Interfacial depletant partitioning for lattice-tunable colloidal assembly

Abstract

The depletion force is a perfect example of entropic action where electron binding is not involved to assemble matter. Here we demonstrate that depletion forces can also be used to modulate dimensional changes in a colloidal lattice. The modulation is possible by introducing the concept of depletant partitioning, where depletants can freely penetrate semipermeable colloids during self-assembly. The reversible and programmable absorption of depletants gives rise to the emergence of a dynamic depletion potential that stabilizes colloidal assemblies. The range energies and amplitudes of the colloidal lattice variations are modulated by the temperature dependent concentration and affinity of depletants in solution. This colloidal platform exhibits a finely tunable self-reinforcing effect where rising temperature fortifies the expansion of colloidal superlattices, creating future avenues for assembly and action in colloidal-based materials.

Colloidal systems have the potential to mimic complex self-assembly processes found in nature, which can lead to innovative materials with tailored properties. This study introduces an approach using depletant partitioning to dynamically modulate the size and stability of colloidal assemblies through temperature changes, achieving finely tunable lattice variations and expanding the possibilities for colloidal-based materials.

Introduction

The powerful self-assembly of life carries detailed instructions at the molecular level that direct the growth of functional systems over time. Mimicking such instructions has shown significant promise from the colloidal perspective, given similarity of the length and time scales of the interactions¹.

Advances in colloidal fabrication over the past years, have made possible to create submicron colloids as unique model systems with fine control over their size, shape, and binding interactions² to the point that a simple colloidal particle can encode ~100 binding types (e.g., DNA flavors) on its surface³. Various colloidal crystals (e.g., FCC, BCC, AlB₂, etc.)⁴ and arbitrary structures (e.g., belts, rings, chains)⁵ continue to demonstrate the strength of colloidal approach to fabricate materials with tailored properties⁶.

The richness of the colloidal self-assembly approach could further grow if the interaction rules (i.e., magnitude, directionality, range) are temporally modulated and respond to stimuli. Hence, the assembly process should incorporate more than “constant” pair-wise interactions, shifting to a system-level approach where the energy landscape is adapted as the assembly grows.

Here, we introduce a colloidal approach (dynamic, reversible and entropic in nature), where assemblies are energetically reinforced while they grow in size during heating. This enables the creation of colloidal assemblies, whose size and energy states become finely tunable with temperature. The modulation of depletion interactions is the basis for this approach.

The depletion force, first described in 1954 by Asakura and Oosawa⁷, arises when depletants (nanoscopic rigid bodies) create excluded volumes on the surface of colloidal particles, creating an osmotic pressure imbalance that pushes colloids towards to each other^8–10 (Supp. Text.). The entropic nature of the depletion attraction force is simple and abundantly present in molecularly crowded environments yet, oftentimes inadvertently ignored¹¹.

In more recent years, the understanding of depletion forces has facilitated applications from self-assembly of mimicking atomic structures like diamond¹² to purification of atomic/molecular liquid mixtures^13,14. Depletants like nanoparticles¹⁵ and large macromolecules like PEO⁸ have offered great stability to crystalline assemblies by adjusting the magnitude of the depletion force solely based on their concentration.

If depletants remain on the exterior of the colloids (non-absorbing depletants), as so far conventional, the range and magnitude of the depletion attraction can only be regulated by adjusting the size and the relative concentration of depletants to colloids in suspension. In this direction, the use of microgels of poly(N-isopropylacrylamide) (PNIPAM) as depletants has offered an opportunity to adjust depletion forces with temperature¹⁶. PNIPAM depletants respond with a marked size contraction around their critical solution temperature (~32 °C); such a transition permits to decrease the depletion force when heating around 32 °C at a constant depletant concentration^17–19.

However, a fully dynamic depletion potential emerges if we grant the capacity of depletants to permeate colloids, introducing the concept of absorbing depletants. Here, temperature can also serve as a continuous parameter that modulates the production and partition of depletants – the ones that remain in the bulk phase acting as “true” depletants versus the ones that permeate and swell colloids. In consequence, colloidal assemblies that grow by absorbing depletants with temperature will simultaneously increase their Asakura and Oosawa attractive binding potential in a temperature-dependent form:

$$\frac{U_{dep}^{max}}{K_{B}T}=-{\varphi }_m\left(T\right)\cdot \left[1+\frac{3}{2}\left(\frac{R\left(T\right)}{R_m}\right)\right]$$ 1

where $U_{dep}^{\max }$ is the pair-wise depletion potential (maximum at surface-to-surface distance), $T$ is the absolute temperature, $k_B$ the Boltzmann constant, $R_m$ is the depletant size, ${\varphi }_m\left(T\right)$ and $R\left(T\right)$ are the temperature-dependent volume fraction of depletants and the colloid size, respectively (see Supp. Text).

We demonstrate that the reversible nature of an emerging dynamic depletion interaction (possible by interfacial depletant transport) enables colloids to expand (heating) or contract (cooling) while attraction is still active, leading to depletion-stabilized colloidal assemblies. Overall, the transport of absorbing depletants between phases is fundamental and perhaps unaccounted for in other dynamic systems where macromolecules are actively transported to the interior of semipermeable structures²⁰. Powerful molecular applications that exhibit dynamic behavior, conventionally involve the transport of small units that induce much larger periodic responses like ions permitting oscillations in chemical reactions^21–24. Here, molecular transport is translated into fine variations at the colloidal level, holding future promise to finely regulate the self-assembly of matter^25–27 and to tune photonic band gaps in colloidal crystals²⁸ with temperature.

Results

Micelles as absorbable depletants

Our colloidal system consists of a mixture of only two elements: polymers (Pluronic® L61) capable of forming micelles with changes in temperature²⁹ and semi-permeable colloids (3-(trimethoxysilyl) propyl methacrylate TPM colloids)^30–32 capable of absorbing polymeric micelles (See Methods, Materials). All mixtures of polymers and TPM colloids are highly stable under a wide variety of conditions (e.g., temperature and relative polymer-to-colloid volume fractions) in the absence of additional surfactants³³. Much of the colloidal stability stems from the particularly high charge density of silanol groups on the TPM colloids' surface (e.g., zeta potentials <−30 mV for pH values greater than 3.5; here we work neutrality pH~6.7)³⁴.

As in most absorption-driven chemical systems, a dynamic equilibrium exists at a given temperature, in this case, between micelles absorbed into colloids and those remaining in solution. Interestingly, the unabsorbed micelles in the bulk phase can act as micellar depletants, while the absorbed micelles increase the size of colloids as depicted in Fig. 1. The distribution of micelles between the two phases (Fig. 2a) can be approximated by combining classical calorimetry with an isotropic swelling of a polymer network in equilibrium using only the temperature and volume fractions of polymer $\left({\varphi }_p\right)$ and colloid $\left({\varphi }_d\right)$³⁵ as input parameters (Methods, Supp. Text, Figs. S1–S5).

Fig. 1. Schematic mechanism of depletant partitioning creating expanding colloidal assemblies.

a Micellable polymers and semipermeable colloids are mixed at volume fractions, ${\varphi }_p$ and ${\varphi }_d$, respectively. Polymers exist as free coils in equilibrium with colloids if mixed at a temperature $\left(T_0\right)$ below the onset micellization temperature of the polymer $\left(T_{om}\right)$. Initial colloidal size $\left(R_0\right)$ remains unchanged upon mixing with polymers –coils are not rigid enough to act as depletants and have little interaction with the colloids. b An increase in temperature $T_1$ above $T_{om}$ results in an entropic coil collapse in which free coils aggregate into micelles. These micelles are highly hydrophobic, which allows them to absorb into the colloids, resulting in colloidal swelling. Not all micelles will absorb into the colloids; they partition between the bulk phase and the colloids. Micelles in the bulk phase activate a dynamic depletion attraction, creating crystalline lattices that simultaneously grow over time. c A further increase in temperature $T_2$ above the end of polymer micellization transition $\left(T_{fm}\right)$ will preserve crystalline assemblies with no additional growth or energy reinforcement. Circular insets show bright field microscopy images of our colloidal mixture (TPM colloids and Pluronic® L61 polymers, see Methods), where TPM colloids grow and form an expanding crystalline assembly by partitioning of L61 depletants (micelles) when heating. Scale bars 10 µm.

Fig. 2. Thermal tuning of depletant partitioning.

a Temperature-dependent volume fraction of micelles $\left({\varphi }_m\right)$ (black) along with their partition between bulk (depletion, blue) and colloid (swelling, red) phases for a mixture of TPM colloids $\left({\varphi }_d=0.0025\right)$, and Pluronic® L61 polymers $\left({\varphi }_p=0.05\right)$. Estimations obtained by fitting calorimetry and optical microscopy data sets (See Supp. Info. Model, Figs. S1–S3). b Estimated increase in depletion attraction (black) upon combined increase in both colloid size (red) and depletant volume fraction (blue) when heating. Experimental data for the colloidal swelling ratio (mean (red solid circles), standard deviation (shaded red region)) used to validate predictions in (a and b). c Diffusion of semipermeable colloids at various relative polymer-to-colloid concentrations $\left({\varphi }^{pd}={\varphi }_p/{\varphi }_d\right)$ as function of temperature –normalized at 30 °C. A drop in diffusion is obtained during heating due to the accelerated colloidal expansion and viscosity increase below ~35 °C. A more typical behavior of diffusion increasing with temperature is obtained after ~35 °C. TPM colloid fraction $\left({\varphi }_d\right)$ varied while L61 fraction kept constant $\left({\varphi }_p=0.01\right)$. Solid lines represent predictions (See Supp Info) and circles experimental data sets at respective colors (mean (solid circles), standard deviation (shaded region).

Whereas the range of the depletion potential remains nearly constant (given by the depletant size thermal stability), its magnitude can now be dynamically modulated by adjusting micelle production with temperature (Fig. 2b). The magnitude of the attraction can be fully suppressed when only polymer coils are present and no micelles are formed (i.e., below the onset micellization temperature, $T_{om}$). For temperatures above $T_{om}$, polymer micelles will form and rapidly partition between the interior of the colloids and the bulk phase. Here, the overall polymer concentration becomes instrumental dictating the temperature dependence of the micellization transition (number of available micelles and their temperature range: onset, and end of transition)³⁵, where increments in polymer concentrations increase micelle production and shift the onset of micellization towards lower temperatures (e.g., $T_{om}$ decreases from 23 to 19 °C, when ${\varphi }_p$ increases from 0.01 to 0.05, Fig. S2).

The nature of the micellization transition is also highly dependent on the chemical nature and molecular weight of the polymer used as demonstrated for Pluronics L31³⁵. With just twice the molecular weight and the same PEO/PPO ratio, L61 creates ~25% more micelles than L31 at any temperature³⁵, and lowers the onset of the transition by ~10 °C (Fig. S2). This chemical modification is fundamental to creating strong depletion attraction at low temperatures and facilitating the formation of colloidal crystals.

The magnitude of the depletion potential gradually increases with micelle production and higher concentrations of unabsorbed micelles acting as depletants in the bulk phase during heating. The maximum magnitude of the depletion potential is delimited by the end of the micellization transition $\left(T_{fm}\right)$ and the tendency of depletants to aggregate into larger molecular structures (coacervates) in the medium³⁶, caused by the highly hydrophobic nature of depletants.

Surprisingly, the resulting change in colloidal size upon micelle absorption creates a somewhat counterintuitive effect in their translational diffusion (D) where colloids slow down when heating within a temperature region, ~30-35°C, Fig. 2c. In the absence of polymer, colloids diffuse proportional with increments in thermal energy (Stokes-Einstein), here, the size expansion rate of the colloids can surpass the thermal energy increment due to the large and asymmetric production of micelles being absorbed by colloids (Fig. S2, Supp. Text Eqs. S30-S36). Our diffusion measurements (Fig. 2c) were performed at low colloid volume fractions $\left({\varphi }_d\le 0.002\right)$ to better isolate the effect of micelle absorption from confinement and particle-particle interactions (Methods Section). Once the production and absorption of micelles balance the increments in temperature, the diffusion of colloids increases with temperature as conventional (non-expanding) systems (>35°C, Fig. 2c).

At a given temperature, the emerging depletion attraction between colloids is counterbalanced by the electrostatic repulsion coming from the colloids surface in the medium (See Supp. Text). The particularly high charge density on the surface of TPM droplets³⁴ avoids irreversible binding effects driven by van der Waals forces³⁷. Our colloids remain stable and reversible despite the electrostatic repulsive force decay, while the depletion force grows with increasing temperature (Eq. 1, Fig. 2b, Fig. S6). Adjusting the relative polymer-to-colloid concentration $\left({\varphi }^{pd}={\varphi }_p/{\varphi }_d\right)$ and temperature creates dynamic regimes, where the total colloidal interaction transitions from repulsive to attractive as shown in Fig. 3.

Fig. 3. Free energy model for fluid-crystal assembly at different polymer-to-colloid concentrations.

a The contour plot presents the predicted total free energy ($\frac{U_{total}}{k_{B}T}$) as a function of temperature and polymer-to-colloid concentrations $\left({\varphi }^{pd}\right)$ (Model Supp. Text). Regions of net attraction ($U_{total}<0$, red) favor crystallization and regions of net repulsion ($U_{total}>0$, blue) favor a fluid state. b Brightfield microscopy images show the corresponding experimental outcomes of varying ${\varphi }^{pd}$ while keeping L61 polymer fraction at ${\varphi }_p=0.05$. At moderate concentrations (${\varphi }^{pd}<20$), the system transitions from a fluid to an ordered crystal upon heating. Past ${\varphi }^{pd}>20$, the system undergoes coacervation while the system remains strongly attractive at low colloid concentration (crystallization is minimized in polymer-rich regions). Scale bars 10 μm.

The higher the concentration of depletants relative to the number of colloids $\left({\varphi }^{pd}>25\right)$, the higher the coacervation capacity of micellar depletants. Temperature cycling in such regions will, in principle, favor attraction with large variations of colloidal size. While extensive coacervation and low number density of colloids can hinder the formation of well-defined crystalline assemblies, we observe that crystallization can still occur under moderate conditions, although with slower kinetics and reduced structural uniformity (Fig. 3).

Increasing the concentration of colloids in the medium $\left({\varphi }^{pd}\approx 20\right)$, favors rapid absorption of depletants to effectively decrease micelle coacervation. This region represents the upper bound for colloidal assemblies with large variations in size and strong colloidal attraction. Further increase in the relative concentration of colloids $\left({\varphi }^{pd}<10\right)$ suppresses coacervation at the expense of minimal attractive forces. This region represents the lower bound for our colloidal assemblies with small variations in colloidal size (Fig. 3).

Lastly, the overall quality and rate of crystallization transition will arise from a balance between the total interaction potential ($U_{total}$) (Fig. 3) and the frequency (kinetics) of the binding interactions. The latter is controlled by colloidal diffusion (Fig. 2c) and colloidal concentration $\left({\varphi }_d\right)$. The number of colloids becomes particularly important to control the onset of crystal formation (Fig. S7, Movie S1). The total attractive energy required for colloidal crystal nucleation is typically below $~-3k_{B}T$, where colloids find enough attraction to overcome the nucleation barrier and time to create a stable nucleus. For energies above that energy level, various nuclei form, rapidly evolving into polycrystalline assemblies (moderate attraction) or gels (strong attraction).

In our colloidal platform, that attraction energy level $\left(~-3k_{B}T\right)$ is tunable by temperature and achieved at $~$36, 38, 43, and 49°C, for ${\varphi }^{pd}$ 25, 20, 10 and 5, respectively (Fig. S7). The kinetic effect is visualized in Movie S1, where the onset of crystallization is delayed by reducing the frequency of binding interactions (lowering diffusion and colloidal number density). There, nuclei can form almost instantaneously around the predicted temperature for a diffusive concentrated system $\left({\varphi }^{pd}=5,{\varphi }_d\approx 0.01\right)$, whereas a period of time is necessary for nuclei to emerge for a less diffusive dilute system $\left({\varphi }^{pd}=25,{\varphi }_d\approx 0.002\right)$.

Mapping crystal regimes

A range of equilibrium states can be achieved by selecting the directionality of the thermal stimulus (Fig. 4). During a standard heating (Fig. 4a), colloids expand while transitioning from a fluid-like to a crystal-like state (Movie S2). This transition is quantified using the hexatic bond order parameter (∣ψ∣)³⁸, where ∣ψ∣ ≥ 0.8 indicates robust hexatic order driven by emerging depletion attraction surpassing the electrostatic repulsion. Crystal formation is only possible after the colloids expand and diffuse long enough to break the nucleation barrier (Fig. 4a-III). Thus, the dynamic emergence of crystals is only visible after some time is allowed, if the energy state is low for a given temperature and polymer-to-colloid concentration.

Fig. 4. Time-dependence of crystallization during thermal ramping.

Temperature profiles for mixtures of TPM colloids and L61 polymers with corresponding brightfield and Voronoi tessellations colored by the local hexatic bond order parameter $\mid \psi \mid$. The color map quantifies local order, from disordered (yellow, low $\mid \psi \mid$) to crystalline (purple, high $\mid \psi \mid$). a During heating at 1 °C/min (${\varphi }_p=0.05$, ${\varphi }^{pd}=20$), the system transitions from a fluid to a crystalline state as the temperature-activated depletion attraction strengthens (Movie S2). b Upon cooling at 1 °C/min (${\varphi }_p=0.03$, ${\varphi }^{pd}=15$), the crystal persists but contracts as the colloids deswell, demonstrating the stability of the attractive state (Movie S3). c An isothermal hold at 55°C (${\varphi }_p=0.05$, ${\varphi }^{pd}=20$) shows that crystalline order improves over time, highlighting the kinetic nature of the process as particles require time to diffuse and anneal into a more ordered lattice (Movie S4). d An isothermal hold at a lower temperature of 35 °C (${\varphi }_p=0.05$, ${\varphi }^{pd}=20$), where attraction is insufficient to overcome thermal energy, results in the time-dependent dissolution of the crystal (Movie S5). Scale bars 10 μm.

During cooling (Fig. 4b, Movie S3), the depletion attraction still dominates over repulsive interactions, and with no nucleation barrier, colloids collectively contract in size resulting in progressively smaller “shrinking crystals”. Crystallinity also increases as temperature decreases (Fig. 4b-VI), likely due to a combined result from favored spatial rearrangement (colloid size reduction Fig. 4b), a depletion attraction (short-ranged) and a rapidly vanishing long-range contribution created by the polymer release (distant attraction Movie S3).

The effective increase in polymer concentration near colloids surface during cooling could create localized transient micelles (not micelle reabsorption observed leading to small increments of depletion attraction above equilibrium values) and polymer coil gradients radially spanning ~33 µm in 1 s $\left({⟨r^2⟩}^{1/2}=\sqrt{6Dt};D\approx 185{\mu m}^2/s;R_p\approx 1.5nm,T=30^{\mathrm{\circ }}\mathrm{C}\right)$. The non-equilibrium phoretic attraction and the possible transient depletion quickly decay as the polymer gradient vanishes over time bringing the system back to the binding in equilibrium (Fig. 3).

Interestingly, the nature of such a long-range attraction has also been identified in two-dimensional colloidal systems (our TPM colloids are sedimented on the capillary glass substrate forming 2D shrinking lattices during cooling), where surface flows are produced in proximity to a boundary wall, producing an effective hydrodynamic attraction to colloids in close proximity³⁹. The overall observed cooling response of our platform could create future opportunities to reorganize grain boundaries and heal defects with mild temperature reductions in preformed colloidal crystals.

The time dependence of the crystalline assembly also becomes particularly evident during isothermal holds. Assembled crystals kept at high temperatures (Fig. 4c, Movie S4), where both attractive forces and diffusion are high, stimulate crystal growth over time. However, isothermal holds at low temperatures lead to time-dependent melting (Fig. 4d, Movie S5). As time progresses, the depletion force is insufficient to counteract the thermal collisions of water molecules destabilizing the lattice, with crystal edges at the crystal-water interface being the start point for the propagation front of crystal melting.

Tuning colloidal lattice spacing

Temperature cycling creates a rich landscape of colloidal phases, where crystal size variations can only exist if attractive forces dominate the energy potential within the explored temperature range. Temperature variations between 25°C and 55°C reveal three distinct regions (Fig. 5b, Movie S6): (I) a region above a critical temperature $\left(T_c\right)$, defined as the temperature where depletion becomes dominant and controls the total interaction potential. Thermal cycling above $T_c$ will always preserve crystalline assemblies with small variations in colloidal size; (II) a region between $T_c$ and $T_{\min }$, where $T_{\min }$ corresponds to the temperature at which the total interaction potential becomes negative ($U_{total}<0$), marking the thermodynamic onset for depletion-driven crystallization. Thermal cycling in this region carries transient forces of comparable magnitude to the depletion attraction, creating crystalline assemblies that dissipate and renucleate with temperature variations; and (III) a region below $T_{\min }$ (Fig. S8), where depletion-driven crystals cannot form, but long-range attraction during cooling produces metastable crystals (Fig. 5a, b, Movie S6). Interestingly, the nucleation rate of the expanding crystals can still be modulated by adjusting heating ramps, where crystal size increases at lower ramps (Fig. 5c).

Fig. 5. Thermal Modulation reveals distinct 2D colloidal phases.

a Schematic of a system with one depletant that can absorb and partition between the colloidal phase and solution. b Correspondent thermal cycling along with regions of interest that depict distinct colloidal responses (${\varphi }_p=0.05$, ${\varphi }^{pd}=20$) (Movie S6). Temperature cycling in Region I (above $T_c$) exhibits crystalline lattice variations. Cooling in Region 2 shrinks crystals while heating melts crystals. Crystals can be reformed over isothermal thermal holds over time. Temperature cycling in Region III (below $T_{\min }$ lts crystals upon heating and shrinks crystals while cooling yet crystals are not formed with isothermal thermal holds over time. c Brightfield images of final crystal achieved by either doing a slow or fast ramp. Slowing the ramp (i.e., increasing nucleation time) increases the chance of obtaining a uniform 2D colloidal crystal. d Brightfields images show small variations (~1%) in colloidal size (pristine crystalline change) since temperature variations (above $T_c$) occur past the point of bulk swelling, (Movie S7). Predictions of total free energy (blue line) and colloid size (prediction (purple line); experimentally measured mean (purple solid circles) and purple error bars) for a temperature protocol (dashed orange) (${\varphi }_p=0.05$, ${\varphi }^{pd}=20$), permit small lattice variations for energies above $~-8k_{B}T$ and $T_c\approx 43^{\circ }C$. e) Crystal dissolution within the time-dependent region with a slight increase in temperature. Brightfields show hysteresis on how crystals dissipate with an increase in temperature and quickly recrystallize (Movie S8). Scale bars 10 µm.

Crystal lattice variations are only possible within region I $\left({T>T}_c\right)$. These colloidal systems offer a pristine amplitude (Fig. 5d), provided by the strategic selection of polymer (micellization range) and colloids (micelle absorption affinity). Unfortunately, this region is above the point at which most of the colloidal swelling has already occurred, resulting in a modest variation of the lattice parameter of ~1%.

Although fluid-to-solid transitions are relatively well explored in attractive colloids, our transient energy landscape found in region II $\left(T_{\min }<{T<T}_c\right)$ creates a hysteretic behavior between states (Fig. 5e, Movie S8). There, non-equilibrium contributions coming from absorption/desorption of polymers during temperature cycling seem to create transient concentration gradients capable of driving long range forces^40–42. Such nonequilibrium interactions become more evident in thermal intervals where micelle production is accelerated (Fig. S2, Region II Fig. 5e) and the collective magnitude seem to surpass the short-ranged depletion attraction in equilibrium, at least for short periods of time (Movie S8). The observed asymmetry in binding response is likely due to the directionality of the concentration gradients, where polymer release during cooling drives a phoretic attraction (with possible transient depletion strengthening), whereas micelle influx during heating creates an osmotic repulsion (with possible transient depletion weakening). Both, long-range and transient short-range contributions rapidly disappear as polymer gradients vanish from the colloid surface reaching the bulk concentration in equilibrium (Fig. 3).

In consequence, reheating a crystal from ${T<T}_c$ causes a uniform melting, characterized by directed non-Brownian colloidal motion away from crystallite centers (Figure S9). A similar behavior has also been observed in active colloidal systems where osmotic flows are produced if molecular flows occur in proximity of colloids lacking no-slip surfaces⁴³. A crystalline phase can reemerge when the sample is held isothermally in this region (II) after reheating since the depletion attraction is high enough to dominate the potential (~5 $k_{B}T$) in the absence of polymer gradients that rapidly disappear over time (Movie S9).

More interestingly, crystals can rapidly reform from the melt with rapid cooling (no isothermal step needed) within region II (Fig. 5e, Movie S8). The rapid release of polymers reverses the direction of long-range forces, becoming attractive and adding to the depletion force (although decreasing in magnitude) to drive the formation of a new crystal. In region III $\left(T<T_{\min }\right)$ depletion forces are weak (~1.1 $k_{B}T$), preventing isothermal colloidal crystallization. However, rapid cooling still induces crystallization due to the same gradient-driven colloidal attraction created by the substantial and asymmetric release of polymer coils near the wall still present at low temperatures (Movie S6).

Amplifying lattice parameter variations

Adding an “amplifier” can substantially extend the amplitude of the colloidal lattice variation (Fig. 6a, Movie S10). We select a polymer (Pluronic® F108, see Methods) with a similar micellization capacity with temperature but decreased affinity towards colloids (larger molecular weight and higher hydrophilic character). Thus, any additional micelles formed will remain in solution (non-absorbing depleting micelles), effectively increasing the attractive depletion potential. This guarantees the presence of colloidal crystals at low temperatures, where the depletant absorption is maximum.

Fig. 6. Amplified tuning of colloidal lattice parameter.

a A system with two depletants: one depletant that can absorb and partition between the colloidal phase and solution, and one that only remains in solution. Correspondent thermal cycling will amplify variations on lattice spacing by accessing the bulk of the colloid swelling due to the additional entropic reinforcement provided by the non-absorbing depletant at lower temperatures (Movie S10). b Brightfield images of a colloidal crystal undergoing a thermal cycle along with corresponding phase/orientation Voronoi diagrams. The phase remains closely preserved during cycling. Scale bars 50 µm. Inset scale bars 25 µm. (Movie S10). c Macroscopic view of crystal expansion/contraction in a capillary tube. Crystal boundaries return to initial positions. Scale bars 2 mm. d Change in lattice parameter versus temperature of the system in panel b (experimentally measured mean (purple solid circles) and standard deviation (shaded purple region); prediction (purple line)). Thermal tuning of the lattice parameter $\left(a\right)$ reversibly modulates a 25% change in the original interplanar distance $\left(a_0\right)$. Estimated photonic band gap thermal shifting of a monocrystal with the same lattice dimensional change (shaded red region: top edge in dashed red line, bottom edge in dashed blue line).

Additional energy effects are obtained if the amplifier (non-absorbing polymer) can additionally adsorb on the colloidal surface (Fig. S10). In such cases, the colloidal electrostatic repulsion can be decreased⁴⁴, but the added steric stability obtained on the colloids’ surface becomes valuable to preserve the functionality of the lattice-tunable system, especially in the presence of electrolytes that typically drive instabilities (e.g., coalescence, colloid binding) by screening surface charges.

Maximizing the depletion force with the use of a depletion amplifier (non-absorbing depletant) effectively removes the effects of long-range forces and extends the range of action for crystalline size variations at any temperature (Fig. 6a). Our model (Supp. Text) of an amplified lattice-tunable colloidal crystal allows for lattice variations of ~25% the reference size, within a 20°C temperature window with preserved crystalline integrity (arg(ψ), Fig. 6b–d, Movie S10). This phenomenon is strikingly evident on a macroscopic scale, where heating and cooling cycles produce highly reversible swelling and deswelling of crystalline domains (Fig. 6c).

Increasing the bulk depletant level further pushes the attraction beyond the crystallization window, and the assembly percolates into a polycrystalline gel (Fig. S11), a sample-spanning network built from locally hexagonal strands. Notably, this gel still undergoes reversible, temperature-driven reconfiguration, providing an alternative “size-tunable colloidal gel” regime complementary to the crystalline lattice modulation in Fig. 6 (Movie S11).

Leveraging the exceptional control and properties of this fluid colloidal platform holds unique promise to tune photonic bandgaps via temperature modulation. For instance, estimates of an inverse colloidal crystal with lattice-dimensional variations similar to those shown in Fig. 6b could effectively modulate the forbidden propagation of light for wavelengths ranging from 5.7 to 7.4µm (Fig. 6d, Fig. S12). Ongoing technological trends in colloidal lattice fabrication^12,45 and colloidal size control^33,46 offer unique avenues to facilitate requirements for optical properties of the building blocks (e.g., low dielectric constant) and shift the modulation capacity within the visible range of the spectrum (Fig. S13).

Discussion

Colloidal systems offer foundational elements to understand complex processes in both life and physical sciences. Here, we revealed how an emerging entropic action can induce reversible and continuous size variations in soft matter assemblies. The partial absorption of depletants (polymeric micelles) in equilibrium across a semipermeable barrier (mesoscopic colloids), can create a transient depletion force that forms crystals that simultaneously expand and contract upon thermal stimulation. Crystalline assemblies that follow this approach can modulate their interplanar distance with mesoscopic resolution by solely adjusting temperature. The lattice spacing variation can be amplified by reinforcing the assembly with non-absorbing depletants. This effect permits the formation of crystalline assemblies at lower temperatures before the bulk of the colloidal expansion occurs. The overall thermal modulation and entropic reinforcement of the colloidal platform create a finely controllable route to regulate growth in size-changing colloidal superlattices – an aspect of unique interest to create future tailored interactions in colloidal-based materials. The reach of this study is also of considerable interest to biological and materials sciences, where dynamic consumption or emergence of components is typical and could induce unaccounted-for depletion.

Methods

Materials

PEG-PPG-PEG (Pluronic® L-61, Mₙ ∼ 2000 Da), PEG-PPG-PEG (Pluronic® F-108, Mₙ ∼ 14,600 Da), and sodium chloride (NaCl, ≥99%) were purchased from Sigma-Aldrich and used as received. 3-(trimethoxysilyl) propyl methacrylate (TPM, ≥97%) and ammonium hydroxide (28% NH₃ in H₂O) were also purchased from Sigma-Aldrich and stored at 4°C. All solutions were prepared with ultrapure water (18.2 MΩ·cm resistivity, ≤5 ppb TOC) from an ELGA Purelab® Flex 3 system.

Semipermeable Colloid Synthesis

Monodisperse TPM colloids were prepared via a one-pot reaction. In a 1 L glass beaker, 3 mL of TPM monomer was added to 100 mL of ultrapure water containing 75 µL of 28% ammonium hydroxide. The solution was agitated with a 1” stir bar at ∼200 RPM for 60 minutes. This process typically yielded semipermeable colloids with a radius in the range of 1–2 µm, with the final size outside of this range being dependent on the initial TPM monomer concentration and the base concentration.

Formulation of colloidal mixtures: semipermeable colloids + micellable polymers

Aqueous stock solutions of 10% (v/v) Pluronic® L-61 and 10% (v/v) Pluronic® F-108 were prepared. For experiments, these stocks were mixed with the TPM colloids and ultrapure water in 1.5 mL microcentrifuge tubes for a total sample volume of 1 mL, to achieve the desired final concentrations. Polymer-to-colloid volume fractions $\left({\varphi }^{pd}={\varphi }_p/{\varphi }_d\right)$ ranging from 4 to 25 were used, with total polymer concentrations $\left({\varphi }_p\right)$ typically held constant at 1-5% (v/v). When required, the swelling agent (Pluronic® L-61) was added to the TPM colloids first, followed by the steric stabilizer (Pluronic® F-108). If salt was used as a screening agent, it was added last to prevent premature coalescence. After each addition, solutions were homogenized for ∼5 minutes at 25°C and 350 RPM using an Eppendorf® ThermoMixer C.

Microscopy sample preparation and thermal control

The formulated colloidal mixtures were loaded into glass capillaries (VitroCom; 25.0 mm length, 0.1 mm path length, 2.0 mm width), and the ends were sealed with UV-curable resin to prevent evaporation. The sealed capillary was mounted onto a glass slide, which was then placed on a conductive indium tin oxide (ITO) coated slide (Sigma-Aldrich; 15-25 Ω/sq) within a custom-built thermal chamber on a Nikon Eclipse Ti2 inverted microscope. Prior to imaging, the colloids were allowed to sediment. For experiments on 2D systems (e.g., Fig. 5), dilute samples were used, resulting in the formation of a quasi-two-dimensional monolayer at the bottom of the capillary. For experiments requiring 3D crystals (e.g., Fig. 6), higher particle concentrations were used to form multilayered, three-dimensional structures. Temperature was monitored with a 100 Ω resistive temperature sensor (TH100PT) placed centrally beneath the capillary. The entire system was controlled by a THORLABS TC200 temperature controller, which was programmed using a Raspberry Pi to execute precise heating/cooling ramps (1–5 °C/min) and isothermal holds. Temperature datasets are reported for temperatures above room temperature (e.g., 30-60 °C) to guarantee $\pm$0.1°C temperature control. A schematic of the experimental setup is provided in Fig. S5.

Colloid size measurement: image acquisition and particle segmentation

Automated bright-field imaging was performed under 10, 20, 40, or 60x air-air objectives. Perfect focus and automatic light adjustment were employed throughout the duration of imaging. After imaging, the TPM size was analyzed using FIJI⁴⁷. To generate masks of the particles to calculate centroid locations, an advanced segmentation algorithm was required. Stardist 2D⁴⁸ and LABKIT⁴⁹ were employed with an in-house data set for training to perform the segmentation. To improve segmentation results, the in-house data set was artificially expanded by using UTILE-Gen⁵⁰. The results were then analyzed using the built-in analyze particles function of ImageJ to get the resulting ferret diameter, centroid, and occupied area. If particle tracking was needed, two frames were masked over a known time and compared using TrackMate⁵¹.

Crystallinity analysis

To determine crystallinity, Voronoi Tessellation and Delaunay Triangulation were performed, given the centroid locations of each particle. A neighbor was only classified as a nearest neighbor if and only if the midpoint of the Delaunay Triangulation edge lay on the corresponding Voronoi edge. An example of how these masks are generated is shown in Fig. S14. The local hexatic bond orientation parameter, $∣{\psi }_k∣$, could then be calculated using³⁸:

$$∣{\psi }_k∣=\frac{1}{N_j}\cdot {\sum }_{j=1}^{N_j}{e}^{{6\cdot i\cdot \theta }_{jk}}$$ 2

where N_j is the number of nearest neighbors to particle k and θ_jk is the angle between the x-axis and particle k with one of its nearest neighbors. A $∣{\psi }_k∣$ value of 1 indicates perfect order. To determine the orientation or the phase of a particle, the argument of ψ_k was taken²³.

Colloidal diffusion measurements

For diffusion measurements, samples were prepared under dilute conditions to ensure that colloids remained freely diffusing throughout the duration of the recordings. Samples were formulated at a fixed polymer concentration $\left({\varphi }_p=0.01\right)$ at four relative polymer-to-colloid concentrations, ${\varphi }^{pd}$ = 5, 10, 20 and 25. The resulting colloid volume fractions, ${\varphi }_d=$ 0.002, 0.001, 0.0005 and 0.0004 are lower (five-fold) than those used in experiments where crystallization is observed (e.g., ${\varphi }_p$ = 0.05, ${\varphi }_d$ = 0.01). Thus, colloids remained well separated and particle trajectories reflected single-particle diffusion without contributions from crowding or collective effects as swelling occurred during heating. Samples underwent a temperature ramp from 30 °C to 60 °C. At 5 °C intervals, and additionally at 32°C, the system was held at a constant temperature to reach thermal equilibrium. At each isothermal step, 30-second videos were recorded at 5 frames per second, with an exposure time of 50 ms. Particle trajectories were extracted after particle segmentation using the TrackMate plugin in Fiji. The workflow employed a Laplacian of Gaussian (LoG) detector to identify particles and a Linear Assignment Problem (LAP) tracker to link them between frames. The resulting trajectory data were analyzed using custom Python scripts to calculate the ensemble Mean Squared Displacement (MSD). The translational diffusion coefficient, D, was determined by a linear fit to the initial portion of the MSD curve using the 2D Einstein-Smoluchowski’s relation, MSD(t) = 4Dt. For modeling of the diffusion measurements, see Supp Info. Translational Diffusion.

Differential scanning calorimetry

To determine the critical micelle temperature (CMT) for the Pluronics^®, a TA Instruments Differential Scanning Calorimeter (DSC) Q2000 was employed. A direct ramping method of 1 °C/min was used over an interval of 0-70°C with TZero Hermetic Lids and Tzero Pans, each supplied by TA Instruments. The CMT was calculated by taking the peak of the heat capacity curve for four different polymer concentrations (Pluronic® L-61, ${\varphi }_p=$ 0.01, 0.02, 0.03, and 0.05). The energy of micellization was calculated by integrating the area under the heat capacity curve after baseline subtraction (further details in the Supp. Text).

Dynamic light scattering & zeta potential

The initial size for Pluronic^® micelles were determined by utilizing a Malvern Panalytical Zetasizer Nano ZS instrument. Samples were diluted to their required concentrations and placed in plastic cuvettes with 1 cm pathlength. Hydrodynamic measurements were performed with three total replications, each containing 11 runs. Electrophoretic mobility measurements were carried out in a DTS-1070 folded capillary cell supplied by Malvern Panalytical, utilizing a Malvern Panalytical Zetasizer Nano ZS instrument with a 633 nm red laser. Samples remained undiluted to maintain Pluronic^® concentrations and were flushed through the folded capillary cell, with the measurement being carried out on the third filling. Measurements were only carried out if there was good wetting/contact and there were no inclusions or air bubbles in the cell. Measurements were performed at a set temperature interval, with three replications, each containing 11 runs. Before each measurement, samples were left to equilibrate at their desired temperatures for 90 s. Zeta Potential was then calculated using Smoluchowski’s approximation.

Photonic bandgap for size-changing crystals

To demonstrate the thermal dependence of photonic bandgaps we selected a classical system with relative permittivity ($\epsilon$) = 13 assembled following the irreducible Brillouin zone of a face-centered cubic (FCC) crystalline system. We used the inverse lattice configuration (inverse opal) to calculate the photonic band diagram for various wave vectors using COMSOL. The frequencies of the eigenstates were calculated as a function of the wave vector $\left(k\right)$ in the range of 0.0 <$\frac{a}{\lambda }$ < 1.0. The width of the bandgap (lower gap edge in W direction and upper gap edge in the X direction) approximately centered at ($\frac{a}{\lambda }\approx 0.794$) is connected with the thermal modulation of lattice parameter ($a$) as a function of colloid size $\left(R\right)$, for an FCC lattice $a\left(T\right)=\sqrt{8}R\left(T\right)$. Changes in temperature effectively modify the lattice parameter ($a$) by adjusting colloidal size due to the relative partition of micelles inside colloids and the medium. The presence of polymeric micelles is assumed to minimally contribute to relative permittivity.

Author contributions

T.F.N, L.G.T and J.A.D.A conceptualized the research. L.G.T and T.F.N designed the methodology. T.F.N and L.G.T performed the research. T.F.N, L.G.T and J.A.D.A wrote the manuscript. All authors read, discussed, and approved the content of the manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Funding

J.A.D.A, T.F.N, and L.G.T were supported by the Defense Advanced Research Projects Agency (DARPA) - Young Investigator Award (YFA) D24AP00324.

Data availability

All data generated in this study are available in the Supplementary Information and Source Data file. Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-75071-z.