Network confinement and heterogeneity slows nanoparticle diffusion in polymer gels

Polymer gels, crosslinked chains swollen by water or organic solvents, are of fundamental and practical interest.^1,2 By incorporating nanoparticles (NPs) into polymer gels, material’s properties and function, such as high swelling, flexibility, and light weight, can be improved and/or added while maintaining attractive gel qualities. For example, NPs have been incorporated into gels to increase mechanical properties without significant reduction in swelling capacity,³ allow controlled release of drugs,⁴ and enhanced filtration capabilities.⁵ Additionally, understanding the environment experienced by drugs, viruses, and gene vectors within synthetic and biological polymer networks can be advanced by characterizing NP movement within these systems, which could improve the efficacy of drug delivery.^6–10 These applications and others^11,12 would benefit from a fundamental understanding of how polymer gel characteristics, such as network structure and confinement, affect NP diffusion.

In addition to recent growing interest in polymer diffusion within complex polymer media,¹³ NP diffusion in polymer solutions and melts, particularly when the probe and matrix have similar characteristic length scales, has received significant interest by theorists and experimentalists. Cai and Rubinstein¹⁴ developed a scaling theory to describe the mean-squared displacement (MSD) of NPs in polymer solutions by comparing the NP size to the correlation length in polymer solutions or the tube diameter, d_t, in polymer melts. When the NP is smaller than the correlation length, the MSD scales as t¹ at all times; however, if the NP is larger than the correlation length and smaller than d_t, the particle’s movement is subdiffusive at short time scales, ∝ t^1/2, and diffusive, ∝ t¹, at longer times. If the particle is larger than d_t, its movement will become diffusive at times longer than the reptation time of the polymer chains in solution. This crossover in dynamics from subdiffusive to diffusive has been observed in polyelectrolyte solutions for particles with diameters from hundreds of nm to μm.^15,16 Brochard-Wyart and de Gennes¹⁷ developed the initial scaling model to describe the diffusion of a single NP in a polymer melt. For a NP diameter smaller than d_t, particles experience only the local viscosity characterized by a section of the polymer chain similar to the particle size, as opposed to the macroscopic viscosity of the polymer. As NP size increases, this scaling theory predicts a sharp change in behavior from dynamics dictated by local to macroscopic viscosity when the NP diameter is equal to d_t. Recent theoretical work has expanded these scaling approaches.^18,19 Using a self-consistent, generalized Langevin equation theory to describe NPs in polymer melts, Schweizer and Yamamoto predicted a recovery of Stokes-Einstein (SE) behavior, where NPs experience the macroscopic viscosity when NP diameter, 2R, is 10 times larger than d_t.¹⁸ Gold NP diffusion in an entangled polymer melt with confinement ratios, 2R/d_t, from 1 to 4, was 10 times faster than the SE prediction.²⁰ While this study did not recover SE behavior, the authors state that their results predict complete coupling of NP motion to polymer entanglement relaxation at 2R/d_t values between 7 and 10.²⁰ Diffusion studies of silica NPs in a poly(2-vinylpyridine) matrix found that SE diffusion was recovered for 2R/d_t ≥ 5.²¹ 

Dynamic theoretical models that describe NP mobility in both crosslinked polymer networks and entangled polymer melts^19,22 focused on the localized constraints placed on NPs when NP size is comparable to the confining mesh diameter. In Schweizer and Dell’s theory of localization and activated hopping of NPs,¹⁹ regimes of polymer behavior were observed with respect to a critical confinement ratio, (2R/d_t)_c ∼ 1, which defines the condition for NPs to become confined by the entanglement or gel network. In systems with confinement ratios below the critical value, NPs are not trapped by the entanglement network. For crosslinked systems with larger confinement ratios, activated hopping is the only mechanism of motion available to a particle after becoming localized. For polymer melt systems in this regime, however, two mechanisms of NP motion exist: activated hopping and passive motion linked to polymer constraint release. The relative dominance of these two mechanisms was determined by comparing the time scale associated with hopping, related to the free energy barrier height determined by relative confinement, to the time scale of polymer reptation, related to the degree of polymerization, length of the entanglement strand, and friction coefficient of the polymer. From this comparison, hopping was determined to be the primary mode of transport only when both the systems are highly entangled and 2R/d_t is approximately 1 to 1.8. Using molecular dynamics simulations of weekly interacting NP-polymer melt systems that were weakly entangled, Kumar et al. found that constraint release from fluctuations in the entanglement mesh was the dominant mode of NP transport.²³ Rubinstein and Cai’s model for hopping diffusion of NPs²² also predicts coupling between NP and polymer dynamics, though with quantitative differences in the hopping step size and the hopping energy barrier. In this model, confinement can come from chemical cross-linking and physical entanglements, though the model predicts that the chemical cross-links impact NP diffusion within gels more than physical entanglements.²² While these models and theories propose various mechanisms for NP diffusion, they agree that the main factor determining NP diffusion in polymer systems is the relative confinement defined by the ratio of the NP size to the mesh or d_t. As described in the Results section, confinement alone was unable to completely capture the slowing of NP diffusion in crosslinked and collapsed gels in this study.

Models for describing solute diffusion in hydrogels have also been developed. These models attribute slower solute diffusion to a decrease in available space, an increase in the hydrodynamic drag on the solute, or an increase in the diffusion pathway caused by polymer chain obstacles.²⁴ In polymer solutions and gels, the diffusion of a variety of probes, from linear macromolecules to spherical particles, has been described by a hydrodynamic model where diffusion coefficients were dependent on polymer concentration through a stretched exponential.^25,26 While these models hold particular relevance for biological polymer networks, such as mucus and the extra cellular matrix (ECM), factors such as chemical interactions and heterogeneity add complexity that is not included in the models. For example, Schuster et al.⁶ showed that polyethylene glycol (PEG) coated particles with a diameter less than 200 nm were able to diffuse through respiratory mucus, whereas 500 nm PEGylated and 100, 200, and 500 nm carboxylated particles were immobilized, clearly showing that NP-polymer interactions can affect NP diffusion in biological polymer networks.

NP diffusion in biological systems can provide insight into the structural and chemical nature of networks.^6,8,9 NPs have also been used to probe the organization of biopolymer networks, such as the ECM, where particle diffusion decreased as collagen concentration increased (i.e., increasing confinement).¹⁰ The motion of probes in hydrated and biological media can be measured by various methods including fluorescence recovery after photobleaching (FRAP), fluorescence correlation spectroscopy (FCS), and single particle tracking (SPT). Because confinement and hopping of NPs occur on sub-diffraction length scales, the resolution of the method must be in the nanometer range. SPT is one of the few methods with sufficient resolution to resolve hopping and distinguish between populations of NP mobility. Although FRAP and FCS are robust techniques for determining ensemble diffusion, SPT allows for distinct mechanisms of diffusion to be identified within an ensemble by tracking single molecules or particles.⁷ This is increasingly important as the assumption of Gaussian diffusion, often evoked to quantify FRAP and FCS data, becomes questionable for complex fluids and systems with heterogeneous structures such as gels, ECM, and mucus.²⁷ 

Polyacrylamide gels (PAGs) are hydrophilic polymer networks found in applications from gel electrophoresis to soft contact lenses. Previous studies of particle dynamics in PAGs have mainly used micron sized colloidal particles to investigate gel properties such as dynamics (i.e., microrheology) and heterogeneity.^25,28–30 Network structure heterogeneity resulting from cross-linking is common to many gels;³¹ heterogeneity that evolves during cross-linking of PAG has been studied by small angle neutron scattering (SANS) and small angle light scattering.³² Only recently have studies of NP diffusion in PAGs been published.^25,29 Dynamic light scattering (DLS) was used to characterize the diffusion of silica NPs during PAG cross-linking.²⁵ In this study, NPs became attached to the PAG network, which impacted diffusion. While this study did not follow single particle trajectories, the analysis assumed Gaussian displacements, which may not be accurate for crosslinked gels.²⁵ Another study used the motion of quantum dots (QDs) with a hydrodynamic radius of 10 nm in PAGs to calculate a trapping site size of about 120 nm for the range of crosslinker concentrations from 0.3 to 1.5 mol. %.²⁹ While this study followed individual NPs and applied a random trapping model, the average mesh size, and thus confinement ratio, as a function of crosslinker concentration was not determined.

While prior studies provide insight, the diffusion of NPs moving through a network is not completely understood in part because systematic experimental studies are lacking, particularly those that investigate systems where the mesh and NP are of comparable size. This study analyzes NP diffusion in PAGs where the mesh size was reduced from larger than to smaller than the NP size (21 nm). The confinement ratio (CR), defined as the NP hydrodynamic diameter relative to the network mesh size, is a parameter that captures probe size and average matrix confinement size. The mesh size of PAGs was reduced by increasing cross-linking density or collapsing the gel network by adding a poor solvent³³ at fixed cross-linking resulting in CRs from 0.4 to 3.8 and 0.4 to 2.1, respectively. For both confinement types, NP motion was confined (localized) to regions smaller than 200 nm, mobile with displacements on order of microns, or a combination of these termed “intermittent localization.” Mean squared displacements (MSDs) of mobile NPs decreased as CR increased. In collapsed gels, the localized NP population increased and MSD of mobile NPs decreased in comparison to crosslinked PAGs. For all CRs, van Hove distributions exhibited non-Gaussian displacements, indicating intermittent localization of NPs. The non-Gaussian parameter increased from a maximum of 1.5 for crosslinked PAGs to 5 for collapsed PAGs, consistent with greater network heterogeneity in collapsed gels. The diffusion coefficients for both confinement types decreased an order of magnitude when the CR became greater than 1. Diffusion coefficients decreased exponentially as CR increased for crosslinked gels, but in collapsed gels, diffusion coefficients decreased more strongly, suggesting that the CR alone was insufficient to capture diffusion in these gels. Collapsing the gel resulted in an increasingly tortuous pathway for NPs, slowing diffusion at a given CR. Elucidating gel characteristics that impact NP mobility, such as structural heterogeneity, will allow for more efficient filtration and drug delivery technologies.

CdSe core, ZnS shell QDs, Qdot 655 ITK Amino-PEG QDs were purchased from Invitrogen. Polyacrylamide and bisacrylamide were purchased in aqueous solutions from Sigma Aldrich. N,N,N′,N′-Tetramethylethylenediamine (TEMED), ammonium persulfate, 3-aminopropyltriethoxysilane (APTES), and glutaraldehyde were also purchased from Sigma.

Piranha (7v/3v sulfuric acid to hydrogen peroxide) cleaned glass slides and coverslips were placed in separate nitrogen purged containers. The container with glass slides was exposed to 1 ml of dichlorodimethylsilane for at least 45 min to make the glass slides a hydrophobic, non-adherent surface. The container of coverslips was exposed to 1 ml APTES in an oven at 70 °C for 6 h. Coverslips were subsequently incubated in 0.5% (v/v) glutaraldehyde in water.

Solutions of acrylamide (71 g/mol) and bisacrylamide (154 g/mol) were prepared in water at 3/.06, 5/0.03, 4/0.1, 5/0.15, and 5/0.3 w/v % acrylamide/bisacrylamide.³⁴ To each of these solutions, ammonium persulfate (10 μl at 10 w/v % per ml of solution) and TEMED (1 μl per ml of solution) were added to initiate cross-linking. Immediately following, 25 μl of each gel solution was pipetted onto a glass slide and a coverslip was placed on top. The gel solution completely wet the area of the coverslip, resulting in approximately 20-40 μm thick gels. After gelation, samples were swollen in DI water for at least 36 h before incubation with NPs to allow excess monomer to leach out. Gels with varied crosslinker concentration were then partially dried and NPs in water were added topically at a concentration of 0.4-0.8 nM to fully rehydrate. The partial drying prior to rehydration with NPs in water created a stronger driving force for the NPs to enter into the gels than simply adding them to fully swollen gels. Samples of 3/.06 w/v % acrylamide/bisacrylamide were prepared in water then exposed to 0, 5, 10, 20, and 40 vol. % acetone containing concentrations of 0.4-0.8 nM quantum dots to volumetrically collapse the gel network. Experiments were performed 12-36 h after incubation, once NPs were distributed equally throughout the gel. As the concentration of TEMED was low, time before experiments short, and pH of the DI water maintained around 5.5-7.5, the gels maintained a non-hydrolyzed, neutral state.³² Samples for rheometry were prepared as discs with diameters of 25 μm and allowed to swell for 36 h before testing.

Thermogravimetric analysis, TGA, was performed on a TA Instruments Q600 SDT. Rheometry was performed on a TA Instruments RFS using a parallel plate geometry. The gels were maintained within the elastic regime as determined prior to the frequency test (0.5%-2% strain, 100 Hz to 0.01 Hz) by a strain dependent test at 1 Hz.

SPT experiments were performed on an inverted Nikon Eclipse Ti optical microscope using an oil immersion Nikon × 100, 1.49 NA objective. Quantum dots (QDs) NPs were excited with a 532 nm laser. Blinking served as an indication that QDs observed were not aggregates, which was confirmed by SEM imaging (supplementary material). The exposure time was set to 40 ms (∼25 fps) in an effort to minimize static and dynamic error. Each video was collected for 30 s using a CCD camera (Cascade-512B, Photometrics). Post-processing of SPT trajectories was performed using a MATLAB based program, FIESTA (Fluorescence Image Evaluation Software for Tracking and Analysis), which was developed to track QDs attached to molecular motors walking along filaments.³⁵ Two-dimensional Gaussian fits of fluorescent intensity were used to determine particle location with subpixel accuracy that had a resolution on the order of the NP size, 20 nm. Drift was evaluated using immobilized QDs and found to be less than the positioning error. Particle tracks with a positioning error greater than 25 nm were excluded from analysis. The full width half max of the fluorescence was initialized at 900 nm, approximately 8 pixels. The minimum number of frames was set to 6 and the max break to 4 frames. A jump tolerance between individual frames of 2000 nm was added to exclude connecting adjacent particles. Particles were separated based on distance traveled and diffusion behavior. NPs moving less than 200 nm from the center-of-mass of their trajectory (localized) were separated from those moving more than 200 nm (mobile). Mean squared displacement (MSD) for each particle was determined using the MATLAB program, msdanalyzer.³⁶ Particle trajectories were fit to Equation (1),

Only MSD curves that were well fit by a power law equation, R² > 0.7, and had an alpha of 0.75 to 1.25 were used to determine diffusion coefficients. MSD is the expectation value for the distance traveled by a particle in a given amount of time, given by Equation (2) where τ is the time between positions, r,

It should be noted that this equation results in more positions being averaged for short time separations between r(t + τ) and r(t) than when the longest time separation is approached. Since longer times of the average MSD contain less data than shorter times and long-time data can be skewed by slower moving particles, shorter time data have more statistical significance, and as a result data were typically compared at times of 1 s or less. For all cross-linking conditions, between 1000 and 3000 mobile particle trajectories were collected. For all acetone systems, particles’ trajectories were collected until mobile particles exceeded 600. MATLAB msdanalyzer as well as personal scripts was used for plotting.

This study analyzes PEG functionalized NP mobility in PAGs as a function of network confinement. The network size was reduced by (1) increasing the cross-linking density and (2) adding a poor solvent, acetone, to collapse the gel as described in Section III A. Our previous work has shown that polymer diffusion through a polymer nanocomposite (PNC) collapses on a master curve when plotted against the single confinement parameter of interparticle spacing between NPs relative to the polymer size.^37–39 The reduction of the normalized diffusion coefficient was observed for hard particles, soft particles, and particles that have weak or moderate interactions with matrix polymer.^37–39 While NP spacing defined confinement in PNCs, confinement in gels is imposed by the network or mesh through which particles diffuse. One objective of this study was to determine if a single confinement ratio (CR) can describe NP diffusion in the crosslinked and collapsed gels. Section III B describes SPT trajectories and the corresponding mean square displacements (MSDs) in PAGs as a function of increasing cross-linking density (B1) and increasing poor solvent concentration (B2). In Section III C, the NP displacements are presented as van Hove distributions, which show deviations from Gaussian distributions. Finally, in Section III D, diffusion coefficients of mobile NPs are plotted versus CR to determine if this simple parameter captures NP diffusion similar to polymer diffusion in PNCs. In contrast to the NPs in the PNC, network heterogeneity in gels appears to play a significant role with increasing heterogeneity resulting in slower NP diffusion at fixed CR.

The average CR is the ratio of the hydrodynamic size of the NP relative to the average mesh size, CR = 2R/ξ. The mesh sizes were reduced by increasing the bisacrylamide crosslinker concentration from 0.06 to 0.3 w/v. Based on our previous studies, these acrylamide and bisacrylamide concentrations were selected to produce mesh sizes above and below 2R.⁴⁰ For PAGs prepared by simultaneous copolymerization and cross-linking in solution, the average mesh size can be determined by scaling models.¹ This model assumes that segments between cross-links behave as a freely rotating chain modified by the Flory characteristic ratio, C_n, which is 8.5 for PAGs in water.⁴¹ The mesh size, ξ, is given by,

where φ is the polymer volume fraction at equilibrium swelling and $(r02)12$ is the end to end distance between cross-links for a carbon-carbon backbone chain,⁴² 

where M_r is the molecular weight of a repeat unit, M_c is the molecular weight between cross-links, and l is the carbon-carbon bond length, 1.54 Å. Using rubber elastic theory, M_c can be related to the zero frequency shear modulus, G′₍₀₎,^1,43

Using dry polymer density, ρ, the universal gas constant, R, and temperature, T = 295 K, the mesh size given by

can be determined from φ and G′₍₀₎ measured by TGA and rheometry (supplementary material), respectively. For each combination of acrylamide and bisacrylamide preparation concentration, Table I lists average G′₍₀₎, polymer volume fraction at equilibrium swelling, average mesh size, and the CR. The hydrodynamic diameter (2R = 20.8 nm) of PEG grafted quantum dots was determined in water/glycerol solutions using the Stokes-Einstein relation and published viscosities (supplementary material). All numbers represent the average of at least three samples. As crosslinker concentration increased, the mesh size decreased from about 50 to 5 nm and correspondingly the CRs increased from 0.4 to 3.8.

For the least confining PAG, CR = 0.4, the mesh size was also reduced by adding acetone after synthesis which collapsed the gel, resulting in a continuous volume phase transition, VPT.^33,44 For the DI water (pH ∼ 6) in this study, PAGs were in a fully swollen state.⁴⁵ While hydrolyzed PAGs exhibit a discontinuous VPT, the gels in this study exhibited a continuous decrease in the swelling ratio, Q, as acetone concentration increased, consistent with neutral PAGs.³³ Table II gives the values of Q, the volume of the gel at equilibrium swelling divided by the volume immediately after cross-linking. Table II also includes the average mesh size and CR. While the chemical cross-links remain separated by the same number of monomers, the effective mesh size of the PAGs was changed as acetone was added to collapse the gels. Namely, the network collapses between cross-links that are chemically fixed, making the free space between crosslinked points decrease and increasing the polymer volume fraction, decreasing Q. Using Equation (7),⁴⁶ mesh sizes were calculated from

While this scaling is for good solvent conditions,⁴⁷ we have chosen to use it to systematically scale the effective mesh sizes despite changing solvation conditions as it appropriately corresponds to the changes we observe in NP mobility described in Secs. III B–D. Similar to the effect of increasing crosslinker content (Table I), the CRs of collapsed PAGs span the less (CR < 1) to highly confined (CR > 1) regimes.

The average mesh size of a gel is a single value that is meant to capture the nanoscale network structure, although a distribution of mesh sizes is typical. Since one goal of this study was to relate NP diffusion to confinement imposed by the gel, this average mesh size gives a quantitative description of the “mean field” mesh. However, a single length scale does not capture the heterogeneity of the network. Published data show that the PAGs’ mesh size depends on the concentration of polymer and measurement technique. From static light scattering, the mesh size for PAGs with similar acrylamide/bisacrylamide ratios used in this study ranged from 5 to 30 nm, similar to values in Table I.⁴⁸ Using diffusing-wave spectroscopy to determine the shear modulus of PAGs with bisacrylamide concentrations from 0.05 to 0.5 mol. %, mesh sizes ranging from 10 to 25 nm were reported.²⁵ SANS and Small Angle X-Ray Scattering (SAXS) methods yield smaller mesh sizes, from 2 to 7 nm.³² Overall, PAG mesh sizes in this study are consistent with literature values.

SPT was used to measure NP trajectories and MSD as a function of mesh size in PAGs. Figure 1 shows NP trajectories and MSDs for a CR of 0.4, where the mesh size is approximately twice the NP diameter. In Figure 1(a), the dark red bullseye represents “localized” NPs with displacements less than 200 nm and the multi-colored traces are “mobile” NP trajectories with much larger displacements (microns). By plotting MSD versus time, Figure 1(b) shows that NPs fall into two populations: (1) mobile NPs (top lines) and (2) localized NPs (dark red). The MSD of the localized NPs is relatively constant, about 0.001 $μm2$⁠. Additionally, the MSD shows that the time for tracking localized NPs is much longer, the majority over 1 s, compared to the mobile NPs, which rapidly diffuse out of plane. At a time of 0.08s, Figure 1(c) displays a histogram of the MSD, showing a small population of localized NPs and greater number of faster mobile NPs with MSD values centered at approximately 10³ nm² (∼32 nm)² and 10⁶ nm² (∼1000 nm)² defined by the maximum counts. For all CRs (Table I), NPs exhibit both localized and mobile behavior similar to Figure 1. The population of localized NPs was 10% for CRs of 1 or less, and increased to 15% and 25% for CRs greater than 1 (supplementary material). Therefore, while both populations were observed for all crosslinked PAGs swollen in water, the majority of NPs were mobile and exhibited displacements greater than 10 times their diameter. In the remainder of this paper, the diffusion of the mobile NPs and caging of the localized NPs will be considered separately.

For the mobile NPs, Figure 2 shows the NP trajectories (top row) and MSD (bottom row) in crosslinked PAGs swollen in water as a function of increasing CR. For the 500 trajectories shown in Figures 2(a)–2(c), the area explored by NPs decreased as the CR increased from 1 to 3.8. As shown in Figures 2(d)–2(f), the MSD values also decreased as CR increased. The MSD followed a log normal distribution (as shown in the mobile NPs of Figure 1(c)); therefore, the geometric ensemble average was determined. Figures 2(d)–2(f) show that the geometric averages (solid black lines) are shifted by two orders of magnitudes to lower MSD values as the CR increased from 1 to 3.8. The fast diffusion of mobile NPs limited their time in the field of view to only one to two decades of time. For example, at a CR of 0.4 (Figure 1(b)), NP trajectories were measurable for an average of about 0.5 s. As CR increased, NPs could be tracked for longer times; for example, the average tracking time for NPs in PAGs with CR = 3.8 was twice as long, 1 s, as the least confined case. Even though the trajectories were followed for a longer time at CR = 3.8, the displacements were significantly less as noted by comparing the areas explored in Figures 2(c) and 2(a). For example, after 0.4 s, NPs explored average spatial areas of 2.4 μm², 0.69 μm², and 0.16 μm² as the CR increased from 0.4 to 1.6 to 3.8, respectively. Preliminary data on crosslinked gels with a confinement ratio of approximately 8 showed localization near the surface of the gel for all NPs (not shown), indicating that while CRs greater than 1 result in both mobile and localized NPs throughout the gels, an upper limit for NP infiltration exists.

NP diffusion was also studied in PAGs collapsed by adding acetone to a water swollen PAG. The addition of a poor solvent resulted in an increase in the percentage of localized NPs and a slowing down of the mobile NPs. For PAGs with 10 vol. % acetone having CR = 0.65, NPs exhibited both mobile and localized behavior as shown in Figure 3. The MSD curves in Figure 3(a) show that mobile NPs have lower MSDs compared to the mobile NPs in water swollen PAGs with the same crosslinker content (Figure 1(b)). Figure 3(b) shows a histogram of the MSD values at 0.08 s with the MSD values for localized NPs centered at 10³ nm² (32 nm)², similar to localized NPs in the water swollen gels. In Figure 3(b), the MSD values of mobile NPs at 0.08 s are centered at approximately 10⁵ nm² (320 nm)², an order of magnitude lower than the MSD values of the crosslinked PAG with CR = 0.4. While localized NPs were the minority in water swollen PAG with CR = 0.4 (Fig. 1(c)), mobile NPs only accounted for 20% of the population in the collapsed PAG with CR = 0.65 (Figure 3(b)). These trends persisted in all acetone collapsed PAGs, with mobile NPs making up only 20 to 25 of the population (supplementary material). The slowing down of the mobile NPs and increase in localized population cannot be attributed solely to an increase in CR but rather reflects differences between the local environment in the crosslinked and acetone collapsed PAGs.

To further investigate NP mobility, Figures 4(a)–4(c) show the mobile NP trajectories in collapsed PAGs with 0 (CR = 0.4), 10 (0.65), and 40 (2.1) vol. % acetone. The spatial coverage explored by NPs decreased as confinement increased in qualitative agreement with the crosslinked PAG results in Figure 2. As shown in Figures 4(d) and 4(e), upon increasing CR from 0.4 to 0.65, the MSD decreased and the tracking time increased. A comparison of Figures 4(e) and 4(f) shows similar behavior as CR increased from 0.65 to 2.1. Figures 4(d)–4(f) show that the geometric average MSD (black lines) decreased as the CR increased from 0.4 to 2.1. Similar to the crosslinked PAGs, mobile NPs are able to diffuse in collapsed PAG when the CR is greater than 1 and can exhibit displacements on the order of microns. In summary, for NP diffusion in both crosslinked and collapsed PAGs, the special coverage explored by NPs and the MSD of NPs decreased as the confinement ratio increased.

Insight into the mechanism of NP diffusion in PAGs can be gained by analyzing NP displacement distributions. The van Hove correlation function, ΔX, describes the probability that a particle moves a distance along one direction within a specific time interval, τ, and is given by,⁴⁹ 

Figures 5(a)–5(c) show the van Hove displacement distributions of mobile NPs in crosslinked PAGs with CRs 1, 1.8, and 3.8, where t = 0.08 s (blue), 0.4 s (red), and 1 s (black). For each CR, non-Gaussian behavior is exhibited by an increase in small displacements (near ΔX = 0) and a broad tail corresponding to larger displacements. At constant CR, the spread in the displacements of mobile NPs broadened as time increased, consistent with NPs exploring more of the gel at longer times. As CR increased, the displacements at all times decreased, for example, after an interval of 0.08 s, the maximum displacement decreased from 1800 nm to 750 nm as CR increased from 1 to 3.8, respectively. In contrast, the distribution of displacements for the localized NPs is independent of time as shown in Figures 5(d) and 5(e). A comparison of Figures 5(d) and 5(e) shows that the distribution is also independent of CR. The spread of the distribution of localized NP displacements is ±100 nm at both CRs. For CR = 3.8, Figure 5(f) shows the XY position trajectory of a mobile NP at 40 ms intervals for a total time of approximately 2.5 s. The mobile NP exhibited many small displacements followed by much longer displacements. The periods of “intermittent localization” are colored as light blue, yellow-green, and red-orange in Fig. 5(f). Taken together, the non-Gaussian behavior of mobile NPs along with the intermittent localization observed in single NP trajectories indicates that NPs experienced different environments within the same gel (i.e., at fixed CR).

The van Hove distributions were also determined for NPs diffusing in the acetone collapsed PAGs. Figures 6(a)–6(c) show the NP displacements in PAGs with 0 (CR = 0.4), 10 (0.65), and 40 (2.1) vol. % acetone. The displacement distributions of mobile NPs in collapsed PAGs (main graphs 6b-c) clearly display non-Gaussian behavior, whereas Figure 6(a) shows that the distribution at 0 vol. % (least confining network) is approximately Gaussian. The mobile NPs in collapsed PAGs have centrally peaked displacements with long exponential tails. These tails correspond to NP displacements that are longer than predicted by Gaussian behavior and become longer as time increases from 0.08 s (blue) to 1 s (black). As acetone content increased and the mesh collapsed, the displacements of the NPs become shorter. Similar to the crosslinked PAGs, intermittent localization was observed in the collapsed gels as shown in Figure 6(d). Localization of the NP was observed at 0 s (dark blue) and between 1.5 and 2 s (green to orange). The insets of Figures 6(a)–6(c) show that the displacement distributions for the localized NPs are independent of time for all three CRs. However, the spatial area explored by the localized NPs decreased as the vol. % acetone increased from 10 to 40 vol. % as shown by comparing the insets in Figures 6(b) and 6(c). Defining the localization region as the spread encompassed by 90% of the displacements, the localized region decreased from 130 nm to 90 nm as acetone increased from 10 to 40 vol. %. This increase in confinement suggests areduction in the local mesh size that confines localized NPs. While the non-Gaussian shape and decreasing displacement with increasing CR of the mobile NPs in the collapsed PAGs are qualitatively similar to those in the crosslinked PAGs, a reduction in the space explored by the localized NPs was only observed in collapsed PAGs, suggesting a difference in local structure even at similar average mesh sizes.

The characteristic length of NP displacements at each time interval can be quantified by fitting the tails of the van Hove distributions to an exponential function, y = A*exp(−x/t) (supplementary material). To include both positive and negative sides of the van Hove distributions, the absolute values of all negative displacements were taken prior to fitting. For both the crosslinked and collapsed gels, the characteristic length initially increased rapidly, and then more slowly at later times as shown in Figure 7. The characteristic length scales as approximately t^0.44 (solid lines) similar to the t^0.5 dependence observed in other systems, such as colloidal particles in entangled actin, where non-Gaussian van Hove distributions were observed.⁴⁹ At fixed time, the characteristic length decreased as the CR increased for both the crosslinked (Fig. 7(a)) and collapsed (Fig. 7(b)) PAGs. One important difference between confinement types is that small changes in CR produced a large decrease in the characteristic length for the collapsed PAGs. For example, at t = 0.4 s, the characteristic length decreased from 440 to 180 nm as the CR increased from 0.6 to 0.7. These large changes suggest that other factors, in addition to CR, impact NP diffusion in the collapsed gels. Another insight is that NPs exhibit smaller characteristic lengths (i.e., more strongly confined) in collapsed PAGs, than in crosslinked PAGs at similar CRs. Thus, the characteristic lengths of NPs decreased more sharply with CR in collapsed PAGs, suggesting that CR alone is unable to capture NP diffusion. The exponential nature of the tails in our study indicates the NPs exhibit dynamic heterogeneity, which we propose is due to the heterogeneity caused by the collapsing network due to the poor solvent. The characteristic lengths describe the large displacements NPs take between being localized, and thus give an indication of the degree of hindrance the NPs are experiencing in the different local microenvironments.

Whereas NPs in a homogeneous environment exhibit Gaussian behavior, the displacement distributions in the crosslinked and collapsed PAGs display a central peak with a broad tail at long displacements. This has also been observed for particle diffusion in complex systems such as cells and f-actin solutions.^49,50 In gels, this deviation from Gaussian behavior was attributed to NPs confined to a trapping well, the size of the central peak.²⁹ Thus, particles are localized or trapped for a time interval and after escaping, either continue diffusive motion or become localized again. As previously noted in Figures 5(f) and 6(d), SPT shows that NPs exhibit intermittent localization, suggesting that the local PAG environment is heterogeneous. Other studies that have observed similar non-Gaussian behavior analyze the localized and mobile populations together leading to central peaks that are confounded by an increase in small steps attributed to localized particles.²⁹ Recognizing that NPs exhibit both diffusive and localized dynamics, only the mobile NP displacements were analyzed in the present study. As discussed in the Introduction, the random cross-linking during gelation can produce heterogeneities in gels.³¹ Using SANS, PAGs were determined to have network dimensions ranging from 5 to 250 nm.³² Because similar synthesis methods are used in this study, PAGs likely exhibit similar network heterogeneity which in turn provides the environment leading to the distinct NP population dynamics and intermittent localization. The localization of NPs is attributed to NPs entering and becoming caged within tight mesh regions, as proposed previously.²⁹ 

The degree of dynamic heterogeneity in a system can be quantified by the non-Gaussian parameter, N_g, given by,⁵¹ 

The N_g characterizes the deviation of the displacement distribution (Figures 5 and 6) from a Gaussian distribution. Quantitatively, the N_g compares the fourth moment to the second moment, namely, the breadth of the distribution relative to the variance. For crosslinked PAGs at all CRs, the N_g of the mobile NP displacements was relatively small, N_g ≤ 1.5. Figure 8 shows N_g as a function of time for the collapsed PAG and includes the water swollen crosslinked PAG at CR = 0.4 (green solid circles). Whereas N_g is only approximately 0.2 for a CR = 0.4 (0 vol. % acetone), N_g increases from 0.9 to 4.5 as the CR increases from 0.6 to 0.7 (5 to 20 vol. % acetone), respectively. At 40 vol. % acetone, which corresponds to complete macroscopic collapse of PAG, N_g decreases to 1.7, suggesting more homogeneous network. In summary, the N_g was systematically larger for the collapsed PAGs compared to crosslinked PAGs, indicating that the addition of acetone increases the dynamic heterogeneity of NPs diffusing through PAGs. Further, the dynamic heterogeneity increased strongly even when the CR only increased by a small amount in collapsed PAGs.

For the acetone collapsed gels, the behavior of the N_g (Figure 8) indicates that the extent of dynamic heterogeneity increased as the poor solvent concentration increased. Using positron annihilation to probe local free volume, PAGs exhibited a decrease in free volume prior to the macroscopic VPT, which is evidence that the changes in the nanoscale environment can occur before the macroscopic collapse of the gel.³³ In the current study, the increase in N_g prior to complete collapse at 40 vol. % acetone is consistent with this previous study. To gain further insight, the macroscopic average mesh sizes can be compared to the nanoscale confinement surrounding the NPs. For all CRs, the FWHM of the displacement distributions for localized NPs (Figures 5(d)–5(e) and insets of Fig. 6) was approximately 30 nm, which is comparable to the average mesh size determined by macroscopic methods (Table I) for the water swollen PAG having the lowest CR value, 0.4. However, by comparing the breadth of the distributions, the localized displacements decreased from 130 nm to 90 nm from CR = 0.65 and 2.1, indicating that the localized NPs can traverse 3 to 4 meshes. This reduction in the spatial coverage of localized NPs (insets of Figures 6(b) and 6(c)) is consistent with localization regions becoming smaller with increasing acetone content. Areas connecting these localization regions can be quantified by analyzing the mobile NPs. The characteristic length of the mobile NPs in the water swollen gel (CR = 0.4) at a single frame step, 0.04 s, was approximately 370 nm, which is more than seven times the average mesh size, 47 ± 7 nm (Table I). As the gel collapsed, however, the characteristic length decreased to approximately 45 nm, which is on the order of the average mesh size, 10 nm. Thus, movement of NPs in the collapsed gel was reduced to a few average meshes within the frame rate of the experiments. Overall, localized NP motion was comparable to the average mesh sizes determined from macroscopic methods and NP mobility decreased with decreasing average mesh size. Thus, by combining the average mesh size with the spatial area explored by NPs, a nanoscopic understanding of local gel structure can be developed.

To interpret the NP behavior in the acetone collapsed system, an understanding of the continuous VPTs in PAGs is needed. When a poor solvent is introduced into the swollen gel, both contraction and microphase separation can be induced.⁴⁴ The phase separation of dense polymer regions from solvent rich regions is caused by the contraction of neighboring strands, expelling the poor solvent and reducing contact free energy, while strands in initially less dense regions remain swollen. The impact this separation and contraction has on NP diffusion is schematically shown in Figure 9. Figure 9(a) shows NPs diffusing through water swollen PAGs, which provides weak confinement (CR = 0.4). However, as acetone is added, regions of the gel collapse into a tighter network, preventing some NPs from diffusing and increasing the tortuosity of the pathway for mobile NPs, Figure 9(b). After the gel has fully collapsed, NPs are highly confined by collapsed polymer strands. NPs inside the most compact, dense regions do not diffuse, while those in less compact regions have reduced diffusion due to increased friction with the collapsed strands and increased confinement, Figure 9(c). Using nuclear magnetic resonance, the dense regions in PAGs have been found to be 6-12 nm, although the authors indicated that the actual dense regions are somewhat larger.⁴⁴ In the present study, the dense polymer regions appear to be on the order of 90 to 130 nm. Overall, NP mobility is sensitive to both the average mesh size and nanoscale heterogeneity caused by the collapsing of the gel.

Recent studies show that particle diffusion in gels, actin solutions, and colloidal solutions exhibit non-Gaussian displacement behavior,^29,49,51 calling into question the validity of the Gaussian assumption and the simple linear relationship between MSD and diffusion coefficient. While it has been suggested that displacement distributions should be characterized, to our knowledge, no further method has been developed to extract diffusion coefficients (D) from non-Gaussian data. Thus, a protocol for determining D from MSD data was employed to allow for a systematic comparison of NP diffusion as a function of CR. To determine ensemble behavior, the geometric average MSD for particles that exhibited mobile behavior was determined and plotted in Figures 2(d)–2(f) and 4(d)–4(f) (solid black lines). Particle trajectories were analyzed to determine power law, t^α, scaling over the initial 80% of MSD traces. Traces fit with an r² greater than 0.7, approximately 70% of mobile NPs, were used to determine the ensemble average of α, which equals approximately 1 for all cross-linking conditions over a time interval of 1 s. The ensemble geometric MSD was then calculated from traces where α = 1 ± 0.25. This standard deviation represents the same range for NPs diffusing in homogeneous glycerol/water solutions. For the crosslinked and collapsed PAGs, the ensemble MSDs were fit with Equation (1) to determine D. In this study, gel relaxation was fast (less than milliseconds),²⁵ the shortest experimental time delay was 40 ms, and NPs traveled many mesh distances, the NP diffusion coefficients correspond to center of mass diffusion and the longest relaxation time.

Diffusion coefficients are expected to decrease as confinement increases. Using scaling relationships proposed by Phillies, probe (both macromolecular and particle) diffusion coefficients in polymer solutions decrease as an exponential or stretched exponential function as concentration or polymer molecular weight increases.⁵² While these studies focus on polymer solutions, the scaling behavior for NPs as a function of crosslinker concentration²⁵ or network collapse has received little or no attention. For crosslinked PAG, NP diffusion coefficients decreased by over an order of magnitude as the CR increased from 0.4 to 3.8, as shown in Figure 10(a). The solid line is a best fit to the exponential function with an argument given by -CR/1.1, where CR = 2R/ξ. The NP diffusion coefficients in the collapsed gels were not well fit by a similar exponential dependence, which we attribute to the impact of dynamic heterogeneity discussed previously. In the crosslinked PAG at the highest CR (=3.8), the average mesh size was only 5 nm, nearly 4× smaller the NP diameter. Nevertheless, NPs exhibit diffusive motion for CR > 1 with no apparent sharp decrease near CR = 1. Although most theories assume that confinement (e.g., tube size, correlation length, and mesh size) is homogeneous, PAGs are known to exhibit polydispersity in mesh size and are therefore heterogeneous. As a result, NPs can move rapidly in open mesh regions while being confined in tighter mesh regions. In the case of crosslinked PAGs, the heterogeneity is relatively small and therefore the confinement due to average mesh size is similar to that of the correlation length in a polymer solution.

Figure 10(b) plots the normalized diffusion coefficients for crosslinked and collapsed PAGs as a function of CR. The SE diffusion coefficients D₀ use the viscosity of in pure water or the water/acetone mixtures to account for the increase in viscosity as vol. % acetone increases. Similar to crosslinked PAGs, NP diffusion in collapsed PAGs decreases with increasing CR; however, the NP diffusion coefficients decrease much more strongly in the collapsed PAGs. The slower diffusion of NPs in collapsed PAGs is also observed by comparing the characteristic lengths shown in Figure 7. For example, compared at similar CR (1.8 versus 2.1) at 0.4 s, the characteristic length in crosslinked PAG was 500 nm, whereas it is only 140 nm in the collapsed PAG. These trends indicate that CR alone is unable to fully describe NP mobility in heterogeneous PAGs. To understand why NPs move slower in collapsed PAGs, the average N_g values were plotted versus the CR in Figure 10(c). These data show that the dynamic heterogeneity increased as the CR increased to about 1 but then decreased for CR > 1 (i.e., 40 vol. % acetone), indicating a more uniform mesh in fully collapsed PAGs. Moreover, relative to the crosslinked PAGs, the dynamic heterogeneity was always much greater in the collapsed PAGs when compared at the same CR. This larger heterogeneity suggests that the collapsed PAGs exhibit regions with smaller mesh size and therefore greater local confinement. New models are needed to quantitatively account for the effect of dynamic heterogeneity on NP diffusion in heterogeneous gels.

In contrast to the crosslinked PAGs, the reduction in NP diffusion in collapsed PAGs is attributed to dynamic heterogeneity, which results from intermittent localization of NPs and a dispersity in mesh size (Figure 9). Other mechanisms that cause slowing should also be considered. Even though attractive mesh-NP interactions could lead to intermittent localization, the NPs were grafted with a neutral PEG brush which does not hydrogen bond with PAG at pH values used in this study, specifically pH between 5.5 and 7.5.⁵³ Similarly, studies of the mechanical properties of PAGs loaded with silica NPs have shown that despite the charged surface of the silica NPs, PAG chains do not adhere to the NP surface.⁵⁴ The slowing down in collapsed PAGs could be attributed to an increase in solution viscosity upon mixing acetone and water. To account for changes in viscosity, D was normalized by the SE diffusion coefficient in water/acetone mixtures as shown in Figure 10(b). However, it is possible that the local viscosity felt by the NPs differs from the bulk viscosity if the acetone is partitioned towards the center of the less dense regions. Furthermore, the collapsed PAG likely contributes a higher segmental frictional drag on the passing NP. Both contributions, if they occur, require advances in modeling to understand their impact on chain and NP dynamics. Additionally, the PEG brush conformation could be collapsed upon introduction of a poorer solvent, although a smaller effective hydrodynamic radius would increase D, opposite to the observed trend. Moreover, because the change in χ is small, going from 0.31 to 0.29 upon adding 10 to 20 volume fraction of acetone to water, the PEG brush is not expected to change significantly, much less expand, as acetone content is increased.⁵⁵ While other factors may contribute to the differing dynamics at the same CR, these studies indicate that the driving force for reduced mobility in collapsed PAGs is dynamic heterogeneity introduced into the PAGs by the addition of a poor solvent, acetone.

The diffusion of NPs within PAGs was studied as a function of network confinement. By increasing crosslinker concentration or adding acetone to collapse PAG, the average mesh size was reduced to produce confinement ratios less than and greater than 1. In both PAGs, there was a corresponding decrease in NP mobility, yet NP diffusion was retained even when the NP was nearly 4× larger than the average mesh. NP mobility could be separated into localized and mobile populations. The mobile NPs displayed either intermittent localization through diffusive motion interspersed with random localization or continual diffusive motion. Diffusion coefficients decreased with confinement ratio, although the NP diffusion in the collapsed gels was slower than diffusion in the crosslinked gels when compared at a constant confinement ratio. This difference was attributed to dynamic heterogeneity, which is supported by analysis of the displacement distributions and non-Gaussian parameters. Displacement distributions showed exponential tails that decreased in length and likelihood as confinement increased. This reduction occurred more rapidly in the collapsed PAGs than in the crosslinked PAGs, which is consistent with the greater dynamic heterogeneity observed in the collapsed PAGs. These results suggest that the local structure of the collapsed PAGs is quite heterogeneous, resulting in dense regions with a small mesh size that localizes NPs and/or acts as obstacles for NP mobility. Further experimental studies of NP diffusion through model gels with a monodisperse mesh size would provide important insight into the significance that dynamic heterogeneity plays in reducing NP diffusion. Additionally, single particle tracking using imaging methods that provide faster frame rates (hundreds of frames per second) and greater spatial resolution (∼1 nm) is needed to determine the existence and impact of the hopping mechanism on NP diffusion in gels. Current models successfully capture the diffusion of NPs in polymer solutions, melts, and gels by assuming homogeneous non-interacting systems; however, more advanced models are needed to capture the effect of gel heterogeneity on NP diffusion and the impact of interactions between NP and polymer. Ultimately, by understanding how the macroscopic and nanoscopic structure of polymer gels affects NP dynamics, the performance of membranes, separators, and drug delivery gels can be tailored with unprecedented control.

See the supplementary material for quantum dot characterization, details of mesh size characterization including rheometry data, and table of NP populations.

Support was provided by the NSF PIRE OISE-1545884 (R.J.C), ACS/PRF 54028-ND7 (R.J.C and E.P), NSF/MWN DMR-1210379 (R.J.C and E.P), and NSF/DMR 1507713 (R.J.C). M.C. acknowledges support from the National Institute of Health, 5T32HL007954-15 (Peter F. Davies PI), via an Interdisciplinary Cardiovascular Training Grant. The work was performed at and supported by the Nano Bio Interface Center (NBIC) at the University of Pennsylvania through NSF NSEC DMR08-32802 and NSF MRI DBI-0721913. The authors thank Robert Ferrier for SEM images, Ben Lindsay for Matlab coding assistance, and Anne van Oosten for training on the rheometer. We also thank Dr. Matt Brukmann for NBIC instrument support.