segment length (b) could be determined to be 60 ± 5 nm, as the slope at low-q depends on b. Meanwhile, the model of a wormlike cylinder deviates in the high-q region, indicating that the local structure is different from the model. That is, the wormlike cylinder model does not consider the local structure (i.e., constrictions), while the actual structure should be equivalent to spheres (SNPs) that are connected similar to strings of pearls. In fact, the high-q region of the SAXS profile can be fitted by a model form factor [Pds(q)] in which spheres with a radius of 7.5 nm align discretely on a straight trajectory (i.e., each sphere is touched) (Fig. 1); the equation is given by

$${P}_{{{{{{\rm{ds}}}}}}}\left(q\right)=\left\{\frac{1}{x}+\frac{2}{{x}^{2}}{\sum }_{m=1}^{x-1}\left[(x-m)\frac{\sin (2{mqR})}{2{mqR}}\right]\right\}{P}_{{{{{{\rm{sph}}}}}}}\left(q\right)$$
(2)

However, this model deviates in the low-q region because it does not take into account the curvature or flexibility of the chain contour. In addition, we calculated a model of summation of the wormlike cylinder [8] and sphere (i.e., we supposed unreacted SNPs were present); however, this model failed to replicate the experimentally obtained data (Fig. S1).

Considering these findings, the structure in the final state should be where the SNPs are linearly assembled with the semiflexibility at b = 60 ± 5 nm. The electrostatic repulsion among each segment in the chain is presumed to lead to this semiflexibility. We believe that the model of the discretely aligned spheres on a trajectory of a wormlike chain should potentially fit our experimentally obtained data of the entire q-range. However, the calculation of this form factor is considerably intricate, and thus, was not performed in this study. Please note that the touched-bead model by Burchard and Kajiwara [9] is not a precisely discrete model and is only applicable when the spheres are adequately small.

The chain-like structure was also confirmed by transmission electron microscopy (TEM), as shown in Fig. S2. The cross-sectional diameter was estimated to be 18 ± 4 nm, agreeing with the findings from the SAXS data. However, due to the tendency for aggregation during the drying process on the copper grid for the TEM observations, we did not compare the flexibility obtained from the SAXS results with that from the TEM results.

Figure 2a shows the results from the time-resolved SAXS during the self-assembly into a chain-like structure. The measurements started immediately following the addition of ethanol to the aqueous dispersion of the SNPs; here, the ethanol concentration was increased from 6 to 74 wt%. The time point at which ethanol was added was considered zero. As time passes, the intensity increased, and the slope became steeper in the low-q region; this result indicated that the self-assembly process was successfully tracked. Meanwhile, the local structure of the nm-scale did not change, as suggested by the unchanged scattering intensity at q > 0.3 nm–1. Moreover, using the Berry plot (Fig. 2b), the weight-average degree of polymerization or number of SNPs per chain (xw) as well as the z-average radius of gyration (〈S2〉z1/2) were obtained at each time point, as shown in Fig. 3a. Additionally, the double-logarithmic plot of 〈S2〉z1/2 against xw (Fig. 3b) indicated an exponent of 0.68, and this value is characteristic of semiflexible chains.

Fig. 2
figure 2

Representative SAXS profiles during the chain formation as a double-logarithmic plot (a) and Berry plot (b). t indicates the time point after adding ethanol to the aqueous dispersion of SNPs

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Fig. 3
figure 3

a Weight-average degree of polymerization (number of the silica nanoparticles per self-assembly; xw) and z-average radius of gyration (〈S2〉z1/2) as a function of time. b Double-logarithmic plot of 〈S2〉z1/2 against xw. The solid curves in (a) represent the kinetic model (see the text), and the solid curve in (b) represents the wormlike cylinder model

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Throughout the process of chain formation and elongation, one may deduce that the edges of two chains collide into one chain, analogous to the scenario encountered in the step-growth polymerization of bifunctional monomers. That is, each SNP has a pair of nearly antipodal adhesive sites, akin to the functional groups of monomers. This reaction is expressed as [10]

$$\frac{{{{{{\rm{d}}}}}}C}{{{{{{\rm{d}}}}}}t}=-\frac{1}{2}k{C}^{2}$$
(3)

with C denoting the molar concentration of the adhesive sites and k representing the rate coefficient. If k is independent of x, this model corresponds to Flory’s classical model [11, 12]. However, k should inherently depend on x, as k is proportional to the diffusion constant (D) [13, 14]. According to the literature, a wormlike cylinder [15] (or a short wormlike cylinder in which both edges are capped by hemispheres [16]) follows D ~ x–α, where α = 0.53 when b = 60 nm (Fig. S3). Thus, k may be expressed as k = k0x–α with k0 denoting a constant, and Eq. 3 is reformulated as follows [5]:

$$\frac{{{{{{\rm{d}}}}}}C}{{{{{{\rm{d}}}}}}t}=-{\frac{1}{2}k}_{0}\frac{{C}^{\alpha +2}}{{C}_{0}^{\alpha }}$$
(4)

Here, C0 stands for the initial molar concentration of the adhesive sites. Therefore, xw can be expressed as a function of time:

$${x}_{{{{{{\rm{w}}}}}}}\left(t\right)=2{\left[1+\frac{1}{2}(\alpha +1){C}_{0}{k}_{0}t\right]}^{\frac{1}{\alpha +1}}$$
(5)

The experimentally obtained data indicated that immediately after ethanol was added to the SNP dispersion (t = 2.2 min), the xw(t) value was 2, signifying the immediate formation of the dimers within 2.2 min because the unimer can approach other unimers from any direction. For this reason, during the derivation of Eq. 5, the constant was adjusted to ensure that xw(0) attained a value of 2 (the Supplementary Material). This model (with k0 = 0.12/C0 mol g–1 min–1) aligned with the experimentally obtained data, represented as the solid black curves in Fig. 3a. Notably, the adjustable parameter in this model is only k0, and this model quantitatively reproduced the time evolution of the degree of polymerization, xw(t) ~ t0.63. If it were Flory’s classical model [11] of α = 0, then xw(t) would follow xw(t) ~ t. The xw and 〈S2〉z1/2 values fluctuated, particularly at approximately t = 30 min. This was likely caused by the concentration fluctuation since the sample dispersion was not homogenized/stirred during the time-resolved SAXS measurements.

Furthermore, the relationship between xw and 〈S2〉 was fitted by the model of a touched-bead worm-like chain (Fig. 3b), where b = 60 ± 5 nm. The equation is expressed as follows [17, 18]:

$$\left\langle {S}^{2}\right\rangle =\frac{{Lb}}{6}-\frac{{b}^{2}}{4}+\frac{{b}^{3}}{4L}-\frac{{b}^{4}}{8{L}^{2}}\left\{1-\exp \left(-\frac{2L}{b}\right)\right\}+\frac{{R}^{2}}{2}$$
(6)

With L = 2Rx. By combining Eqs. 5 and 6, the time evolution of 〈S2〉z1/2 can also be reproduced (Fig. 3a). Therefore, the relationships among xw, 〈S2〉, and t obtained by the time-resolved SAXS measurements were in concordance with the combination of the kinetic model of step–growth polymerization and the structural model of a wormlike chain.

In conclusion, we successfully monitored the formation process of chain-like self-assembly of SNPs with a diameter of c.a. 15 nm through time-resolved SAXS. The data obtained aligned well with the kinetic model, wherein the SNP was regarded as a monomer undergoing a self-assembly process analogous to step-growth polymerization. This insight is beneficial for tuning the average length of the chains. According to a previous study [6], this chain-like structure can be stabilized/hardened by adding tetraethyl orthosilicate, thereby suggesting its potential as a novel class of inorganic nanomaterials. Our future work will concentrate on the mechanical properties of the polymer nanocomposites by employing chain-like silica as a filler.