Anomalous Diffusion Inside Soft Colloidal Suspensions Investigated by Variable Length Scale Fluorescence Correlation Spectroscopy.

The diffusion of molecules and particles inside the aqueous suspension of soft colloids (polymer microgels) is investigated using variable length scale fluorescence correlation spectroscopy (VLS-FCS). Carbopol 940 is chosen as the model matrix system, and two factors affecting diffusion are investigated: the spatial hindrance and the diffusant-matrix interaction. By studying diffusion of molecules and particles with different sizes inside the suspension, VLS-FCS reveals the restricted motion at a short length scale, that is, in the gaps between the microgels, and normal diffusion at a larger length scale. The information on the gap's length scale is also accessed. On the other hand, by tuning the pH value, the diffusant-matrix electrostatic attraction is adjusted and the results expose a short-time fast diffusion of probe molecules inside the gaps and a long-time restricted diffusion because of trapping inside the microgels. It is proved that VLS-FCS is a powerful method, investigating anomalous diffusion at different length scales and it is a promising approach to investigate diffusion in complex soft matter systems.

Introduction

Understanding mass transportation inside complex soft matter systems such as concentrated polymer solutions and gels is important for many reasons, both academically and practically.[1,2] For example, it is related to a number of important processes in nature, such as biopolymers moving through crowded cellular environments, which can greatly influence cell functions such as the kinetics of enzymatic reactions, the formation of DNA, the self-assembly structures, and so forth.[3] In regenerative medicines and drug delivery, clinical applications of hydrogels rely largely on the diffusion of solutes across the polymer network.[4] It is also an important and basic aspect in soft matter physics and materials science, which, for example, can help develop novel composite materials.[5,6] However, the diffusive motion inside soft matter systems cannot always be attributed simply to a normal diffusion process, owing to the structural heterogeneity and diffusant–matrix interaction.[7]

Dynamical heterogeneity emerges because of structural inhomogeneity when its size is comparable to the size of the diffusant, expressed by the anomality in diffusion, having a nonlinear relation between the mean square displacement (MSD) and time.[8−11] Sophisticated theories, including scaling theory,[12,13] obstruction model,[14,15] hydrodynamic theory,[16,17] and multiscale combining model,[18,19] have been developed, and many investigations using molecular dynamics simulations have been performed.[20,21] Considering the coupling between structural relaxation of the matrix and the mobility of particles, scaling theories also have been successful in explaining the physics behind experimental data on diffusion, showing that the particles follow normal diffusion at long time, at which the polymer matrix relaxes, while it undergoes subdiffusion at short time before structural relaxation is accomplished. Experimental observation of nanoparticles diffusing in aqueous semidilute solutions of high molecular weight hydrolyzed polyacrylamide suggests the relaxation of a local “cage” structure in response to the nanoparticles’ dynamics.[22] It is believed that the structural heterogeneity in gel can be probed at different length scales using tracers of various sizes, owing to the fact that the mobility of diffusants depends on the relation between the diffusant’s size and the medium’s structure. Besides, the diffusant–matrix interaction is another important factor,[21,23−25] which can, in some cases, be even more influential on the diffusivity.[26] It is anticipated that the combination of the spatial hindrance and interaction can greatly enrich the diffusive process and an investigation into this issue is quite desirable.

In the current study, a heterogeneous gel system was created by choosing a concentrated suspension of soft colloids—the microgels made of a synthetic negatively charged acrylic polymer, Carbopol 940. The suspension consists of microgels with a strongly cross-linked core and overlapping dangling chains.[27] Small-angle light-scattering experiments have shown the existence of multiple length scales inside the Carbopol 940 suspension, that is, a shorter length scale of ∼400 nm of its cross-linked core and a larger length scale of 6 μm, corresponding to the highly expanded shell with a low monomer concentration.[28] Therefore, the sample has relatively large obstacles of a high segmental density with big spatial gaps in between, serving as a perfect system, providing large enough voids to trap bigger particles and at the same time allowing small molecules to diffuse through. Additionally, the polyacrylic acid (PAA) inside the microgel can be charged depending on the pH value and can be a good system to adjust diffusant–matrix interaction.

Fluorescence correlation spectroscopy (FCS) with variable excitation-detection volume (the lateral radius ranging from 0.2 to 0.4 μm), so-called variable length scale FCS (VLS-FCS), is adopted as the method.[29−31] By investigating the dependence of diffusing time across the variable confocal volume, this method is effective in accessing information on diffusion of multiple length scales and can be considered as a promising method to investigate the anomalous diffusion inside heterogeneous systems. FCS is very efficient in investigating diffusion using fluorescence-labeled probes with a single-molecule sensitivity,[32−35] allowing measurements at extremely low concentrations, that is, 4–5 orders of magnitude lower compared to other methods such as dynamic light scattering or fluorescence recovery after photobleaching. This enables measurements in the manner that the particle–particle interaction can be neglected. Another advantage of FCS is that only the fluorescence-labeled diffusant is visible because the nonfluorescent matrix polymer does not contribute to the signals. This can guarantee a better signal-to-noise ratio, so that mere information of the diffusant is provided. Most importantly, by systematically changing the excitation-detection volume, spatiotemporal information on the single-particle dynamics can be collected using the analysis by the FCS diffusion law—the apparent diffusion time across the confocal volume versus the transverse area.[36]

Nonadsorbing molecules and particles with different diameters, ranging 1–300 nm, are adopted to investigate the effect of spatial hindrance on diffusion. Additionally, the effect of diffusant–matrix interaction is investigated by choosing a positively charged fluorescent molecule, rhodamine 6G (R6G), which has adjustable probe–matrix attraction depending on pH values. The results expose the microscopic picture of the diffusion of particles with dependence on their size and the molecule–matrix interaction. The data have suggested that for the same matrix system, the multiple length scales at which the diffusion process shows heterogeneity can be switched based on diffusant–matrix interaction.

Results and Discussion

Diffusants with Different Sizes Inside the Microgel Suspension

FCS measurements of molecules’ and particles’ diffusion inside the suspension of Carbopol 940 microgel were conducted at pH 3.2. At this pH value, the PAA polymer is neutral and therefore, the electrostatic interaction is mostly suppressed and attention is paid to the spatial hindering effect. Figure displays a few typical normalized autocorrelation functions of particles’ diffusion when the lateral radius (w0) of the confocal volume was adjusted to be ∼0.25 μm. The data cannot be well-fitted using the ideal Brownian motion model expressed as , where z0 is the half length of the confocal volume along the optical axis, ⟨c⟩ is the average concentration of the fluorescent diffusant, and D is the diffusion coefficient.[35] The unsatisfactory fitting indicates the absence of normal diffusion. Considering the uneven distribution of segmental density inside the microgel suspension, the diffusion of the particles inside the microgel suspension should experience structural heterogeneity, making the diffusive motion deviate from the normal diffusion. Instead of obtaining the values of the apparent diffusion coefficient, the diffusion time (τD) for the particle across confocal volume is determined by taking the time at which the correlation function decays to half of its amplitude at zero time lag.

Figure 1

Typical autocorrelation functions of particles with different sizes diffusing inside the solution of Carbopol 940. The lateral radius of the confocal volume is ∼0.25 μm. The solid lines denote numerical fittings using a three-dimensional Brownian motion model.

The data of τD as a function of the square of the lateral radius (w02) are displayed in Figure a. Apparently, the τD value increases with w02 and the data can be well-fitted by the equation τD = w02/4D, in which D is the diffusion coefficient. Attention is paid to the intercept of the fitted line with the vertical axis, as demonstrated in the inset of Figure a—the intercept (τ0) is positive for all particles and it increases monotonically with the particle’s size. According to the FCS diffusion law,[36,45] the positive intercept indicates that the diffusant undergoes restricted motion at the smaller length scale and the motion turns to normal diffusion at the larger length scale. In Figure a, most data can be well-fitted (although the data of the 126 nm particle have bigger errors), implying that the particles with a diameter up to 126 nm undergo normal diffusion within the length scale of all beam spot’s size (larger than the diameter of 460 nm) and subdiffusion occurs at a smaller length scale. (The data of particle 8, i.e., with 308 nm diameter, do not exhibit linearity and are not plotted in Figure a.)

Figure 2

(a) Data of translational time of different particles diffusing across the lateral dimension of the confocal volume as a function of the square of the lateral radius (w0). The value of diameter of each type of particles is displayed in the figure with matching color to the corresponding data set. The solid lines denote the result of fitting using the relation of τD = w02/4D. Inset: The value of the intercept of the fitting with the vertical axis (τ0) as a function of the particle’s diameter. The data are also plotted in the log–log scale, as displayed in the Supporting Information. (b) Value of the diffusion coefficient deduced from fitting in (a) denoted by the red data point and from correlation function fitting, taking into account heterogeneity as denoted by blue data points. The dash line denotes the relation of D–d–1. The error bars in the figure are determined by data of multiple rounds of measurements.

The D values of the normal diffusion deduced from the fitting as a function of the particle’s diameter (d) are plotted in Figure b, and it is seen that up to a d of 5 nm, the D value scales well with the inverse of d, showing that the particle experiences hydrodynamic friction from the solvent within the lateral dimension of 860 nm. The data of small particles agree with the results of the scaling theory,[12] which predicts that a diffusing particle smaller than the correlation length of heterogeneity will experience viscosity of a pure solvent. This is further supported by the data fitting of the small particles, which yields a value of viscosity of 1.07 mPa·s, close to water. However, beyond a size value between 5 and 26 nm, the particle’s motion starts to experience structural heterogeneity, that is, the obstacle of the polymer segments of the microgel exerts an effect.

By fitting the autocorrelation function, introducing heterogeneity can also deduce the value of the diffusion coefficient together with the index showing heterogeneity. In the fitting, the autocorrelation function is expressed as , where α is the index of heterogeneity. For the case of normal diffusion, α equals to unity, while its value is below unity for anomalous diffusion.[29,30] When heterogeneity exists in the medium, the perfect randomness of diffusion is no longer valid and diffusion becomes anomalous. The data fitting shows that the α value is constantly lower than unity and it gets further smaller when the diffusant gets bigger. (The details of the fitting are provided in the Supporting Information.) The fitted diffusion coefficient data are displayed in Figure b (the blue data points) as a comparison. The D values are generally lower than those determined by VLS-FCS and also deviate from the D–d–1 relation even for the smallest particle. This comparison shows that the D value obtained by VLS-FCS is more meaningful than that by fitting the autocorrelation function by introducing heterogeneity. The data have demonstrated that the particles can probe the local viscosity, depending on its size, while it may never probe the macroscopic viscosity because it is considerably smaller than the size of the microgel and the related structural heterogeneity.[46,47] Taking a few basic parameters of Carbopol 940 provided by the manufacturer and published data,[48,49] such as the molecular weight of 4 × 106 g·mol–1, the average distance between the microgels is calculated to be approximately 70 nm. This value is very small compared with the measured size of the microgel, that is, 400 nm of the dense core and 6 μm of the fluffy corona, and it shows the interpenetration of the dangling chains of neighboring microgels.[49]

The restricted motion depends on the particle’s size—the bigger the particle is, the longer time its motion is restricted. The data of MSD of each particle as a function of time lag calculated from the autocorrelation function[50] at the w0 value of ∼0.25 μm are displayed in Figure , in which a general feature of trapped motion (subdiffusion) at short time lags and normal diffusion at longer time lags is visualized somehow. (Although some of the data are scattered, it is reasonable to see the subdiffusion turns to normal diffusion at long time because after multiple rounds of averaging and randomization, the normal diffusion process should be recovered.) This is in agreement with the observation by VLS-FCS. The time of transition from subdiffusion to normal diffusion (τc) can be determined as the time for MSD data to start to follow normal diffusion relation, that is, the solid line in the figure. The value of τc as a function of the particle’s diameter (d) is displayed in the inset. It is observed that the τc value increases monotonically with d and it experiences a vast increase between 61 and 126 nm. This gives an indication that the possible gap (void) size inside the gel suspension is at a similar length scale. This value also agrees with the prediction from the data of bulk rheology measurement, that is, a mesh size of 73 nm.[51,52] The diffusion of neutral particles inside the suspension of Carbopol 940 microgels at their uncharged condition is described as the subdiffusion at short time, when the diffusant is trapped inside the gaps between the microgels, and the normal diffusion beyond the gaps at longer time. The trapping, as expected, depends on the size of the particles: as the larger particles have to overcome a higher energy barrier in order to diffuse beyond the gaps, a longer time of trapping they will experience.

Figure 3

Data of MSD as a function of time lag of particles with different sizes diffusing inside the gel made of Carbopol 940. The value of the diameter of each type of particles is displayed in matching color to each data set. Inset: The time of transition from subdiffusion to normal diffusion as a function of the particle’s diameter.

Diffusion of Fluorescent Molecules under Different Probe–Matrix Interactions

The effect of diffusant–matrix interaction was investigated by taking the positively charged R6G molecule as the probe and varying the pH value so that the charge state was adjusted. Figure a displays the typical autocorrelation function of R6G diffusing inside Carbopol 940 suspension at different pH values, and Figure b shows the value of the apparent diffusion coefficient of the diffusant as a function of the pH value. The fitting of these autocorrelation functions has taken into account the heterogeneity. The apparent diffusion coefficient of R6G experiences a monotonic increase beyond pH 4.2 (the determination of the apparent diffusion coefficient is detailed in the Supporting Information). There should be two effects of pH increase: (1) the dissociation of the R6G probe and (2) the ionization of PAA microgel. The dissociation of the probe molecule (R6G) should lead to the weakening of probe–matrix attraction, bringing about the increase of the R6G’s diffusion rate and this is supported by the agreement between the pH range of D value increase and the reported pKa of R6G, that is, 7.0.[53,54] As a comparison, the diffusion rate of negatively charged Alexa 532 does not have pH dependence. The ionization of PAA microgel PAA microgels should also contribute to the diffusion acceleration because it makes the microgel swell further and larger free space is provided.[55,56]

Figure 4

(a) Typical autocorrelation functions of R6G diffusing inside the solution of Carbopol 940 at different pH values. (b) Value of the diffusion coefficient of R6G by fitting the autocorrelation function as a function of the pH value inside the Carbopol 940 solution. The data of Alexa 532 are displayed for a comparison. The diffusion coefficient data determined by VLS-FCS are also provided using blue data points for comparison.

The detailed picture of R6G’s diffusion in Carbopol 940 can be revealed by data of VLS-FCS. The translational time for R6G crossing the confocal volume as a function of the square of the lateral radius is shown in Figure . By fitting the data with the FCS diffusion law, all data show a negative intercept of the fitting with the vertical axis. This indicates that at all pH values, the R6G molecule undergoes a faster diffusion at a smaller length scale and a slower diffusion at a larger length scale. A scenario is proposed about the diffusion of R6G in the suspension. Being attracted to the segments of PAA, the positively charged R6G molecules diffuse slowly inside the dense microgels. On the contrary, in the gaps between adjacent microgels, the much lower segment density makes the molecule diffuse faster, expressed by the negative intercept in Figure . The diffusion coefficient of R6G molecules inside the suspension of Carbopol 940 is calculated by fitting the data in Figure and displayed in Figure b. It is noticed that their values are consistently lower than those by fitting the autocorrelation function.

Figure 5

Data of translational time of R6G across the lateral dimension of the confocal volume (τD) as a function of the square of the lateral radius of the confocal volume (w02) at different pH values. The solid lines denote the result of fitting using the relation of τD = w02/4D. Inset: The value of the intercept of the fitting with the vertical axis (τ0) as a function of the pH value.

MSD data of R6G diffusing inside Carbopol 940 suspension are calculated from the autocorrelation function for a w0 value of ∼0.25 μm (data shown in Figure ). The feature of anomalous diffusion is clearly seen—all of the data at long time lags deviate from the normal diffusion relation (denoted by the solid lines). The R6G molecule undergoes faster diffusion at short time lags and its motion gets much more restricted at longer time lags. This is in agreement with the results of VLS-FCS, showing that the molecule diffuses freely inside the loose gap between the microgels for short time lags and moves slowly inside the dense microgel core at longer time lags. The diffusion at short time lags can be fitted by normal diffusion. The diffusion coefficient at short time lags can be deduced from the MSD data and the value at pH 8.2, at which the R6G molecule should have the weakest attraction with the PAA segments, is 235 μm2·s–1. This value corresponds to a local viscosity of 1.5 mPa·s, very close to pure water, showing that in the gaps between the microgels, the R6G molecule is experiencing hydrodynamic drag mostly from water.

Figure 6

Data of MSD as a function of time lag of the R6G molecule inside the Carbopol 940 solution calculated from autocorrelation for a w0 value of ∼0.25 μm. The solid lines denote fitting of the short time lag data using the normal diffusion model. Inset: The display of the data of pH 3.2 and 4.2 for a better view.

By assuming that the R6G molecule undergoes normal diffusion inside the gap between the microgels, especially at high pH, where the attractive interaction is weak, and by taking the diffusion coefficient data deduced from MSD data, the size of the gap can be estimated using the FCS diffusion law, that is, by plotting the transverse time as a function of w02 (the VLS-FCS data) together with the calculation based on normal diffusion with a D value of 235 μm2·s–1. The result indicates that the size of the gap between Carbopol 940 microgels at a concentration of 0.5 wt % is ∼280 nm at pH 8.2. The detailed description of the estimation is provided in the Supporting Information.

Conclusions

VLS-FCS has demonstrated its powerfulness in investigating diffusion processes of multiple length scales. For diffusion inside the suspension of soft colloids, this method has quantitatively recognized the difference of the diffusion mode at shorter time and at longer time. For the Carbopol 940 microgel suspension, the spatial hindrance makes the motion of big noninteracting particles restricted between the gaps or inside the voids among the microgels and the time of restriction is estimated, providing information on the size of the gaps or voids. On the other hand, the dynamical heterogeneity can be switched by changing the diffusant–matrix interaction. For a diffusant having electrostatic attraction with the microgels, its diffusion is hindered by the core of microgel, while its motion between the microgels is recognized as closer to normal diffusion, as illustrated in Figure .

Figure 7

Schematic demonstration of particles and molecules diffusing inside the suspension of polymer microgels with the emphasis of spatial hindrance and diffusant–matrix interaction.

Experimental Section

Materials

Carbopol 940 (Lubrizol) was chosen as the model matrix system. The sample was a suspension of microgels, consisting of PAA cross-linked with a polyalkenyl polyether. Fluorescent molecules and particles were chosen as diffusants. Fluorescent diffusants include R6G, Alexa Fluor 488 NHS ester (Alexa 488), Alexa Fluor 532 NHS ester (Alexa 532), quantum nanocrystals (Qdot 655ITK carboxyl quantum dots), and carboxylate-modified microspheres (FluoSpheres) with diameters of 0.02, 0.04, 0.1, and 0.2 μm (measured by transmission electron microscopy). The particles are carboxylate-modified on the surfaces. All except R6G (Sigma) were purchased from Invitrogen, USA. Other water-soluble quantum dots with a carboxylate-modified surface were purchased from Janus New-Materials Company, China. The parameters of these diffusants are listed in Table .

Table 1

Parameters of Diffusants Adopted in the Current Study

diffusant	sample description	d (2Rh, nm)a
R6G	fluorescent molecule	1.2
Alexa 532	fluorescent molecule	1.2
Alexa 488	fluorescent molecule	1.1
Particle 1	CdTe quantum dots	2.0 ± 0.1
Particle 2	CdTe/ZnS quantum dots	3.0 ± 0.1
Particle 3	CdTe/Cds/Zns quantum dots	5.0 ± 0.10
Particle 4	quantum nanocrystals	26 ± 1
Particle 5	FluoSpheres	32 ± 2
Particle 6	FluoSpheres	61 ± 2
Particle 7	FluoSpheres	126 ± 7
Particle 8	FluoSpheres	308 ± 4

d = 2 × Rh, where Rh is hydrodynamic radius in water measured by FCS.

Sample Preparation

The microgel suspension was first prepared by mixing the Carbopol 940 polymer with deionized water at a concentration of 2 wt %. The aqueous solution or suspension of fluorescent diffusants was prepared separately at a concentration of ∼1.1 × 10–8 M for fluorescent molecules and ∼10–4 wt % for fluorescent particles. Afterward, a designated amount of the preprepared Carbopol solution was slowly added to a certain amount of solution of the fluorescent diffusant under continuous stirring, followed by ultrasonication. The final concentration of Carbopol 940 was 0.5 wt %. Small amplitude oscillatory shear measurements show that such a suspension exhibits a gel-like bulk property—its storage modulus (G′) is always higher than the loss modulus (G″) and no crossover is observed, indicating a very long terminal relaxation time of the system (data provided in the Supporting Information). Considering the fact that the sample is composed of separate microgels, these data show that a network with high elasticity spanning across the whole sample has formed, which effectively hinders stress relaxation.[37] The formation of such a percolated elastic network by microgel suspension is due to the mutual interpenetration of the dangling chains outside the microgel core.[38,39]

The prepared sample was then transferred into a sample cell with a bottom window made of a 0.15 mm-thick coverslip. After sealing, the sample was incubated for more than 8 h before measurements. The prepared microgel suspension had a pH value of ∼3.2, at which the microgel is believed to be uncharged, considering that the pKa value of PAA is 4.3.[40] The pKa value of PAA microgel should be even higher.[41] At such a pH value, the carboxylate-modified particles are neutral too. For experiments with pH adjustments, the pH value of the gel was tuned by adding a certain amount of triethanolamine (99.0%, Aladdin). No buffer solution was used here to avoid the effect of ions.

Variable Length Scale Fluorescence Correlation Spectroscopy

The FCS setup is a home-built version based on an inverted microscope (IX71, Olympus). The detailed description of the setup can be found in previous publications.[42,43] Briefly speaking, the 473 or 532 nm output of solid lasers was beam-expanded and collimated and guided into the microscope. A water-immerse objective lens (Plan Apochromat 60×, numerical aperture = 1.2, working distance = 0.25 mm) was used for this study. An optimized set of dichroic mirror and optical filters were installed for the purpose of introducing excitation into the sample and illuminating the background light so that a high enough signal-to-noise ratio was achieved. After passing a pinhole of 50 μm diameter, the fluorescence signal was split into two parts of roughly identical intensity, which were recorded separately using two single photon counting modules (H7421-40, Hamamatsu). The outputs were then input into a commercial FCS board (ISS) and the autocorrelation functions were generated. In order to make the size of the confocal volume at the sample stage adjustable, an iris diaphragm was installed after the beam expander so that the diameter of the beam in front of the back aperture of the objective lens can be tuned. By this approach, the VLS-FCS setup was accomplished.[31,44] Calibration experiments demonstrated that the lateral radius of the confocal volume can be adjusted within 230–430 nm, as detailed in the Supporting Information (Figure S1).