Universal progression of structure and dynamics in colloidal nanocrystal gels during salt-accelerated aging

Abstract

Controlling the structure and function of colloidal gels requires a detailed understanding of how the various components govern network formation and aging. In particular, molecular additives like salts are widely used to tune interparticle interactions, yet their influence on gelation pathways in complex systems such as colloidal nanocrystal gels remains inadequately understood. Here, we investigate how noncoordinating salts modulate the evolution of gels formed using chemically linked tin-doped indium oxide nanocrystals. Through combined structural, dynamic, and kinetic analyses, we demonstrate that increasing salt concentration accelerates gelation. When rescaled by salt-dependent characteristic times, the evolution collapses onto universal trajectories, revealing a time-salt superposition principle. The universality extends across length scales, suggesting a consistent salt-dependent mechanism that controls both local structuring and macroscopic network formation. This observed salt modulation of structure and dynamics provides a predictive basis for controlling the kinetics of nonequilibrium nanocrystal gel assembly, enhancing the rational design of functional nanomaterials with tunable properties.

Salt modulates gelation in colloidal nanocrystal networks, revealing a universal pathway across time and length scales.

INTRODUCTION

Because of the resonant interaction of their free charge carriers with light, plasmonic colloidal semiconductor nanocrystals (NCs) have emerged as versatile building blocks for optically active assemblies (1, 2). While their individual properties are size and composition dependent, assemblies of NCs can exhibit emergent, structure-dependent behaviors, such as coupling of their localized surface plasmon resonances (LSPRs). The opportunity for tailoring optical responses has motivated extensive research into post-synthetic assembly techniques (3–8). One promising approach is gelation, whereby NCs form space-spanning percolated networks that exhibit solid-like mechanical properties. These NC gels offer precise and reproducible structural control, making them highly attractive as tunable and responsive optical materials (9). Despite advances in assembly techniques, NC gels are fundamentally out-of-equilibrium systems, raising open questions about their formation and evolution (10–12). As kinetically arrested materials, they age through local and cooperative rearrangements driven by thermal fluctuations and residual stresses (13, 14), resulting in a slow evolution of viscoelastic properties across broad time and length scales. This physical aging is strongly influenced by sample history and preparation protocols (15).

Recent studies of tin-doped indium oxide (ITO) NCs cross-linked via dynamic covalent hydrazone bonds show that the structural evolution of the gels is modulated by salt concentration (16). In particular, the structure factor, obtained via small-angle x-ray scattering (SAXS), after a given waiting time, evolves systematically with the amount of tetrabutylammonium hexafluorophosphate (TBAPF₆) added to the gelation solution. Increases in the primary peak magnitude and deepening of the correlation well suggest a salt-dependent coarsening behavior consistent with structural evolution toward arrest (17). The structural evolution and optical modeling of the gels supports a correspondence between salt addition and structural aging in time. A time-salt superposition relationship, reminiscent of phenomena observed in other spontaneously aging soft matter systems where salt modulates structural evolution (18–21), is thus hypothesized to describe the evolution of these colloidal NC assemblies.

However, comprehensive studies addressing how salt affects the microscopic dynamics and structure of colloidal NC gels remain limited. This gap arises from several challenges. First, early-stage cluster formation occurs at the nanometer scale—a length scale challenging to probe using conventional approaches such as rheology or optical microscopy. Second, the multicomponent nature of NC dispersions gives rise to complex fluid behavior, characterized by multiple interaction types and length scales that generate micro- to mesoscale structural heterogeneity, making the interpretation of experimental data nontrivial (22–25). Third, most macroscopic probes cannot resolve non-ergodic behavior, where the time-averaged dynamics of a given NC or structural domain deviate from the ensemble average (26), thereby requiring experimental data to be both ensemble-averaged and sufficiently time-resolved to capture the detailed evolution of structure and dynamics (27). Yet, a detailed understanding of how salt influences the microscopic mechanisms governing structure and dynamics during colloidal NC gelation is essential both from a fundamental physics perspective and for the rational design of functional gel-based nanomaterials.

Herein, we ask: How does salt addition influence gelation and aging in chemically linked colloidal NC gels? To address this question, we systematically trace the entire gelation pathway in situ—from stable dispersion to aged gel—at varying salt concentrations. By leveraging hydrazone chemistry as a linking scheme, we establish a controlled model NC gel system and monitor its evolution over broad time and length scales using SAXS, optical absorption spectroscopy, and x-ray photon correlation spectroscopy (XPCS), the latter of which probes NC dynamics within evolving fractal networks.

We find that salt addition accelerates network formation and modulates the timescales of structural and dynamic evolution, yet the final steady-state structure remains salt independent. When rescaled by salt-specific characteristic times, structural and dynamic data across salt concentrations collapse onto master curves, thus indicating a time-salt superposition phenomenon and revealing a universal kinetic pathway of gelation. Further analysis of the dynamic data reveals that this universality spans the accessible length scales, indicating a consistent salt-dependent mechanism that governs both local structuring and macroscopic network formation. Measurements of the effective pairwise interactions without chemical linking, and simulations of the gelation kinetics, identify electrostatic screening as a key driver of faster gelation. Furthermore, kinetic nuclear magnetic resonance (NMR) analyses of the chemical bonding underpinning network formation also confirm that reaction rates increase with salt concentration. Analysis of the microstructural evolution during gelation, together with insights from the kinetics of chemical linking, indicates that gelation kinetics are limited by formation of bonds between NCs. Consequently, the observed superposition implies that in colloidal systems governed by this mechanism—where the barrier that controls the gelation kinetics, whether electrostatic or chemical in origin, is short ranged—a universal superposition of structure is expected once the kinetic effects of salt reducing the barrier are accounted for by rescaling time. Altogether, this work demonstrates that additives such as salt modulate gelation through molecular- and nanoscale mechanisms, providing guidelines for fabricating colloidal networks with targeted structural and dynamic properties.

RESULTS

Preparation of NC building blocks and gelation

Linker-mediated NC gels were fabricated by first synthesizing oleate-capped indium tin oxide (OL-ITO) NCs using a modified slow-injection method (28), yielding stable and uniform NCs with a narrow size distribution. Bright-field scanning transmission electron microscopy (STEM; Fig. 1B) and SAXS (Fig. 1D) analyses confirmed their spherical morphology and low size dispersity. The NC form factor $P\left(q\right)$, obtained from SAXS measurements with the scattering wavenumber $q=4\mathrm{\pi }\text{sin}(\mathrm{\theta }/2)/\mathrm{\lambda }$ (where $\mathrm{\theta }$ is the scattering angle and $\mathrm{\lambda }$ is the x-ray wavelength), was accurately captured by an analytical spherical model with a core diameter of ${\mathrm{\sigma }}_{\text{core}}=15.9$ nm and a narrow size distribution (9% SD). The tin doping concentration was 5.2 at %. To prepare the NCs for gelation experiments, the oleate ligands on the ITO NCs were replaced via direct ligand exchange with aldehyde-terminated ligands (AL) (Fig. 1A), which were synthesized using established protocols (fig. S1) (7). These ligands contain an aldehyde terminal group for linking, a polyethylene glycol (PEG)–like solubilizing backbone, and a tri-carboxylate anchoring group for binding to the NC surface. Dispersions of AL-capped ITO NCs (AL-ITO) in N,N-dimethylformamide (DMF) were prepared at various TBAPF₆ concentrations ranging from 0 to 100 mM. Gelation was initiated by introducing a bifunctional molecular linker with four PEG repeat units, hydrazide-PEG₄-hydrazide, at a fixed linker-to-colloid ratio, $\mathrm{\Gamma }$ ($\mathrm{\Gamma }=200$), to the AL-ITO dispersions containing varying salt concentrations. This linker promotes dynamic covalent bonding via hydrazone formation between the aldehyde end groups on the ligands and the hydrazide groups on the linker, driving a time-dependent transformation from a free-flowing dispersion to a viscoelastic gel. Scanning electron microscopy of dried gels revealed a network of interconnected NCs (Fig. 1C). The structure factor $S\left(q\right)$, which characterizes the mesoscale organization of the NCs, was extracted from the total x-ray scattering cross section using the form factor obtained from dilute OL-ITO dispersions in toluene at an NC core volume fraction $\mathrm{\Phi }=0.0005$. Notably, $S\left(q\right)$ profiles of gels aged for 12 months were indistinguishable across all TBAPF₆ concentrations (Fig. 1E), indicating that the final structure of the gel is independent of the initial salt concentration.

Fig. 1. Preparation of AL-ITO NCs and gels.

(A) Schematic illustration of the ligand-exchange process, where OL-ITO NCs are functionalized with ALs to form AL-ITO. The AL-ITO NCs are then linked via the chemical linker hydrazide-PEG₄-hydrazide to form the gel network. STEM image of the ITO NCs (B) before ligand exchange (OL-ITO), and SEM (C) after ligand exchange (AL-ITO) and linker addition, showing gel network formation (scale bars, 100 nm). (D) SAXS form factors $P\left(q\right)$ of dilute (core volume fraction $\mathrm{\Phi }=0.0005$) OL-ITO in toluene and AL-ITO samples in DMF at 0 mM and 100 mM TBAPF₆ concentrations. Spherical model form factor fits $P\left(q\right)$ (red curve) to the data confirm the spherical morphology, uniform size (core diameter ${\mathrm{\sigma }}_{\text{core}}=15.9$ nm), and small size dispersity (9%) of the NCs. (E) Structure factors $S\left(q\right)$ of the aged (12-month-old) gels made from AL-ITO dispersions at an inorganic core volume fraction $\mathrm{\Phi }=0.03$ prepared at TBAPF₆ concentrations of 0, 33, 66, and 100 mM.

Evolution of microstructure

Following the addition of the molecular linker to AL-ITO dispersions at NC core volume fraction $\mathrm{\Phi }=0.03$, SAXS measurements were performed for a series of salt concentrations at various wait times $t_w$ throughout the gelation process, with $t_w$ defined as the time elapsed after linker addition. Structural evolution was characterized via $S\left(q\right)$ analyses (Fig. 2A and figs. S2 to S4). Consistent with previous studies (16, 29), the temporal evolution of $S\left(q\right)$ as a function of $q$ shows a growing signal intensity at the lowest accessible $q$ over time. This increase indicates the development of long length-scale density fluctuations, corresponding to the formation of inhomogeneities associated with network formation or spinodal phase separation. Experimentally, the AL-ITO and linker mixture undergoes visible phase separation over time, evidenced by the emergence of a clear, NC-poor supernatant in the capillaries containing the aged gels. To systematically compare the effects of salt concentration on the gelation mechanism at structurally equivalent states, it is important to define a system-dependent time corresponding to a point at which the system reaches a common structural state during its evolution. Because macroscopic phase separation can lag behind microscopically detectable changes, we estimate this time point based on the low-$q$ behavior of $S\left(q\right)$. Specifically, we extrapolated $S\left(q\right)$ to $q\to 0$ by fitting the low-$q$ region of the scattering profile to the Ornstein-Zernike form (figs. S3 and S4) (30)

$$S\left(q\right)\approx \frac{S\left(0\right)}{1+{\left(q\mathrm{\xi }\right)}^2}$$ (1)

where $\mathrm{\xi }$ is the thermal correlation length representing the spatial scale over which density fluctuations are correlated. Conditions where phase separation occurs can be approximately located by the condition $S\left(0\right)\ge 10$ (31, 32). Therefore, we use the time, $t_{10}$, at which $S\left(0\right)\approx 10$ as a threshold, marking the onset of large-scale structure formation, consistent with spinodal decomposition. Using this threshold, we define a dimensionless timescale, $t_w/t_{10}$, which facilitates direct comparison of gel evolution across salt concentrations. Characteristic times marking the onset of structural changes during gelation have similarly been identified based on thresholds monitored by rheology (33), SAXS (19, 34), or other techniques in previous studies.

Fig. 2. Microstructure evolution observed via SAXS.

(A) $S\left(q\right)$ evolution of AL-ITO NCs containing 100 mM TBAPF₆ after linker addition at a constant linker-to-NC ratio ($\mathrm{\Gamma }=200$) over a 60-hour gelation period. Inset shows the $S\left(q\right)$ at the characteristic time point for the various TBAPF₆ concentrations, defined as the time where $S\left(0\right)\simeq 10$. (B) Characteristic transition time $t_{10}$, defined according to inset of (A), as a function of TBAPF₆ concentration. Correlation length ($\mathrm{\xi }$) and mass fractal dimension ($D_f$) of the heterogeneous domains formed during network formation at varying salt concentrations (C and E) over wait time ($t_w$) and (D and F) over normalized wait time ($t_w/t_{10}$) showing universal progression of microstructure.

We find that $t_{10}$ decreases approximately linearly with increasing salt concentration (Fig. 2B), demonstrating that salt accelerates the gelation rate. Moreover, the $S\left(q\right)$ at $t_{10}$ across different salt concentrations collapse onto a single curve (Fig. 2A, inset), indicating that the mesoscale structure of the gel at this point is independent of the amount of salt added. Correspondingly, $\mathrm{\xi }$, which physically describes the typical size of clusters or correlated domains within the gel network, increases with wait time, reaching larger values more quickly for higher salt concentrations. For all salt concentrations investigated, the increase in $\mathrm{\xi }$ is exponential with time, similar to the evolution of structure in systems controlled by reaction-limited cluster aggregation (RLCA) kinetics (35), where the linking rate governs the aggregation and network formation. The power law behavior of a diverging $S\left(q\right)$ at low $q$ is consistent with a fractal network structure (7, 16), with $I\left(q\right)\sim S\left(q\right)\propto q^{-D_f}$ fits yielding the mass fractal dimension ($D_f$) values (Fig. 2, E and F). Like $\mathrm{\xi }$, the rate of increase in $D_f$ increases with added salt, before eventually plateauing at $D_f\approx 2.2$ at long wait times—a value typical of colloidal gels and RLCA aggregates (Fig. 2, C and D) (8, 16, 35, 36). When normalized by their respective $t_{10}$ values, the time dependencies of both $\mathrm{\xi }$ and $D_f$ collapse across all salt concentrations over the full temporal evolution of the gels, revealing a time-salt superposition consistent with a universal progression of microstructure. This universality of the structural evolution holds regardless of the value of $S\left(0\right)$ used to specify the characteristic time, confirming the robustness of the observation (figs. S5 and S6).

Structural evolution of the NC gel networks also produces temporal optical modulation of plasmonic optical spectra (16, 36). The red shift in the absorption peak is a manifestation of enhanced LSPR-LSPR coupling as NCs become nearer to each other (Fig. 3A and fig. S7) (5, 7, 8, 16, 37, 38). Like the structural parameters from SAXS, we find that the time-dependent optical extinction spectra of the gels also depend on the salt concentration (16). The result for [TBAPF₆] = 0 mM is not shown, as the gelation kinetics were too slow for gelation signatures to be observed optically.

Fig. 3. Microstructure evolution observed via optical spectroscopy.

(A) Normalized absorbance spectra recorded over time (0 to 270 hours) for an evolving NC gel ($\mathrm{\Phi }=0.003$) containing 100 mM TBAPF₆ and $\mathrm{\Gamma }=200$, showing a red shift in the spectra during gelation. Inset shows the normalized absorption spectra at the respective half-point of the LSPR peak position versus wait time $t_w$ for the various TBAPF₆ concentrations. For the optical studies, $\mathrm{\Phi }$ was reduced by an order of magnitude relative to the SAXS studies to maintain absorbance within detectable limits, which likely contributes to the slower time evolution of the gels. (B) Half-point ($t_{0.5}$) of the progression of the LSPR peak position as a function of salt concentration, illustrating the role of the salt in accelerating the gelation process. The result for $t_{0.5}$ at [TBAPF₆] = 0 mM is not shown, as the gelation kinetics were too slow for gelation signatures to be observed optically. (C) LSPR peak positions extracted from absorbance spectra at different salt concentrations as a function of waiting time, $t_w$, with (D) showing the half-point normalized relationship. (E) Temporal evolution of the full width at half maximum (FWHM) of the LSPR at varying salt concentration, with (F) showing the half-point normalized relationship. Both the LSPR peak position and FWHM, when plotted as a function of normalized time, corroborate the universal structural evolution during gelation.

Specifically, the absorbance spectra and LSPR peak positions progressively broaden and red shift, respectively, upon linker addition (Fig. 3, A and C, and fig. S7). The gels continue to age after arrest and do not reach a static optical value within the experimental timescales. For all salt concentrations, the temporal LSPR evolution was fit with a sigmoidal curve, from which the half point $t_{0.5}$ was obtained and used to define a characteristic timescale for each sample at a given salt concentration. While sigmoidal fits can accurately determine this characteristic $t_{0.5}$ value, they cannot accurately capture the unstable aging plateau values, as sigmoids are symmetric functions. Consequently, we truncated the data where the gelation kinetics dominate the optical evolution to accurately quantify the main gelation process without overfitting with additional parameters to account for aging. Thus, the resulting final plateau values are not intended to match the experimental results in that region, with the error reflecting the deviation due to convolution of gelation and aging (fig. S8). Although the initial dispersion plateaus and final “static” gel plateaus vary slightly across cases, the differences in the fitted plateaus are negligible compared to the overall optical shifts during gelation.

Overall, the half point ($t_{0.5}$) of the LSPR peak position versus time relationship, obtained from a sigmoidal fit (fig. S8), decreases monotonically with salt concentration (Fig. 3B). At all salt concentrations, the full width at half maximum (FWHM) increases with wait time, a consequence of the inhomogeneous broadening of the LSPR peak, reflecting microstructural heterogeneities that emerge during network formation (Fig. 3E and fig. S9) (38, 39). The rate of increase of the FWHM, for a given wait time, increases with salt concentration (fig. S9), consistent with salt-induced acceleration of gelation. Like in the SAXS analyses, an approximate time-salt superposition of each structure-dependent optical response parameter is obtained when $t_w$ is normalized by the relevant $t_{0.5}$ values (Fig. 3, D and F, and figs. S8 and S9), reinforcing the consistency of the structure transformation that occurs in these linked NC networks.

Evolution of microscopic dynamics

The microscopic dynamics of the dispersion-to-gel transition were probed using XPCS. XPCS, an x-ray analog of dynamic light scattering (DLS), analyzes temporal correlations in the scattered x-ray intensity to extract dynamic information in real time on microsecond scales (40–43). Unlike DLS, XPCS can resolve dynamic processes in non-ergodic systems across nano- to micrometer scales, which is critical for obtaining a deeper understanding of colloidal nanoparticle gelation (33, 44). The intensity autocorrelation function of XPCS is defined as

$$g_2(q,\mathrm{\Delta }t)=\frac{\langle I(q,t)I(q,t+\mathrm{\Delta }t)\rangle }{{\langle I(q,t)\rangle }^2}$$ (2)

where $I(q,t)$ is the scattered intensity at wavenumber $q$ and time $t$. Angle brackets denote ensemble averaging over all equivalent times $t$ and detector pixels. The accessible lag times $\mathrm{\Delta }t$ are determined by the detector frame rate and total acquisition duration. The intensity correlation function is related to the intermediate scattering function $g_1(q,\mathrm{\Delta }t)$ through the Siegert relation

$$g_2(q,\mathrm{\Delta }t)=1+b{\mid g_1(q,\mathrm{\Delta }t)\mid }^2$$ (3)

where $b$ is the speckle contrast, determined by factors such as the beam coherence, detector pixel size, and beam size, and is measured independently using a porous ceramic membrane as a static reference sample.

The intermediate scattering function $g_1(q,\mathrm{\Delta }t)$ characterizes the system’s dynamics. After correcting for the baseline and coherence factor post-measurement (fig. S10), we obtain ${\mid g_1(q,\mathrm{\Delta }t)\mid }^2$ (Fig. 4), which is then modeled using a Kohlrausch-Williams-Watts (KWW) function of the form (45–47)

$$g_1(q,\mathrm{\Delta }t)=[1-f\left(q\right)]\text{exp}\left\{-{\left[\frac{\mathrm{\Delta }t}{\mathrm{\tau }\left(q\right)}\right]}^{\mathrm{\beta }\left(q\right)}\right\}+f\left(q\right)$$ (4)

where $\mathrm{\tau }\left(q\right)$ is the characteristic relaxation time, $\mathrm{\beta }\left(q\right)$ is the KWW exponent that reflects the distribution of relaxation times, and $f\left(q\right)$ is the non-ergodicity parameter, representing the plateau value of $g_1(q,\mathrm{\Delta }t)$. The robustness of the KWW model enabled reliable extraction of dynamic parameters despite beamline-induced oscillations in ${\mid g_1(q,\mathrm{\Delta }t)\mid }^2$. Physically, $f\left(q\right)$ quantifies the fraction of density fluctuations at wavenumber $q$ that do not relax over the measurement window. A finite $f\left(q\right)>0$ typically indicates dynamic arrest: Particles are localized or caged over the corresponding length scale $\sim 2\mathrm{\pi }/q$. Overall, the $g_1(q,\mathrm{\Delta }t)$ function describes an exponential decay from unity to $f\left(q\right)$ at a relaxation rate $1/\mathrm{\tau }\left(q\right)$, and modulated by the KWW exponent $\mathrm{\beta }$, where $0<\mathrm{\beta }\le 2$. Representative results for ${\mid g_1(q,\mathrm{\Delta }t)\mid }^2$ illustrate both temporal (Fig. 4A) and wavenumber (Fig. 4B) dependencies. The smooth curves are fits based on Eqs. 3 and 4. Shorter exposure times were necessary to minimize beam-induced damage and dynamics. This constraint led to poorer statistics at higher $q$ (figs. S11 to S14). As a result, the autocorrelation function data are limited to the low-$q$ regime, where $q$ values remain below the SAXS peak position (Fig. 1E). In this range, the measured dynamics primarily reflect structural rearrangements within the growing fractal clusters that form the gel network.

Fig. 4. Evolution of microscopic dynamics probed via XPCS.

(A) Temporal (at $q=0.0031 {\mathrm{\AA }}^{-1}$) and (B) $q$-dependent (at $t_w=8.3$ hours) evolution of the normalized intensity autocorrelation function ${\mid g_1(q,\mathrm{\Delta }t)\mid }^2$ for the evolving system. The smooth curves are fits based on Eqs. 3 and 4 to extract dynamic parameters. In both panels, [TBAPF₆] = 0 mM.

As expected, the characteristic relaxation time $\mathrm{\tau }\left(q\right)$ increases with wait time, reflecting the progressive slowing of particle dynamics during gelation, with $\mathrm{\tau }\left(q\right)$ rising rapidly after a period of modest change (Fig. 5, A and B, and figs. S15 to S18). At short wait times, the microscopic relaxation rate [$1/\mathrm{\tau }\left(q\right)$] exhibits diffusive scaling with $q^2$, but as the system ages, it crosses over to a linear $q$ dependence, consistent with ballistic-like motion arising from collective rearrangements and internal stress relaxation (figs. S19 and S20) (13, 15, 44, 48–53). Concurrently, the KWW exponent $\mathrm{\beta }$, which characterizes the distribution of relaxation times around $\mathrm{\tau }\left(q\right)$, also remains nearly constant during a latency regime dominated by Brownian dynamics ($\mathrm{\beta }\approx 1$), before exhibiting a transition to nondiffusive behavior as reflected in the deviation of $\mathrm{\beta }$ from 1 (Fig. 5, C and D, and figs. S15 to S20) (54). The non-ergodicity parameter $f\left(q\right)$ increases steadily with time after an initial latency period, rising from $f\left(q\right)=0$ (representing an equilibrium, fully diffusive state) to $f\left(q\right)=1$ (indicating a fully arrested, non-ergodic state) (Fig. 5, E and F, and figs. S15 to S18). A simplified picture is that the system comprises two coexisting populations—one dynamically arrested and one still undergoing free motion. A similar emergence of non-ergodicity has been observed in dynamic measurements of gels formed from larger charge-stabilized colloids after salt addition (55), as well as in marginal-solvent colloidal gels where gelation was triggered by temperature-induced attractions (14, 33, 44, 47). Notably, the latency period in the dynamic parameters shown here is similar to that observed in the evolution of $\mathrm{\xi }$ and $D_f$ from the microstructure analysis (Fig. 2, D to F) and also apparent in the optical evolution results (Fig. 3, C to F). This delay suggests that the onset of dynamic arrest occurs only after substantial structural buildup, consistent with previous observations that rigidity in low-volume-fraction colloidal gels can emerge suddenly once compact clusters percolate (56, 57).

Fig. 5. Quantification of the microscopic NC dynamics.

(A) Temporal evolution of the $q$-dependent characteristic relaxation time, $\mathrm{\tau }\left(q\right)$. (B) $\mathrm{\tau }\left(q\right)$ plotted against the gel point–normalized wait time, $t_w/t_{10}$. (C and D) Evolution of the KWW exponent, $\mathrm{\beta }\left(q\right)$, with (D) showing $\mathrm{\beta }\left(q\right)$ as a function of $t_w/t_{10}$. (E and F) Evolution of the non-ergodicity parameter, $f\left(q\right)$, with (F) displaying $f\left(q\right)$ versus $t_w/t_{10}$. All data correspond to $q=0.0031$. The gel point–normalized time, $t_w/t_{10}$, is defined as in Fig. 2 (A and B).

Mirroring the structural evolution observed via SAXS and optical spectroscopy, the time dependence of the gelation dynamics also exhibits a pronounced dependence on salt concentration. Beyond the latency point, the evolution of $\mathrm{\tau }\left(q\right)$, $\mathrm{\beta }\left(q\right)$, and $f\left(q\right)$ shifts systematically with increasing TBAPF₆ concentration, with curves translating horizontally in time while preserving their overall shape. This behavior indicates that salt accelerates the onset of dynamic evolution, apparently without altering the fundamental kinetic pathway. Consequently, rescaling the time axis by $t_{10}$ collapses the data onto a single master curve, revealing a universal dynamic trajectory for the gelation process (Fig. 5, B, D, and F, and figs. S15 to S18). In their investigation of thermoreversible nanoparticle gels, Bahadur et al. (33) demonstrated that a single gel formation timescale collapses the structural, dynamical, and rheological responses across different formation times and quench temperatures. Consistent with this finding, our structural and dynamical results likewise exhibit such a collapse, underscoring the universality of the gelation process. Beyond the collapse across $t_w/t_{10}$, we also observe a comparable time-salt superposition and universal progression when defining the characteristic time using alternative $S\left(0\right)$ thresholds. Thus, the universality of structural and dynamical evolution is not contingent upon the specific choice of structurally defined characteristic time (fig. S21).

Additionally, although initially shown for a representative $q$, the time-salt superposition and universal progression of gelation are evident across the full range of experimentally accessible length scales, supporting the conclusion that a single salt-dependent mechanism governs both local structuring and macroscopic network formation (Fig. 6 and fig. S20). To illustrate, we examine a $q$-dependent analysis of the temporal evolution of the non-ergodicity parameter, $f\left(q\right)$, which we model as

$$f(q,t_w)=f_{\infty }\left(q\right)[1-\text{exp}(-{\mathrm{\Gamma }}_f{t}_w)]$$ (5)

where $f_{\infty }\left(q\right)$ is the long-time localized fraction and ${\mathrm{\Gamma }}_f$ is the growth rate of localization (44). Fits across scattering vectors $q$ show that $f_{\infty }\left(q\right)$ decreases systematically with increasing $q$, indicating that large-scale density fluctuations become nearly arrested, while small-scale fluctuations retain partial mobility, demonstrating that the gelation process exhibits both spatial and temporal heterogeneity. Notably, $f_{\infty }\left(q\right)$ collapses onto a universal curve across all salt concentrations (Fig. 6B), demonstrating convergence to the same long-time steady state at every $q$. The growth rate of the localized fraction ${\mathrm{\Gamma }}_f$ is also $q$-dependent, with higher salt concentrations driving faster localization (Fig. 6C). When normalized by the corresponding $t_{10}$, however, ${\mathrm{\Gamma }}_f$ values collapse onto a single curve, revealing universal length-scale dynamics across all salt conditions (Fig. 6D).

Fig. 6. Length-scale dependence of the emergence of non-ergodicity.

(A) Non-ergodicity parameter $f\left(q\right)$ as a function of wait time $t_w$ at several scattering wavenumbers $q$, representing the probed length scales. The dashed lines are fits to Eq. 5 for each $q$. (B and C) Fit results as a function of $q$. (B) Steady-state localized fraction at asymptotically large wait times, $f_{\infty }$, shown as a function of $q$ for all salt concentrations. (C) Localized growth rate, ${\mathrm{\Gamma }}_f$, as a function of scattering wavenumber $q$ for all salt concentrations. The rate increases with salt across all $q$, with a more pronounced effect at higher $q$ (smaller length scales). (D) The characteristic time normalized localized growth rate ${{\mathrm{\Gamma }}_f}^{\ast }{t}_{10}$ as a function of $q$ showing a superposition across length scales. In all panels, salt concentrations are indicated by different symbols: $○=0$ mM, $\mathrm{\Delta }=33$ mM, $\ast =66$ mM, and $⬡=100$ mM.

Mechanism of salt-accelerated gelation: Structural insights

Given the time-salt superposition observed across length scales, probing the effects of salt on interactions and structuring at the smallest relevant scale—the NC core diameter ${\mathrm{\sigma }}_{\text{core}}$—offers insights into the mechanism of salt-accelerated gelation kinetics. To isolate the effect of salt, we first analyze the NC dispersion structure and effective pairwise interactions in the absence of attractive linkers. Upon ligand exchange to form AL-ITO NCs, structural analysis reveals pronounced differences in $S\left(q\right)$ compared to as-synthesized OL-ITO NCs. Specifically, $S\left(q\right)$ of AL-ITO NC dispersions exhibits stronger oscillations and decreased osmotic compressibility, as quantified by a smaller zero–wavenumber structure factor $S\left(0\right)$, relative to OL-ITO NCs at the same volume fraction and salt concentration (Fig. 7A and fig. S22). A pronounced drop in $S\left(q\right)$ below unity at low $q$ and a shift of the primary peak to higher $q$ with increasing $\mathrm{\Phi }$ are hallmarks of long-range repulsion (Fig. 7B) (58). This behavior is consistent with charged, repulsive particles, where $S\left(q\right)$ can be modeled by Yukawa potentials for which the peak position follows $q_{\text{max}}\approx 2.2{\mathrm{\rho }}^{1/3}$, with the number density $\mathrm{\rho }$ calculated as $\mathrm{\rho }=\mathrm{\Phi }/({\mathrm{\pi }\mathrm{\sigma }}_{\text{core}}^3/6)$ (Fig. 7C) (23, 59). Zeta potential (ZP) measurements confirm a high negative surface charge on the AL-ITO NCs (Fig. 7D and table S1). Control experiments using PEG–phosphonic acid (molecular weight: 414 Da) as a capping ligand—bearing a single phosphonic acid instead of a tri-carboxylic acid—yielded NC dispersions with $S\left(q\right)$ lacking the correlation peak oscillations of charged colloids and a ZP value near 0 mV, consistent with AL-ITO’s negative surface charge originating from the abundant carboxylic acid groups, which may undergo deprotonation (fig. S23).

Fig. 7. Salt effect on effective pairwise interactions of dispersed NCs.

(A) $S\left(q\right)$ of 16.8-nm AL-ITO NC dispersions at $\mathrm{\Phi }=0.025$ with no added salt and with different surface ligands [oleate-capped (OL) versus aldehyde-capped (AL)]. (B) $S\left(q\right)$ of AL-ITO NCs in DMF at different colloid volume fractions ($0.0005\le \mathrm{\Phi }\le 0.025$) dispersed in DMF with no added salt. (C) Plot of $q_{\text{max}}$ versus ${\mathrm{\rho }}^{1/3}$ at increasing salt concentrations from 0 to 100 mM. The solid black line, $q_{\text{max}}=2.2{\mathrm{\pi }\mathrm{\rho }}^{1/3}$, indicates behavior theoretically expected from charged colloids with long-range repulsions that maximize interparticle spacing (23, 59). The red curve represents the exact solution of the Percus-Yevick integral equation for a hard-sphere fluid at a packing fraction consistent with $\mathrm{\rho }$ and a hard-sphere diameter ${\mathrm{\sigma }}_{\text{HS}}$ obtained from a fit to the experimental second osmotic virial coefficient $B_2$ versus salt results (see fig. S24). The legend lists only a subset of the data to represent the range of salt concentrations studied. (D) ZP measurement of OL-ITO and AL-ITO NCs ($\mathrm{\Phi }=0.005$) in toluene and DMF, respectively, without added salt. (E) $S\left(q\right)$ of AL-ITO NCs in DMF with increasing salt concentration 0 ≤ [TBAPF₆] ≤ 100 mM at constant colloid volume fraction $\mathrm{\Phi }=0.025$. (F) Osmotic second virial coefficient $B_2$ as a function of [TBAPF₆]. The dotted blue line represents $B_2$ for the neutral OL-ITO NC dispersion system. The inset shows the reciprocal of the structure factor at zero wavenumber, $1/S\left(0\right)$, versus NC number density $\mathrm{\rho }$, at different salt concentrations. The osmotic second virial coefficient $B_2$ is calculated as one-half the slope of the linear regression (Eq. 6), shown as the dashed black lines. The data points in the inset are color-coded to match the legend in (E).

Salt addition increases the ionic strength and reduces the electrostatic double-layer thickness, thereby screening interparticle repulsions. This screening effect is evident in the evolution of $S\left(q\right)$ with increasing salt concentration at fixed $\mathrm{\Phi }$. As ionic strength rises, low-$q$ intensity increases and the correlation peak decreases in magnitude, broadens, and shifts to higher $q$, indicating that particles can approach each other more closely. These trends are consistent with a reduced electrostatic repulsion between NCs. Since there is no appreciable attraction, a colloidal “gas”-like structure is approached (Fig. 7E), similar to that observed for sterically stabilized OL-ITO NCs. To quantify the screening effect of the salt on the pairwise interactions, we calculate the osmotic second virial coefficient $B_2$ from the osmotic compressibility, which relates to $S\left(0\right)$ as

$$\frac{1}{S\left(0\right)}-1={\left(k_{\mathrm{B}}T\right)}^{-1}{\left(\frac{\mathrm{\partial }\mathrm{\Pi }}{\mathrm{\partial }\mathrm{\rho }}\right)}_T-1=2B_2\mathrm{\rho }+\mathcal{O}\left({\mathrm{\rho }}^2\right)$$ (6)

where $\mathrm{\Pi }$ is the osmotic pressure. By fitting a quadratic function to $S\left(q\right)$ at low $q$, we extract $S\left(0\right)$ (Fig. 7F, inset, and fig. S24) and estimate $B_2$ (Fig. 7F) (58, 60, 61). The observed monotonic decrease in $B_2$ with increasing salt confirms that salt screens repulsive pairwise interactions. The salt-induced reduction in effective pairwise NC repulsion is most pronounced at low salt concentrations (0 < [TBAPF₆] < 33 mM). Above this range, $B_2$ remains positive and nearly constant, indicating that the dispersion remains net repulsive due to the steric effects of the ligands, consistent with observations for OL-ITO (60).

Assuming that interparticle interactions follow Derjaguin-Landau-Verwey-Overbeek (DLVO) theory (62, 63), we fit the experimental salt dependence of $B_2$ using its integral expression to extract the NC charge number $Z$ and thermodynamic hard-sphere diameter ${\mathrm{\sigma }}_{\text{HS}}$ (fig. S25). To minimize the influence of adventitious counterions on the estimated screening length, fitting was restricted to systems with salt concentrations ≥3 mM, where added salt dominates the ionic strength. From this analysis, we obtained $Z=71\pm 6$ and ${\mathrm{\sigma }}_{\text{HS}}=19.4\pm 1.9$ nm. Using these parameters, structure factors computed via integral equation theory (IET) showed qualitative agreement with experimental $S\left(q\right)$ (fig. S26). This agreement, along with the quantitative trends in $B_2$, confirms that added salt screens interparticle repulsions. Screening reduces the configurational constraints of the electrostatic repulsions, which could facilitate faster local structure evolution, thereby enhancing gelation kinetics upon introduction of chemical linkers. However, the fact that the reduction in effective repulsion is most pronounced at low salt concentrations suggests that electrostatic screening alone cannot account for the continued acceleration of gelation across the full 0 to 100 mM salt range.

Kinetic Monte Carlo (kMC) simulations (64) (see Materials and Methods and the Supplementary Materials) offer one avenue for investigating how electrostatic interactions and salt screening may be expected to influence linked colloidal gelation, and whether salt screening is a primary mechanism accelerating the dispersion-to-gel evolution. The interparticle interactions are described by the DLVO potential, with the Debye length (${\mathrm{\kappa }}^{-1}$) determined by the salt concentration, combined with a strong short-range attractive potential mimicking the chemical linking. The simulations and corresponding $S\left(q\right)$ reproduce key features of the experimental data, showing increased NC coordination upon network formation (Fig. 8A) and subsequent coarsening into denser structures, quantified by the increase in $\mathrm{\Delta }S$ (the difference between the primary contact peak and the Porod dip in $S\left(q\right)$; Fig. 8B). We use $\mathrm{\Delta }S$ as the measure of structural evolution in place of $S\left(0\right)$, as the resolution of $S\left(q\right)$ in the low-$q$ region is restricted by the finite size of the simulations (65).

Fig. 8. Simulation of salt-accelerated gelation.

(A) Snapshot of the colloidal gel from a kMC simulation. NC colors indicate their number of nearest neighbors. (B) Schematic $S\left(q\right)$ illustrating the definition of $\mathrm{\Delta }S$, which grows as the colloidal network evolves. (C) $\mathrm{\Delta }S$ as a function of waiting time for different salt concentrations. (D) $t_{1.2}$ versus salt concentration; the characteristic time $t_{1.2}$ is defined as the time when $\mathrm{\Delta }S$ reaches 1.2. (E) Relationship between $t_{1.2}$ and the Yukawa repulsion energy barrier. The dashed line corresponds to an Arrhenius fit of $t_{1.2}$ versus the energy barrier. (F) $S\left(q\right)$ of colloidal gels at $t_{1.2}$ for various salt concentrations. (G) Time-salt superposition shown by plotting $\mathrm{\Delta }S$ against normalized waiting time.

Tracking $\mathrm{\Delta }S$ over simulation time confirms salt-accelerated gelation and coarsening rates (Fig. 8C). When $\mathrm{\Delta }S\ge 1.2$, the $\mathrm{\Delta }S-\text{log}\left(t_w\right)$ curve shows a nonlinear saturation trend, indicating that the kinetics of coarsening attenuates. Hence, we choose to use the simulation time at which $\mathrm{\Delta }S=1.2$ as the characteristic time ($t_{1.2}$). We find that $t_{1.2}$ decreases with increasing salt concentration (Fig. 8D). Similar trends are observed when defining gelation based on other $\mathrm{\Delta }S$ thresholds, confirming the robustness of this observation (fig. S27). For effective linking, the NCs must overcome the Yukawa repulsion energy barrier before approaching closely enough to form a bond. This barrier corresponds to the value of the DLVO potential at the edge of the attractive well, $E_{\mathrm{B}}=U_{\text{DLVO}}({\mathrm{\sigma }}_{\text{HS}}+\mathrm{\delta }),$ where $U_{\text{DLVO}}\left(r\right)$ is the screened electrostatic potential and $\mathrm{\delta }$ denotes the width of the attractive well. We find that the characteristic gelation time, $t_{1.2}$, exhibits an approximate Arrhenius-type dependence on this barrier (Fig. 8E and fig. S31). This Arrhenius-like scaling indicates that salt accelerates gelation by lowering the Yukawa repulsion energy barrier within the DLVO framework. Notably, $t_{1.2}$ drops sharply between 10 and 33 mM, and then levels off from 33 to 100 mM—mirroring the plateau in $B_2$ as the electrostatic repulsion is screened out and approaches zero at high salt concentrations (fig. S30).

The simulated static structure factors at $t_{1.2}$ are approximately the same across the full $q$ range for all simulated salt concentrations, consistent with the experimental observations (Fig. 2A, inset). Minor deviations appear only in the low-$q$ region (Fig. 8F and fig. S27), reinforcing that the mesoscale gel structure at this stage is largely independent of salt concentration. This structure universality further justifies the use of $\mathrm{\Delta }S$ as a representative metric for structural evolution. Normalizing the time axis by $t_{1.2}$ yields approximate time-salt superposition in the $\mathrm{\Delta }S\left(t_w\right)$ trajectories (Fig. 8G). To investigate the effect of particle volume fraction on the observed time-salt superposition behavior, we also carried out the simulations at volume fractions of $\mathrm{\Phi }=0.01$ and 0.003, with the latter volume fraction corresponding to the volume fraction used in the optical measurements (Fig. 3). Although the gelation kinetics were much slower at these volume fractions, both showed similar approximate time-salt superposition behavior (figs. S28 and S29). Irrespective of the $\mathrm{\Phi }$ used, however, the quality of the trajectory collapse depends on the choice of $\mathrm{\Delta }S$ used to define the characteristic time. Better superposition at early times is achieved with lower $\mathrm{\Delta }S$ thresholds (e.g., $\mathrm{\Delta }S=1.1$), whereas later times align better with higher thresholds (e.g., $\mathrm{\Delta }S=1.4$; fig. S27). These observations further point to the approximate nature of the superposition as obtained from the kMC simulations within the relatively simple DLVO framework.

Together, the weak dependence of the characteristic time $t_{1.2}$ on salt concentration above 30 mM and the diminished electrostatic screening effects on $B_2$ at high salt both indicate that electrostatic screening alone cannot account for the accelerated gelation kinetics across the full 0 to 100 mM salt concentration range. While the DLVO theory provides a useful approximate framework, its simplifying assumptions lose validity outside of the dilute regime for both colloids and electrolytes. Moreover, the kMC simulations neglect other interactions known to influence colloidal gelation, such as many-body hydrodynamic forces (66, 67). Furthermore, we have neglected charge regulation—the process by which a particle’s surface charge adjusts dynamically in response to changes in the surrounding chemical environment (68). This phenomenon may be relevant to our system; we find that the charge number $Z$ that best matches experimental $S\left(q\right)$ with theoretical structure factors from IET varies with both salt concentration and colloid volume fraction (figs. S32 and S33 and table S2). Despite these shortcomings, the ability of DLVO theory and kMC simulations to qualitatively capture salt-accelerated gelation kinetics and salt-time superposition effects suggests that they are helpful starting point for interpreting the experimental observations. Yet, this model fails to account for the continued superposition observed at the higher salt concentrations experimentally investigated.

Mechanism of salt-accelerated gelation: Reaction-driven insights

To examine other factors that may influence salt-accelerated gelation beyond electrostatic screening of the colloids, we investigated the underlying chemical linking process: the hydrazone condensation reaction. A simplified molecular model was designed using benzaldehyde and benzoylhydrazine, reacted under the same solvent conditions as the NC gelation system, with benzaldehyde as the limiting reagent (Fig. 9A). Chemical linking kinetics were assessed, while TBAPF₆ salt concentrations were varied from 0 to 66 mM. Using ¹H NMR spectroscopy (Fig. 9B), we monitored the consumption of reactants and formation of the hydrazone product over 23 hours (figs. S34 to S37). Integration of characteristic benzaldehyde and hydrazone peaks enabled calculation of the reaction rate constant $k$ using the integrated second-order rate law (69, 70)

$$\frac{1}{{\left[B\right]}_0-{\left[A\right]}_0}\text{ln}\left(\frac{{\left[A\right]}_0\left[B\right]}{{\left[B\right]}_0\left[A\right]}\right)=\mathit{kt}$$ (7)

where ${\left[A\right]}_0$ and ${\left[B\right]}_0$ are the initial concentrations of benzaldehyde and benzoylhydrazine, $\left[A\right]$ and $\left[B\right]$ are their concentrations at time $t$, and $k$ is extracted as the slope of the linear regression (fig. S38). The variations in $k$ resulting from added salt reflect the impact of chemical rate changes on bond formation between the NCs, but in a reaction environment more amenable to quantitative analysis. The values of $k$ were derived using spectral data up to $80(\pm 3)\%$ of the time required to reach a steady-state product concentration. We observed that the time to 80% completion decreases with increasing salt concentration (Fig. 9C), mirroring the trends in $t_{10}$ and $t_{0.5}$ from SAXS and ultraviolet-visible (UV-Vis) results, respectively (Figs. 2B and 3B). Accordingly, $k$ increases with salt concentration, suggesting that increased ionic strength, achieved through the addition of the salt, enhances the condensation reaction rate—possibly by stabilizing the charged intermediates created during the condensation reaction. Although the microscopic mechanism differs, the net effect is conceptually similar to the lowering of the electrostatic barrier observed at lower salt concentrations. Notably, the salt acceleration of the linking reaction rate continues to grow in magnitude, even at higher salt concentrations where electrostatic screening effects on $B_2$ have saturated.

Fig. 9. Salt effect on kinetics of hydrazone bond formation.

(A) Schematic of the chemical reaction resulting in hydrazone bond formation from reactants, benzaldehyde, and benzoylhydrazine with varied [TBAPF₆]. (B) Representative ¹H NMR spectra of the hydrazone bonded product, benzaldehyde reactant, and benzoylhydrazine reactant (from top to bottom) in the region of interest used to track the kinetics of bond formation. Peak i at ~10.1 ppm corresponds to the aldehyde proton in benzaldehyde, and peak ii at ~8.6 ppm corresponds to the imine proton in the hydrazone product. (C) Time of 80% reaction completion (black data points) and rate of linking reaction, k (red data points), with respect to concentration of TBAPF₆. Error bars on the k values represent 95% confidence interval of the standard fitting errors (see fig. S38).

DISCUSSION

These measurements of the reaction rate constant, considered alongside the structural characterization of the NC dispersions, the evolving gel microstructure and dynamics, and the simulations of gelation kinetics, reveal a multimodal mechanism by which salt accelerates gelation. At low concentrations, salt screens electrostatic repulsions, lowering the free-energy barrier for NCs to approach one another. Once in close proximity, hydrazone bond formation initiates network assembly. Although electrostatic screening saturates at relatively low salt concentrations, the rate of chemical bond formation continues to increase with additional salt, enabling faster linking among NCs and further accelerating gelation.

Beyond the kinetic effect of added salt, a key outcome of this work is the emergence of time-salt superposition in the structural and dynamical evolution of the NC gel. This superposition can be rationalized by considering how salt accelerates both the onset of structural rearrangements during network formation and the underlying chemical kinetics. In our system, both processes appear reaction-controlled: The overall rate is governed by the probability of forming a bond upon collision, and particles must undergo many collisions before sticking because a repulsive barrier suppresses the sticking probability. Increasing the salt concentration enhances electrostatic screening and accelerates the chemical reaction, thereby increasing the likelihood of particle association per collision. For superposition to emerge, the repulsive interaction that must be overcome in this reaction-controlled regime is likely short ranged, with the Debye length ${\mathrm{\kappa }}^{-1}\le {\mathrm{\sigma }}_{\text{HS}}$. Evidence for such short-range repulsion is seen in the sharp reduction of the second virial coefficient, $B_2$, at low salt concentrations, where only a small amount of salt is required to fully screen the NC charges. The chemical reaction driving gelation is also inherently short ranged, as linking can occur only at particle contact. Consequently, within the good-solvent regime—where salting-out and other salt-induced NC destabilization phenomena are absent—one would expect structural and dynamic superposition once the kinetic effects of added salt are accounted for. To the extent that salt continues to lower the barrier to the chemical reaction, this superposition may be expected to persist even at higher salt concentrations, despite the electrostatic barrier being fully screened.

In a corresponding set of combined XPCS and SAXS experiments, we monitored the gelation of this same linker-mediated NC system under slightly modified conditions. The NC volume fraction was kept at $\mathrm{\Phi }=0.03$ with ${\mathrm{\sigma }}_{\text{core}}=14.6$ nm, and the linker concentration was held at 10 mM, corresponding to a linker-to-colloid ratio of $\mathrm{\Gamma }=336$, in an effort to induce gelation within a convenient experimental timescale. The salt concentration range was extended, from 66 mM to 200 mM TBAPF₆. This range was chosen to overlap with the original dataset while extending the upper limit to twice the previously explored concentration. Across this broader salt window, we observe a continued acceleration of gelation beyond 100 mM, with the gelation kinetics exhibiting progressively stronger salt-induced kinetic enhancement (Fig. 10A). Crucially, beyond this accelerating effect, both the structural (Fig. 10, B to D, and figs. S39 to S41) and dynamic (Fig. 10, E to G, and figs. S42 and S43) evolution exhibit robust time-salt superposition throughout the entire extended salt concentration range. These results further corroborate the universality of the physical mechanism of salt-induced gelation acceleration as a reduction of the barrier to NC linking, a phenomenon that persists well beyond the salt concentrations at which electrostatic screening effects become saturated.

Fig. 10. Microscopic structure and dynamics of the temporally evolving gels at high-salt conditions.

(A) Characteristic transition time $t_{10}$, defined as above, as a function of TBAPF₆ concentration. (B) $S\left(q\right)$ at the characteristic time point $t_{10}$ for the various TBAPF₆ concentrations. (C) Temporal evolution of the correlation length $\mathrm{\xi }$ of the heterogeneous domains formed during network development over wait time $t_w$, and (D) over normalized wait time $t_w/t_{10}$, showing universal progression of microstructure. (E) $q$-dependent (at $t_w=t_{10}$) evolution of the normalized intensity autocorrelation function ${\mid g_1(q,\mathrm{\Delta }t)\mid }^2$ for the evolving system at the elevated salt concentrations. (F) Temporal evolution of the characteristic relaxation time $\mathrm{\tau }\left(q\right)$ at the increased salt concentrations versus wait time $t_w$, and (G) versus normalized wait time $t_w/t_{10}$, demonstrating universal progression of microscopic dynamics.

While the final structure obtained at long times may be similar, systems with long-ranged repulsive interactions and short-ranged attractions typically exhibit distinct intermediate states along the evolution pathway, ranging from stable colloidal cluster fluids to clusters of varying morphologies and intermediate-range-order structures (5, 71, 72). Therefore, one might expect the no-salt system to deviate from time-salt superposition relative to the other higher-salt-content systems because the electrostatic barrier is maximal and the repulsions are long-ranged in the absence of added salt, potentially leading to different network structures at the same characteristic time. However, the experimental observation of superposition across the full 0 to 100 mM range suggests that the repulsive interaction is indeed short ranged and relatively weak even at 0 mM, potentially due to adventitious ions or charge-regulation effects. This short-range character explains why the structures remain similar once the barrier-reducing effect of salt is accounted for through normalization by a characteristic time associated with structural equivalence. Since our modeling of reaction-controlled association with electrostatic effects predicts approximate time-salt superposition, we expect a similar collapse experimentally across a broad range of salt conditions, regardless of whether the dominant barrier originates from electrostatics or from chemical reaction kinetics (Fig. 10 and figs. S39 to S43).

The observations of time-salt superposition of linked NC gel formation reported here relate to broader analyses of time-composition superposition in diverse soft matter systems, including spontaneously aging Laponite suspensions (18), protein condensates (19), polymer hydrogels (20), polyelectrolyte complexes (21), and colloidal gels (33, 35, 73–75). In many cases, changing composition modifies physical interactions in a way that affects time-dependent structure and corresponding properties, and the results are examined through techniques like rheology that probe the macroscopic mechanical response under deformation. Our work provides mechanistic insight by connecting microscopic structure and dynamics to corresponding evolution of macroscopic observables, advancing understanding by integrating experimental and computational approaches that probe chemistry, structure, and dynamics across length scales.

By tracing the gelation pathway across a wide salt concentration range using complementary techniques—SAXS, optical spectroscopy, XPCS, kinetic NMR, and kMC simulations—we establish that salt accelerates colloidal network formation and dynamic evolution without substantially altering the structural pathway through the gelation process. Effective pairwise interaction measurements and simulations confirm that electrostatic screening reduces repulsive barriers to assembly, enabling particle approach, with the most notable effects occurring in the low-salt concentration regime. Simultaneously, NMR studies demonstrate that salt also accelerates the hydrazone bond formation kinetics, likely through enhanced proton transfer in a more polar environment, and through this mechanism, salt continues to lower the barrier to bond formation at higher salt concentrations. Despite the success of DLVO theory and kMC simulations as an initial framework to describe these observations, its limitations—including neglect of charge regulation, many-body thermodynamic and hydrodynamic interactions, and salt-dependent chemical reaction rates—suggest that future efforts should aim to incorporate these complexities into models for a more complete description of gelation dynamics. On the experimental side, expanding control parameters beyond salt—such as pH, solvent properties, and particle size and charge density—offers additional avenues to tune particle interactions and gelation kinetics. Ultimately, integrating these insights will help in the rational design of new soft materials including reconfigurable, programmable colloidal NC architectures with tailored mechanical and dynamic behaviors, and optical response.

MATERIALS AND METHODS

Ligand synthesis

ALs were synthesized by modifying established literature methods (7). Briefly, a peptide base was synthesized using solid-phase peptide synthesis (SPPS), with the aldehyde functional group added via solid-phase copper azide alkyne cycloaddition (SP-CuAAC). The resulting ligand was then cleaved from the resin, purified using high-performance liquid chromatography, and then dried for use.

Synthesis and functionalization of NCs

ITO NCs were synthesized on a Schlenk line through a modification of a slow growth procedure (28). A precursor solution containing Sn(IV)acetate and In(III)acetate in 10 ml of oleic acid was degassed (90°C, 1 hour) and heated (150°C, 3 hours) under flowing nitrogen. The solution was then slowly injected into 13 ml of oleyl alcohol at 290°C under nitrogen flow. After synthesis, NCs were washed five times with ethanol and dispersed in hexane. The NCs were functionalized through a direct ligand exchange (7, 16). As-synthesized, oleate-capped NCs were precipitated into a pellet and sonicated in a ligand solution (0.01 M AL in DMF) for 2 hours before reacting overnight. The functionalized NCs were washed three times with a 3:7 (v/v) EtOH:hexane mixture and then dispersed in pure DMF.

Preparation of samples for scattering and spectroscopic measurements

SAXS and XPCS

SAXS measurements were performed in transmission mode at three facilities: the SAXSLAB Ganesha instrument at the Texas Materials Institute (TMI), University of Texas at Austin; beamline CHX (11-ID) at the National Synchrotron Light Source II (NSLS-II; Upton, NY, USA); and sector 8-ID-I at the Advanced Photon Source (APS; Argonne, IL, USA), with sample-to-detector distances of ~1, 16, and 9 m, respectively. XPCS was conducted at CHX using a 12.8-keV beam focused to 40 × 40 μm², with scattering patterns collected on a Dectris Eiger 500K area detector (pixel size 75 × 75 μm²) with a maximum frame rate of 9 kHz positioned 16 m from the sample. After a dose-dependent study (fig. S45), each XPCS series was recorded at full transmission with 1000 frames. At this setting, the short exposure time minimized beam-induced dynamics despite the high transmission. All the data presented here were recorded at a frame rate of 9 kHz. All samples, including a silver behenate standard for $q$-calibration (TMI SAXS), were sealed in flame-closed 1-mm glass capillaries (Hampton Research) with approximately 10-μm wall thickness. SAXS data were processed using the Irena and Nika packages in Igor Pro. XPCS data analysis was done using custom python code for data access and reduction in iPython notebooks running on Jupyter Lab provided by the beamline.

Optical spectroscopy

Gel optical spectra were measured with an Agilent Cary 5000 UV-Vis-NIR spectrophotometer over a wave number range of 4000 to 8000 cm⁻¹. Gel precursors were injected into a custom cells made from two glass slides separated by a Surlyn thermoadhesive spacer with a 25-μm thickness. The cells were placed horizontally in the path of a columnated beam between two fiber optic cables. Cell construction and optical measurement setup is shown in fig. S46. Gels were measured at varying time intervals to ascertain high-time-resolution spectral evolution. Normalized optical spectra of gels at each salt concentration can be found in fig. S7. The peak positions were fit with a Gaussian fitting of the highest 10% of data points, and the FWHM values were calculated using the discrete wave number width of half the peak value. Peak evolutions with time were fit with a decreasing standard logistic function, while FWHM values were fit with an increasing standard logistic function. The time axes of the curves were normalized by the fit $t_{0.5}$ to evaluate the time-salt superimposition of optical properties.

Nuclear magnetic resonance

NMR measurements were performed using a Bruker OXO-500 spectrometer using a cryoprobe. Spectra were calibrated to the DMF solvent peaks. Samples were prepared in a 75:25 v/v of DMF-d7:ethylene glycol-d₆ solvent mixture, containing 20 mM benzaldehyde, 22 mM benzoylhydrazine, 1,3,5-trimethoxybenzene as an internal standard, and a varied concentration of TBAPF₆ of 0, 15, 33, or 66 mM. Peak integrals and calculated concentrations of the reaction were normalized to the internal standard.

DLS and ZP

DLS measurements of dilute NCs ($\mathrm{\Phi }=0.0005$) were performed using a Zetasizer Nano ZS (Malvern) with a He-Ne laser ($\mathrm{\lambda }=633$ nm) and 173° backscattering detection at 25°C. Autocorrelation functions were collected in triplicate (10 s each) and averaged. Diffusion coefficients were obtained from the short-time decay using third-order cumulant analysis, and hydrodynamic diameters were calculated via the Stokes-Einstein equation. ZP was measured using the same instrument and conditions, with a dip cell in a glass cuvette. Measurements were repeated at least three times for reproducibility. Both DLS and ZP results can be found in table S1.

Numerical fitting and calculation of the structural factors of the NC dispersions

We computed structure factors for NCs interacting via the DLVO potential using liquid-state IET (59). We numerically solved the Ornstein-Zernike equation with the hypernetted chain (HNC) closure implemented in the pyPRISM package (v1.0.4) (76). To validate the applicability of IET for this system, we performed Brownian Dynamics (BD) simulations using the Highly Optimized Object-oriented Many-particle Dynamics-Blue Edition (HOOMD-Blue, v4.5.0) toolkit (77) to model NCs with DLVO-type interactions. The hard-sphere interactions in BD simulations are approximated using the Heyes-Melrose potential (78). Comparison between IET predictions and BD simulation results ensures consistency across theoretical and computational approaches (fig. S44). Additional details on the structure factor calculations are provided in the Supplementary Materials.

kMC simulations of the gelation process

The gelation process is simulated with customized kMC simulations (64). Details of the kMC simulations are provided in the Supplementary Materials. Static structural factors are calculated from snapshots of kMC simulations using the Freud package (79).

Kinetics of chemical linking

Further details of the sample preparation and the second-order reaction integrated rate law calculations and the linear regression can be found in the Supplementary Materials.

Supplementary Materials

This PDF file includes:

Sections S1 to S7

Figs. S1 to S46

Tables S1 and S2