Spontaneous shrinking of soft nanoparticles boosts their diffusion in confined media

Improving nanoparticles (NPs) transport across biological barriers is a significant challenge that could be addressed through understanding NPs diffusion in dense and confined media. Here, we report the ability of soft NPs to shrink in confined environments, therefore boosting their diffusion compared to hard, non-deformable particles. We demonstrate this behavior by embedding microgel NPs in agarose gels. The origin of the shrinking appears to be related to the overlap of the electrostatic double layers (EDL) surrounding the NPs and the agarose fibres. Indeed, it is shown that screening the EDL interactions, by increasing the ionic strength of the medium, prevents the soft particle shrinkage. The shrunken NPs diffuse up to 2 orders of magnitude faster in agarose gel than their hard NP counterparts. These findings provide valuable insights on the role of long range interactions on soft NPs dynamics in crowded environments, and help rationalize the design of more efficient NP-based transport systems.


Supplementary
. DDM measurements of for soft NPs in agarose gels: A) C ag = 0.05%, B) C ag = 0.1% and C) C ag = 0.5% w/w. Large soft NPs (r 0 H = 92 nm) were not diffusing in agarose at C ag > 0.5%, and were not shown in panel C but were replaced by small soft NPs (r 0 H = 25 nm). Lines in red, blue, teal and purple are NPs with r 0 H = 25, 40, 50 and 92 nm respectively. Figure 4. Stretch exponent of the ISF for hard and soft NPs in water (gray), agarose C ag = 0.1% (green), 0.5 % (red) and 1 % (black). Figure 5. Representation of the rheological properties of the agarose solutions and gels in pure water at ambient temperature (T≈22 o C). Squares represent the storage modulus (G') and circles represent the loss modulus (G") for each agarose concentrations. Storage modulus data for 0.05% agarose was not shown in this graph because the values were below instruments detection limit. Here, q is normalized by the dilute radius r 0 H of the particle and the peak structure factor would be expected to be located at qr 0 H ≈ 1 which does not appear in this graph.

Supplementary Tables
Supplementary Table 1

Rheological measurements
The rheological properties of the agarose hydrogels were measured under oscillatory shear using Osmotic pressure of agarose was determined by incubating agarose gels (1% w/v) in Ficoll solutions with various concentration / osmotic pressure. Agarose gels were covered with a dialysis membrane of 15 kDa molecular weight to create the pressure on the gels. Agarose was periodically weighted on an analytical balance to screen water gain / loss. The osmotic pressure was determined at the Ficoll concentration at which the volume of agarose does not change, hence the applied external osmotic pressure Π ext , is in equilibrium with the internal osmotic pressure of the agarose.
mechanical properties were at the steady state, a frequency sweep test was performed at the appropriate strains for each sample, following by a strain sweep test at 10 rad s -1 frequency.

Structure factor of soft NPs suspension at ϕ
The structure factor of microgels was measured using the DeGennes narrowing method. Diffusion was measured with DDM and DLS at different q values and normalized with D 0 in the dilute regime (ϕ0), so that τ R (q,ϕ)/τ R (q,ϕ0) ~ D 0 /D(q) ~ S(q) allow to access for the structure factor S(q).
Using microgels = 92 nm, we get S(q) ≈ 1 which is invariant over the studied q range.
DLS data were obtained with an ALV instrument (CGS-3 goniometer and LSE-5004 correlator) as a function of the scattering angle, using a 633 nm laser, to confirm that the increase in diffusion coefficients (interpreted as particle shrinkage in Fig. 2) is not due to a structural effect. Experiments . The exponent β was found to be close to 1.

Volume fraction determination of NP suspensions
PS (hard NPs) were compared to microgels (soft NPs) at similar volume fractions. Because microgels can swell or shrink, calculations of ϕ was different for microgels and hard NPs. For hard NPs, we used the following equation to describe ϕ: (1) And for microgels, we used the approximation provided by Scotti et al. as it is valid for low solid volume fractions 1 : , Where w i is the mass fraction of the NP, ρ solv is the density of the solvent, ρ NP or ρ μGel is the density of microgels. For microgels, the term is added to consider the swelling of microgels.
R swollen is the radius of the swollen microgel and R collapsed is the radius of the microgel shrunk with temperature as measured by DLS.

Supplementary Note 1 -Examples of DDM measurements in agarose gel.
Intermediate scattering function (ISF), f(q,τ), shows the dynamics of the particles and can be extracted from the DDM autocorrelation functions, g(q,τ), as expressed in Eq.1 . Extracting f(q,τ) and applying corrections for sometimes a slow thermal drift in the gel using Eq.11 yields the ISFs reported in Supplementary Fig. 1. All the observed dynamics fits the generalized exponential equation , / with the stretched exponent β close to 1 in water, which indicates NPs are not interacting with each other or the media and polydispersity is not significant, and 0.7 < β < 1 in agarose solutions and gels (with few exceptions of β ≈ 0.5-0.6), highlighting particle-gel interactions depending on the NPs type and size (see Supplementary Fig. 4).
After filling a capillary with agarose gel, we noticed a slow gelation process affecting the particle dynamics. To avoid the effect of slow relaxation, we performed DDM experiments as a function of the waiting time, ranging from 15 minutes to 3 days, with t=0 the time at which the capillary was filled with a solution. We found that a resting time of 16h was necessary prior to any DDM measurement in order to reach data at the steady state in the agarose gel. This is expressed in Supplementary Fig. 2 (B, D, F and H), where the diffusion coefficient of hard NPs is independent of time after a relaxation period of 16h. We also show the relaxation time τ R approximately follows q -2 . Similar results were found for soft NPs as well ( Supplementary Fig. 3).

Supplementary Note 2 -Rheological measurements
Agarose was found to display an apparent Newtonian behavior at very low concentrations (0.05%) as shown by the power-law increase in the loss modulus, G", with frequency in Supplementary   Fig. 5. Above this concentration, agarose demonstrates a non-Newtonian behavior at the studied frequency range because of interchain interactions. An expected sol-gel transition occurs close to C ag = 0.1% characterized by the storage and loss moduli following a power-law scaling of the form G' ~ G" ~ ω n , where n is a critical relaxation exponent. The sol-gel transition occurs at an agarose concentration between 0.1 and 0.5% evidenced by the independency of G' to ω. The storage moduli of these gels are 330 ± 10 kPa and 1890 ± 220 kPa for C ag = 0.5% and 1% agarose, respectively.

Supplementary Note 3 -Effect of the osmotic pressure on microgel diffusion
We investigated the soft NP shrinkage as a function of osmotic pressure by adding Ficoll 400 kDa to solution of NP microgels and evaluate the minimum pressure required to shrink the microgels to half their swollen size, as it was observed in agarose ( Figure 2).
We measured the diffusion coefficients of soft and hard NPs using DDM and the osmotic pressure As Ficoll concentration increases both osmotic pressure and viscosity increase. However, we found that the ratio decreases as the Ficoll concentration is increased (inset of Supplementary Fig. 6C) suggesting that the microgel particle shrinks, i.e.
decreases. The viscosity of the solution can be estimated by using the Stokes-Einstein relation and considering the hard PS particle does not change size so that This allows extraction of the microgel radius r H as a function of Ficoll concentration from solely measured quantities using the following expression Evolution of r H as a function of the measured osmotic pressure confirmed that particles are shrinking as the osmotic pressure is increased. We found that the smallest size attainable by the microgel (r 0 H = 92.5 nm in Ficoll is r H ≈ 60 nm, which is obtained at a concentration of 50 mg mL -1 Ficoll. This concentration yields an osmotic pressure Π osm = 9.1 kPa (see Supplementary Fig. 6A) which shrinks the microgels similarly to other stimuli such as temperature and concentration ( Supplementary Fig. 6D). The measured thermal shrinking observed in this study is consistent with other reports 2, 3 .
The osmotic pressure of agarose was calculated from its swelling / deswelling in Ficoll solutions.
The concentration of Ficoll at which the agarose gels (C ag = 1% w/w) does not swell / deswell (ΔV = 0) was estimated by a linear fit ( Supplementary Fig. 6B). At this point, the internal osmotic pressure, including the elastic and mixing osmotic pressures, is equivalent to the external osmotic pressure. Therefore, the agarose gel (C ag = 1% w/w) has an osmotic pressure of Π osm = 89 Pa equivalent to 3.1 mg mL -1 Ficoll, which is too weak to induce any microgel shrinkage.

Supplementary Note 4 -Zeta potential of hard and soft NPs
Zeta potential was assessed by measuring the electrophoretic mobility of the nanoparticles using DLS for NPs in 4 mM NaCl solutions. Values are presented in Supplementary Table 1 for a range of size for both hard AuNPs and soft microgels at two different temperatures.

Supplementary Note 5 -Dynamics in saline solutions
We investigated the effect of adding salt on the diffusion of the NPs in agarose gel at C ag = 0.5 % w/w. A larger mesh size of the gel is expected with higher ionic strength solutions 4 . Generally, those ions contribute in reducing water-polymer interactions consequently promoting polymerpolymer interactions and leading to larger chain aggregates as well as pores 5,6 . With bigger pores, we expect no overlap of the EDL and thus no microgel shrinking. Supplementary Fig. 7 shows the DDM-measured diffusion coefficient D G of hard Au NPs (two sizes) and soft microgel particle (one size). With salt addition, we found D G values of the soft particles agree well with the prediction of the Kang et al. model for the hard particles suggesting that the soft microgel particle do not shrink.
Using Eq. 4 to calculate the size r H of microgels, we obtained a size that corresponds to the fully swollen microgels r 0 H observed in the diluted regime.

Supplementary Note 6 -Calculations of the agarose mesh size and interaction distance H
The mesh size l was determined based on the first hard immobilized particle in agarose gels.
Because softer gels are not dense enough to trap NPs of 220 nm in diameter, we used the expression * / where L is the length of the fiber, C* ag the overlap concentration of agarose and C ag the concentration of agarose. This approach allows to estimate the mesh size l for C ag 0.25% w/w using the mesh size of agarose 0.5% as a reference to evaluate C* ag = 0.097% w/w. Calculated values of l in Table S3 are in good agreement with previously published values. 7 In agarose gels, the fiber-particle distance H is obtained using the following expression:

Supplementary Note 7 -Elastic energy of a spherical particle
The elastic energy U of a soft spherical particle of elastic modulus E is described by its elastic potential energy per unit volume U v = 1/2Eξ² with ξ being the strain on the particle so that we have (7) and the strain , where R 0 is the initial radius of the particle and r is the shrinked layer of the particle. The potential energy required to compress a soft sphere to a fraction α of its original size (α being the shrinkage ratio of the particle) is given by: Therefore, for a particle of a weak elastic modulus (E = 1 kPa), initial radius of R 0 = 50 nm, and α = 2/3, we get U/k B T = 1.34. The required energy for shrinking by α this type of particle then needs only 10-20 agarose fibers within its vicinity, which is reasonable considering the fractal nature of agarose and the non-homogenous fiber distribution in its matrix.