The impact of nanoparticle softness on its tracer diffusion coefficient in all polymer nanocomposites

The advent of the fourth industrial revolution will require novel materials, where one promising material class is all-polymer nanocomposites. A variety of nanoparticles and fillers have been utilized to enhance desired specific properties of polymeric materials leading to a broader range of applications in the aerospace, automotive, and pharmaceutical industries.^1–6 Most current research focuses on the addition of hard impenetrable nanoparticles to a polymer matrix, which provides enhancements in mechanical and electrical properties, while modifying the polymer dynamics and flow properties.^7–10 Dispersing these hard nanoparticles is always a challenging task, where aggregate formation and limited favorable interactions between polymers and nanoparticles lead to failure in attaining targeted properties. Additionally, the incorporation of the nanoparticle usually deteriorates processing conditions further limiting their use.^11–13 These challenges lead to the demand of a new class of nanoparticles that are organic in nature and can provide beneficial interactions between polymers and nanoparticles to aid in homogeneous nanoparticle dispersion. Interesting and unexpected changes in the dynamics and flow properties of the polymer matrix have been observed with an inclusion of such organic nanoparticles due to their unique topology, and hence their complicated dynamics is still not well understood.¹⁴ 

Several studies have sought to more thoroughly understand the change in the dynamics of polymer melts with the inclusion of soft and hard nanoparticles. For instance, Mackay et al. have shown an unexpected viscosity reduction of an entangled linear polystyrene (PS) matrix with the incorporation of soft nanoparticles. This reduction in viscosity was attributed to a reduction in free volume demonstrated by the change of the glass transition temperature of the polymer–soft nanoparticle nanocomposite. Increasing the concentration of the nanoparticle leads to a further strong increase in viscosity at all frequencies as well as an increase in the plateau modulus, which is attributed to abrupt changes in the entanglement mesh.¹⁵ 

Moreover, previous work in our group has shown an increase in the diffusion of linear PS with the incorporation of 10 nm soft cross-linked PS nanoparticles, which is in stark contrast to the impact of adding hard nanoparticles to a polymer matrix. With the addition of hard nanoparticles, a reduction in diffusion and suppression in the motion of polymer chains is most often observed. Surprisingly, no increase in free volume was observed in our studies, indicating that simple plasticization cannot be the underlying mechanism for this phenomenon. We attribute this unique behavior to constraint release that speeds up the molecular motion of the polymer chain, in a similar manner to star polymers.¹⁶ This behavior is drastically different than what has been reported with hard nanoparticles. For instance, Mu et al reported that the addition of carbon nanotubes (CNTs) decreases the tracer diffusion coefficient of linear PS chains until a minimum is reached at 0.4%, after which the diffusion of the matrix is recovered.¹⁷ Analogous results were also found in systems that incorporate grafted nanoparticles, where Choi et al. reported an even stronger slowdown in the diffusion of the PS matrix with the incorporation of PS grafted silica nanoparticles.⁸ They attributed this effect to the fact that with a high grafting density on the nanoparticles, the free polymer chains cannot penetrate through the grafted chains. In this case, the effective particle diameter is larger than the core size leading to stronger confinement and a slowdown in the matrix dynamics.¹⁸ 

It is clear that a better understanding of the impact of soft nanoparticles on the dynamics of the polymer is needed, as well as the effect of softness, size, and deformability of nanoparticles on the motion of the nanoparticles themselves to enhance their potential use in a range of applications. Although significant research has been implemented to study the dynamics of the polymer in the vicinity of hard nanoparticles, the literature on nanocomposites containing entirely organic nanoparticles is still scarce.^12,19,20 This is due to the fact that it is hard to quantify the motion of these soft nanoparticles which are relatively slow compared to the linear polymer. Also, tuning contrast between the nanoparticle and the matrix can be difficult. Consequently, we have developed a protocol to measure the diffusion coefficient of soft nanoparticles in order to quantify their mobility. Our results show that these soft nanoparticles are mobile, not stationary, and that the overall mutual diffusion of a soft nanoparticle and a linear polymer chain can be analyzed to extract the tracer diffusion coefficient of the soft nanoparticle.²¹ For this previous study, a set of soft cross-linked PS nanoparticles was synthesized through nanoemulsion polymerization using a batch method, where the rate of monomer addition was not controlled.²² We have recently expanded our synthetic control over the soft nanoparticle structure, which enables this study that examines the diffusion of a new set of soft nanoparticles that vary in molecular weight, but retain the same cross-link density, and thus softness, of the original nanoparticles. The study of the diffusive behavior of these nanoparticles provides a pathway to separate the effects of the molecular weight and nanoparticle softness on its diffusive behavior to more precisely define the impact of nanoparticle softness on its translational motion.

The examined samples were all bilayers of deuterated polystyrenes on top and protonated nanoparticles on bottom. A control sample that is a bilayer of deuterated and protonated polystyrene was also prepared and examined under the same conditions to obtain the tracer diffusion coefficient of the linear polymer. The protonated and deuterated polystyrenes were purchased from Polymer Source. Both have number average molecular weight of 535 000 and polydispersity of 1.09. The soft polystyrene nanoparticles were synthesized by nanoemulsion polymerization of styrene where divinyl benzene, DVB, was added to the emulsion as a cross-linking agent. The DVB locks the polymer chain into a nanoparticlelike conformation. The first set of nanoparticles was synthesized by implementing a batch polymerization technique with no control over the rate of monomer addition and only cross-linking density was modified.²² For each nanoparticle, a variation in DVB added to the emulsion provided nanoparticles with cross-link densities of 0.80 mol. % for NP1A, 1.91 mol. % for NP2A, and 4.60 mol. % for NP3A. In these nanoparticles, as the cross-linking increases, the hardness of the nanoparticle increases. The morphology of the particles is best portrayed as a microgel with cross-links from the DVB producing a distinct core and a fuzzy interfacial shell consisting of free chain ends and loops. The increase in the amount of DVB generally decreases the radius of gyration of the nanoparticles and decreases its fuzziness. For the second set of nanoparticles, a semibatch technique was utilized where the rate of monomer addition is adjusted with the same % DVB %.¹⁴ 

This method resulted in nanoparticles with a similar morphology but with varying molecular weight. It is worth mentioning that using a semibatch method and a very low rate of styrene addition with 4.60% DVB resulted in a nanoparticle with a slightly different morphology, referred to as smooth gel with no fuzzy interface. This specific nanoparticle NP3AA exhibits a very small R_g and negligible fuzziness_. The details of the nanoparticle molecular weight and topology of the nanoparticles used in this study are presented in Table I.^14,22 To provide perspectives on relevant length scales for this system, the diameters of the soft nanoparticles range from 17 to 25 nm, 2R_g of the linear polymer is 38 nm, the tube diameter of polystyrene is approximately 8 nm, and the distance between cross-links is ca. 4–5 nm.

All Si wafers were purchased from Wafer World. Prior to bilayer casting, the Si wafers were cleaned in a piranha solution of sulfuric acid and hydrogen peroxide in the ratio of 3:1. The wafers were then rinsed in de-ionized water and dried. To further remove any organic contaminants, the wafers were placed in UV/ozone for 15 min. 1%–1.5% solutions of nanoparticle were prepared in toluene and used for spin casting. The solution was spin cast onto a wafer at 1500 rpm for 30 s to obtain the desired film thickness. For the deuterated layer, a 1% linear PS solution in toluene was spin cast onto a 4 in. wafer and then floated on water. The deuterated film was then picked up on the protonated layer forming a bilayer. All bilayers were then kept in a vacuum oven at room temperature for 24 h to remove residual solvent and water. Neutron reflectivity experiments were then performed on the Liquids reflectometer at the Oak Ridge National Lab. All samples were measured as cast and after annealing for different times. The annealing process was completed in a vacuum oven at 150 °C. After each annealing time, the samples were quickly quenched on a cooling block to stop the diffusion process.

The reflectivity of the sample was then measured and plotted as a function of q_Z, which denotes the scattering vector perpendicular to the surface of the samples and is defined by^23,24

In this equation, θ is the angle of incidence and λ is the wavelength of the incident neutrons. All data were reduced and fitted using the analysis package MOTO FIT in IGOR PRO. The scattering length density (SLD), thickness of the layers, and interfacial roughnesses are all fine-tuned to provide the best fit for the reflectivity curves. The quality of the fitting is assessed statistically through the value of χ² that was less than 10 for all fits and less than 5 for all nanoparticle samples.^23,25 A mass balance check is performed by integrating the area under the SLD profiles for the as-cast and the annealed samples assuring that the variation does not exceed 5%.

Our first experiments monitored the mutual diffusion of a bilayer of linear PS and dPS. This sample served as a control where the tracer diffusion coefficient of the linear matrix is determined and used in the analysis of the diffusion in the soft nanoparticle/linear chain bilayers. Figure 1 shows the reflectivity curve of the as-cast sample with clear fringes that decrease with annealing. The increase in roughness and dampening of fringes is an indication of the interdiffusion of the two layers. The scattering length density profiles in Fig. 2 show a sharp interface that broadens with annealing, where both the deuterated and protonated polymers are moving at the same rate.

Similar experiments were then completed to monitor the interdiffusion of bilayers of dPS and the soft nanoparticle. The mutual diffusion of these samples represents the interdiffusion of the nanoparticle into the linear polymer and vice versa. All nanoparticles diffuse into the linear matrix except for NP3AA. Figures 3 and 4 show the reflectivity curves for the diffusion of NP3AA and NP1B, respectively. Dampening of the fringes with annealing was not observed for NP3AA suggesting that this nanoparticle is essentially stationary, where the lack of fuzziness seems to suppress its motion.

Moreover, to determine the mutual diffusion coefficient of the two components, the time evolution of the bilayer is monitored at different annealing times and fitted using the one-dimensional solution of Fick's second law as shown in Eq. (2),²⁶ 

In this equation, t and h are the annealing time in seconds and the initial dPS thickness, respectively. Fitting the density profile of the deuterated material, $ϕd(z)$⁠, to Eq. (2) provides the mutual diffusion coefficient of the linear polymer and nanoparticle, D_m.

The scattering length density profiles SLD_m (z) extracted from fitting the reflectivity data were used to determine the density profiles of the deuterated material using Eq. (3),

Figure 5 shows the volume fraction profiles for the NP1A and NP3A bilayers as cast and after the longest annealing time of 63 h. NP1A and NP3A exhibit sharp transitions between layers for the as-cast samples that tend to roughen with annealing due to the interdiffusion of both dPS and protonated nanoparticle layers. However, NP1A exhibits a broader interface with annealing in comparison with NP3A. It is qualitatively clear from the volume fraction profiles that the diffusion process is asymmetric, where the polymer and the nanoparticle are moving at different rates. This asymmetry yields a level of uncertainty in the use of Eq. (2) to fit the data, which is estimated to result in ca. 10% uncertainty in the interfacial width.

The mutual diffusion coefficient (D_M) can then be correlated to the Onsager transport coefficient (D_T) using the following equation:^27,28

where $χ$ is the Flory–Huggin interaction parameter between polymers and nanoparticles and ϕ_i is the volume fraction of each component. The segment–segment interaction parameter χ is estimated to be zero since the matrix and the nanoparticle are chemically analogous. Furthermore,$χs$that denotes the interaction parameter at the spinodal can be calculated using Eq. (5),²⁷ 

The mutual diffusion coefficient extracted from this analysis represents the change of concentration gradient of both species. However, in order to quantify the tracer diffusion coefficient of the nanoparticle, which represents the discrete motion of the nanoparticles, two models were considered.^29,30 The fast mode theory developed by Kramer is a model for a system where the overall diffusion is controlled by the fast component and is represented by the following equation:²⁹ 

In Equation (6), NP and PS are the nanoparticle and the linear polystyrene matrix, while D represents the tracer diffusion coefficients of the different components. N is the degree of polymerization and ϕ represents the volume fraction at the inflection point which is 0.5. Analysis of the mutual diffusion coefficient using the fast mode theory leads to negative and unreasonable values for the tracer diffusion coefficient, so it is not considered further.

The slow mode theory presented by de Gennes relates the Onsager transport coefficient to the tracer diffusion coefficients of polystyrenes and the nanoparticle using the following equation:^29,31

With knowledge of the tracer diffusion coefficient of the linear matrix and $χs$⁠, all variables in the equation are known except the tracer diffusion coefficient of the nanoparticle, which can be determined. Table II shows the tracer diffusion coefficients of the first set of nanoparticles as determined using both theories.

Figure S1 in the supplementary material shows the tracer diffusion coefficients as a function of annealing times for both sets of nanoparticles, where the tracer diffusion levels off and equilibrates at long annealing times. At short annealing times, the tracer diffusion coefficient changes rapidly and denotes the transition of the motion of the particles from subdiffusive to diffusive motion. Thus, to evaluate the tracer diffusion coefficient correctly, it is crucial to anneal the samples for long times and confirm that the particles pass the subdiffusive regime. Table III lists the tracer diffusion coefficients that are experimentally determined for all nanoparticles using the slow mode theory at the longest annealing time.

In Fig. 6, the tracer diffusion coefficients of the soft nanoparticles are plotted as a function of molecular weight for each cross-linking density. Interestingly, there is a clear trend with cross-linking density where increasing cross-linking density leads to a decrease in the tracer diffusion coefficient of the nanoparticle for a given molecular weight. This result verifies that decreasing the deformability of the nanoparticle reduces its mobility regardless of nanoparticle’s molecular weight. A more highly cross-linked core increases the nanoparticle hardness, which leads to the nanoparticles being less able to deform and fit into the available spaces within the matrix and hence, their motion in this highly entangled system is suppressed. Increasing the cross-linking density from 0.81 to 1.91 decreases the tracer diffusion coefficient by a factor of ∼2, while increasing the cross-linking density to 4.6% reduces the mobility further. Another interesting trend is the great reduction in the nanoparticle mobility with increasing molecular weight for all cross-link densities.

The molecular weight dependence is further shown in Fig. 7, where log–log plots of the nanoparticle tracer diffusion coefficient as a function of molecular weight are presented for each cross-link density. Qualitative inspection of these plots provides further insight onto the mechanism of diffusion. The molecular weight dependence is stronger for low cross-linked nanoparticles which further validates the assumption that deformable nanoparticles diffuse faster due to its ability to distort and fit into spaces. Another factor that needs to be taken into consideration is the effective fuzziness that is reduced with increasing the cross-linking density. These dangling chain ends could also lead to disentanglement and dilation of the reptation tube of the linear matrix and further facilitate diffusion of the nanoparticle. A NP3AA nanoparticle with no fuzzy interface does not move over similar time scales, suggesting that the smooth nanoparticle interface can lead to increased friction between nanoparticles and linear chains that significantly inhibits its motion.

In Fig. 8, the ratio of the diffusion coefficients of linear analogs to that of the nanoparticle is plotted as a function of cross-linking density for all nanoparticles. The increase in this ratio with cross-linking density for all nanoparticles validates the importance of softness of the nanoparticle on its mobility regardless of molecular weight and exemplifying that the softer nanoparticles diffuse more quickly. Surprisingly, our nanoparticles do not exhibit an exponential molecular weight dependence of the diffusion coefficient that is expected for star polymers.^33,34 Our previous results showed that the incorporation of these nanoparticles onto 535 000 linear dPS leads to an increase in the diffusion of the linear matrix, where this result is explained in terms of constraint release similar to what has been reported in the literature with star polymers.^16,21,32 Thus, while these particles appear to exhibit some similarity in their behavior to starlike polymers, their nanoscale structure is sufficiently different that these nanoparticles diffuse by a different mechanism than stars.

Figure 9 shows the tracer diffusion coefficient of the nanoparticle plotted along with the tracer diffusion coefficient of their linear analogs. The nanoparticles diffuse much slower than the linear matrix, which is consistent with our previous interpretation suggesting that the soft nanoparticle motion is more similar to fractal microgels rather than star polymers and thus, require a cooperative motion of the polymer chain to diffuse.^35,36 The deformability of these nanoparticles can lead to different conformations adopted by the nanoparticles and hence lead to different mechanisms of diffusion, somewhat similar to pinned and unpinned cyclic polymers that may follow linear reptation in some cases and in other scenarios diffuse via arm retraction seen in stars.³² It will be interesting to monitor the diffusion of the nanoparticle in different molecular weight matrices to capture their mechanism of diffusion, where they may deform differently depending on the level of entanglement.

Finally, Fig. 10 compares the diffusion coefficient of the nanoparticles to the theoretical Einstein diffusion for a hard sphere of similar radii (Table IV). Classical Stokes–Einstein diffusive behavior is represented by Eq. (8) where k_b is Boltzmann's constant, T is the temperature, η is the fluid viscosity of the matrix, and d is the diameter of the particle. The viscosity of the neat PS was measured using rheology to be approximately 5.75 × 10⁵ Pa,

Equation (8) is then used to estimate the diffusion coefficient of hard spheres that are the same size as these nanoparticles. These results are shown in Fig. 10, where the Stokes–Einstein value is shown in black, while the experimental values for the soft nanoparticle diffusion coefficient are represented by the blue symbols. It is interesting that these soft nanoparticles all exhibit diffusivities that are slower than that predicted by the Stokes–Einstein equation, indicating that simply accounting for friction fails to capture the motion of these nanoparticles and highlight the importance of the fuzzy interfaces and loops that allow further interactions between the nanoparticle and the matrix. The short polymer chains and loops on the nanoparticle surface may thread with the matrix, leading to further suppression in the nanoparticle motion in comparison to bare spheres.

In fact, theoretical work by Yamamoto and Schweizer³⁷ shows that when the size of the nanoparticle is less than the size of the linear polymer chain and the nanoparticle is only 2–3 times the tube diameter (conditions that fit these experiments), a hard nanoparticle will not fully couple to the linear entanglement network. This results in the nanoparticle diffusing faster than that which is predicted in Eq. (8). Thus, the slower diffusion of the soft nanoparticle is even more unexpected and exemplifies the need to more fully understand the interaction of the soft nanoparticle and linear polymer chains.

We present a novel methodology to determine the diffusion coefficient of organic based nanoparticles in order to elucidate the molecular and nanoscale physics controlling their dynamics in a linear polymer matrix. Monitoring the diffusion coefficient of nanoparticles with identical cross-link density, and thus softness, for multiple molecular weights provides a pathway to tease out the importance of nanoparticle softness on its diffusive properties.

These results show that the motion of the nanoparticle is linked to its softness and therefore deformability. For a given molecular weight, increasing the cross-linking density of the nanoparticle increases its hardness and suppresses its diffusive motion in a linear matrix, emphasizing the importance of the deformability of the nanoparticle as well as their effective fuzziness on the nanoparticle motion. Moreover, the molecular weight dependence of the nanoparticle diffusion varies with nanoparticle softness and deviates from the exponential molecular weight dependence for star polymer diffusion. Thus, it appears that the diffusion of these nanoparticles more closely resembles fractal microgels that can take advantage of the cooperative motion of the matrix to open pathways for the nanoparticle to diffuse. Comparison of the experimental values to those predicted from the Stokes–Einstein theory shows great deviation where the nanoparticles diffuse slower than a hard sphere. This is consistent with the existence of entanglements or threading between the nanoparticle and the linear matrix that further suppress nanoparticle motion. Consequently, the simple friction/viscosity accounted for in Stokes–Einstein does not capture all factors that inhibit the motion of the nanoparticle. Further studies of the diffusion of these nanoparticles in lower molecular weight matrices are underway to provide further insight into the role of the matrix entanglements on diffusion and conformations adopted by the nanoparticle in the linear matrix.

See the supplementary material for the tracer diffusion coefficients as a function of annealing times for both sets of nanoparticles.

This research was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. A portion of this research was also completed at ORNL's Spallation Neutron Source, which was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy.