A new preprint predicts that temperature-sensitive microgel particles may shrink faster than they swell after a sudden temperature change. In the modeled jump scenarios, the particles reached 90% of their total radius change in 1.609 milliseconds while shrinking, compared with 3.382 milliseconds while swelling. The result comes from equations and numerical calculations, not a direct time-resolved experiment, so it describes what the model predicts rather than what has yet been shown in real particles.
The work focuses on pNIPAM microgels, spherical particles made from a temperature-responsive polymer. The theory treats their size as a consequence of solvent moving into or out of the polymer network. It was built to describe both swelling and deswelling away from equilibrium, including cases in which the surrounding temperature does not change instantaneously and cases in which the particle’s size fluctuates during the response.
A model built around solvent flow
The authors describe the motion through a solvent-flux formulation: the particle expands or contracts as water is transported through its polymer network. The model uses Flory–Rehner free energy to represent the osmotic contribution, meaning the part of the system’s thermodynamics associated with solvent and polymer mixing. It also allows the transport resistance to depend on the particle’s state and temperature, rather than treating mobility as fixed throughout the process.
Near equilibrium, the equations recover exponential relaxation of the radius and the expected quadratic dependence of relaxation time on particle size. In that regime, the model connects the usual relaxation parameter γ to solvent transport through the polymer network. This gives the framework a link to the familiar Tanaka–Fillmore description while extending the calculation to larger, nonequilibrium changes.
For its applied parameterization, the study used pNIPAM microgels synthesized in Milli-Q water by precipitation polymerization, with a crosslinking density of 4.8 mol%. Dynamic light scattering measurements supplied equilibrium hydrodynamic radii of 345 ± 25 nanometres at 15 °C in the swollen state and 200 ± 10 nanometres at 48 °C in the collapsed state. The calibration also imposed a collapsed-state bulk modulus of 200 kilopascals, a stiffness parameter used in the model.
The particle’s mobility changes as it changes size
One feature of the calculation is that the modeled swelling diffusion coefficient was not monotonic with radius. Instead, the calculated mobility rose and fell, reaching a maximum at a radius of 262.94 nanometres and a polymer volume fraction of 0.22. In plain terms, the model does not assume that a particle transports solvent most efficiently when it is simply at its largest or smallest; the most mobile state appears in between.
That state dependence helps produce different time courses for the two directions of motion. In the sudden-jump calculations, deswelling had the shorter modeled τ90, the time required to complete 90% of the total radius change. The comparison was 1.609 milliseconds for deswelling and 3.382 milliseconds for swelling. The study also found that the characteristic relaxation time followed a quadratic size dependence in both directions, so the modeled response became longer as particle size increased.
The calculations also examined finite thermalization, meaning the temperature change was allowed to settle over a measurable interval instead of being treated as instantaneous. This lets the model compare the microgel’s own relaxation with the time imposed by the surrounding temperature protocol. The distinction becomes important whenever the particle is changing size while the thermal environment is still moving toward its new state.
When heating or cooling takes time, the path changes
The modeled response shifted between two regimes. When thermalization was fast, the relaxation time approached an intrinsic plateau set by the microgel’s own dynamics. When thermalization was slow, the overall response became controlled by thermalization and increased linearly with the thermalization time. The result means that a measured swelling time could reflect both transport inside the particle and the pace at which the temperature change reaches it.
The same finite-temperature protocols produced loops when radius was plotted against temperature. The loops were smaller in the quasi-static limit, when the imposed temperature variation was slow, and larger when the temperature changed faster. These are dynamic nonequilibrium trajectories: the radius does not follow the same path on the way toward a new state as it does while returning, because the particle is temporarily out of equilibrium with the changing temperature.
The authors present this crossover as part of a framework for interpreting time-dependent microgel measurements. It shows why an apparent relaxation time need not belong entirely to the particle: part of the delay can come from the temperature environment itself. Whether experiments can reliably separate those two contributions remains an open question identified by the study.
A brief increase in size noise during collapse
The study also used a numerical solution of the Smoluchowski equation, a probability equation for how the particle’s radius is distributed over time, to examine stochastic size fluctuations during collapse. In that calculation, the modeled standard deviation of the radius increased from 0.456 nanometres to a maximum of 1.041 nanometres at 11.5 microseconds, when the mean radius was about 24.9 nanometres. It later narrowed to 0.172 nanometres.
The fluctuation calculation used a time step of 10^-8 milliseconds, a spatial grid spacing of 0.02 nanometres and zero-flux boundaries at radii of 17 and 40 nanometres. Those details define the numerical demonstration rather than a measurement range for every pNIPAM particle. The stochastic example also used a size-reduced microgel, which limits how directly its fluctuation values can be transferred to the larger parameterized particle.
What remains to be tested
The evidence is a theoretical framework, a calibrated pNIPAM parameterization, deterministic numerical integrations and a scaled stochastic calculation. No participant sample is involved: the modeled system is a single spherical microgel, and the supplied analysis reports no inferential statistical tests or uncertainty propagation for the dynamic predictions. The work therefore provides model-based predictions rather than direct evidence of how a measured population of particles behaves.
The model represents a single spherical, spatially homogeneous microgel and uses affine swelling assumptions. Water transport is treated with an incompressible, quasi-static approximation and constant solvent number density. Several important quantities, including the effective chain number and bulk modulus, are calibrated or imposed rather than independently measured in this work. These choices make the equations tractable but leave open how sensitive the predictions are to departures from the assumed structure and parameter values.
A key experimental test would be time-resolved dynamic light scattering or single-particle tracking during a rapid temperature jump, looking for the predicted temporary maximum in radius fluctuations. Experiments would also need to determine how reliably intrinsic microgel relaxation can be separated from external thermalization. The supplied analysis identifies core–shell, charged and concentrated interacting microgels as possible extensions, but it does not establish that the present predictions apply to those systems.
The document is an arXiv version 1 preprint posted on 20 August 2026. No journal publication or peer-review status is reported in the supplied material. The authors acknowledge grants PID2022-136540NB-I00 and W911NF-23-1-0099, along with high-performance computing support from PROTEUS and C3UPO.
Paper data and sources
Original title: A solvent-flux theory for nonequilibrium swelling dynamics of thermoresponsive microgels
Authors: Arturo Moncho-Jordá, Alessandro Patti, Fabián A. García-Daza, Alejandro Cuetos
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text