Melissa Hess^{ab},
Eric Roeben^{a},
Patricia Rochels^{a},
Markus Zylla^{a},
Samira Webers^{c},
Heiko Wende^{c} and
Annette M. Schmidt*^{a}
^{a}Institute of Physical Chemistry, Chemistry Department, Faculty of Mathematics and Natural Sciences, University of Cologne, Luxemburger Str. 116, D-50939 Köln, Germany. E-mail: annette.schmidt@uni-koeln.de
^{b}Institute for Complex Systems, IHRS BioSoft, Forschungszentrum Jülich GmbH, D-52428 Jülich, Germany
^{c}Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen, Lotharstraße 1, D-47057 Duisburg, Germany
First published on 28th November 2019
Rheological approaches based on micro- or nanoscopic probe objects are of interest due to the low volume requirement, the option of spatially resolved probing, and the minimal-invasive nature often connected to such probes. For the study of microstructured systems or biological environments, such methods show potential for investigating the local, size-dependent diffusivity and particle–matrix interactions. For the latter, the relative length scale of the used probes compared to the size of the structural units of the matrix becomes relevant. In this study, a rotational-dynamic approach based on Magnetic Particle Nanorheology (MPN) is used to extract size- and frequency-dependent nanorheological properties by using an otherwise well-established polymer model system. We use magnetically blocked CoFe_{2}O_{4} nanoparticles as tracers and systematically vary their hydrodynamic size by coating them with a silica shell. On the polymer side, we employ aqueous solutions of poly(ethylene glycol) (PEG) by varying molar mass M and volume fraction ϕ. The complex Brownian relaxation behavior of the tracer particles in solutions of systematically varied composition is investigated by means of AC susceptometry (ACS), and the results provide access to frequency dependent rheological properties. The size-dependent particle diffusivity is evaluated based on theoretical descriptions and macroscopic measurements. The results allow the classification of the investigated compositions into three regimes, taking into account the probe particle size and the length scales of the polymer solution. While a fuzzy cross-over is indicated between the well-known macroscopic behavior and structurally dominated spectra, where the hydrodynamic radius is equal to the radius of gyration of the polymer (r_{h} ∼ R_{g}), the frequency-related scaling behavior is dominated by the correlation length ξ respectively by the tube diameter a in entangled solutions for r_{h} < R_{g}.
While steadily more complex and smart materials are under investigation, skillful analytical approaches are required to fully characterize such samples with respect to their structure and dynamics. Yet, the idea to exploit microscopic objects as tools for the rheological characterization of the matrix goes back already to the pioneers Heilbronn^{8} (1922), and Freundlich and Seifriz^{9} (1923). During the last decades, microrheology has developed into an extensively used methodology for the characterization of soft and complex materials.^{10–17}
Microrheology opens up the access to how viscoelastic materials flow as a function of length scale.^{13} The motion of micrometer sized spheres, either of passive thermal origin or actively driven by an external force (e.g. electric or magnetic field), is tracked and analyzed. Here, the generalized Stokes–Einstein equation relates the diffusivity of the particles’ motion to the viscoelastic properties of the sample.^{12,13,18} Besides the low amount of sample volume, microrheology benefits from a low degree of invasion to the sample and access to a wide frequency range, as well as the detection of local inhomogeneities.^{12,13,15,19,20}
Since microscopic probing relates the material mechanical characteristics to its microstructure, there is increased interest in decreasing the probe size to the size of the characteristic length scales within the material. Stringent deviations from the rheological performance, as obtained by conventional methods, are expected and experimentally observed.^{21,22}
In particular, in polymer-based fluids and soft solids (e.g. gels, melts or solutions), specific characteristic length scales (e.g. mesh size m, radius of gyration R_{g}, tube diameter a or correlation length ξ) are present. By relating the probe particle size to the size of the characteristic length scales of the material, certain regimes connected to characteristic length- and time scales can be identified, yet their interpretation is still the subject of investigation.
For a nanoscopic tracer particle in a polymer solution, the effective viscosity η responsible for the diffusion of the tracer depends on the ratio of particle size d to the correlation length ξ, as already postulated by De Gennes.^{23} When the polymer chains begin to overlap, the polymer solution can be described as a transient network with the correlation length ξ defined as the average distance between the two adjacent contact points between the chains. On this basis, Rubinstein and his coworkers classified the tracer behavior into regimes according to their size with respect to the characteristic length scales.^{24,25} ‘Small’ tracers have a radius or diameter that is smaller than the correlation length, and their translational diffusion is mainly determined by the solvent viscosity η_{S}. In contrast, the diffusion of ‘large’ particles (meaning larger than the tube diameter a) reaches the (quasi-)continuous case for long observation times. Here the tube diameter a is related to the topological confinement of a polymer chain by surrounding chains to a tube-like region.^{24} For ‘intermediate’ particle sizes, we expect at long lime scales that diffusion related to an effective viscosity η_{eff} occurs. This effective viscosity is lower than the one experienced for a macroscopic object.^{25} Based on scaling approaches for polymer solutions, further prediction of the frequency-dependent behavior is also available.
In other studies, the impact of the radius of gyration R_{g} is addressed by molecular dynamics simulation, and it is found that the classical Stokes–Einstein relation fails if the particle size is below the radius of gyration R_{g} of the polymer.^{26,27} The radius of gyration R_{g} is defined via the mean square distance of the building blocks to the center of gravity of the molecule.^{24} A schematic drawing of the important length scales in polymer solutions is provided in Scheme 1.
Scheme 1 Schematic drawing of (a) correlation length ξ, imaginary tube diameter a and particle size d and (b) radius of gyration R_{g}. |
The influence of the relation of probe size to characteristic length scales in polymer solutions and a correlated breakdown of the Stokes–Einstein equation are focused in several scientific studies.^{13,21,22,28,29} The role of R_{g} as a cross-over quantity for the tracer size is obvious from the majority of studies. The nature of the underlying scaling quantity, however, to date is not fully clear.
In this study, we systematically investigate the rotational dynamics of magnetic nanoparticles using the method of Magnetic Particle Nanorheology (MPN)^{30–32} in a series of model polymer solutions of different molar masses and concentrations, with the aim to better understand specific size-dependent particle–matrix interactions, and to explore the impact of the probe size in relation to the characteristic length scales of the matrix.
As polymer model systems, we employ aqueous solutions of linear poly(ethylene glycol) (PEG) by varying molar mass M_{n} and volume fraction ϕ. In order to obtain a systematic variation of the tracer size, we use magnetically blocked CoFe_{2}O_{4} (CF) nanoparticles that are coated with a silica (SiO_{2}) shell. We use AC susceptometry (ACS) in order to obtain information on the rotational (Brownian) relaxation process of the tracer particles in the complex fluid environment of the solutions. By comparing with a theoretical scaling model^{23} we obtain semiquantitative access to frequency-dependent rheological properties, among these the frequency dependent viscosity η*, loss modulus G′′ and diffusion properties, with an emphasis on the size-dependent particle diffusion.
For a variation in the diameter of the probe particles, the identical magnetic cores are coated with a silica shell. Therefore, magnetic cobalt ferrite particles in water are diluted to obtain a solution with 1 × 10^{−5} m% CF particles and 95 v% ethanol. The solution is treated in an ultrasonic bath for a few minutes. Ammonium hydroxide solution (with a total ammonium hydroxide concentration for the different particle batches of 0.08 mmol mL^{−1}, 0.12 mmol mL^{−1}, 0.25 mmol mL^{−1} and 0.26 mmol mL^{−1}) is added, and the reaction vial is placed on a shaking plate.
Tetraethyl orthosilicate (TEOS, 0.09 mmol mg^{−1} CF) diluted in ethanol (0.09 mmol mL^{−1} EtOH) is added dropwise within 2 hours to the shaking reaction mixture at a rate of 5.0 mL h^{−1}. Subsequently, the CF particles are shaken for an additional 19 hours at room temperature. The reaction mixture is centrifuged at 6500 rpm for 40 min, washed three times with ethanol and twice with water, and finally dispersed in water. In the following, the magnetic cores are labeled as CF_x, while silica coated particles are labelled as CF@SiO_{2}_x, with x representing the hydrodynamic diameter of the respective particles in water, as obtained by ACS.
PEG | M_{n} [g mol^{−1}] | M_{w} [g mol^{−1}] | PDI |
---|---|---|---|
P4k | 4245 | 4308 | 1.01 |
P20k | 19560 | 20810 | 1.06 |
P35k | 31870 | 33100 | 1.04 |
A direct connection between the rotational diffusion^{12,22,32} of the magnetically blocked probe particles, as extracted from the phase lag between excitation and response, and the rheological properties of the matrix is provided by the generalized Stokes–Einstein relation for rotation (eqn (1)).
(1) |
(2) |
(3) |
(4) |
The core particles are subsequently individually coated with a SiO_{2} shell using a modified Stöber process.^{35,36} A variation of the shell thickness is achieved by increasing the particle size through an increase of the base concentration during the synthesis. The TEM images of the resulting particles (Fig. 1) show mainly single coated particles with only one magnetic core. The corresponding particle sizes as obtained by different methods are listed in Table 2. The results confirm that the shell thickness of the tracer particles increases with the base concentration, and that the size distribution of the tracers remains low.
Sample | TEM | DLS | ACS | ||||
---|---|---|---|---|---|---|---|
r_{g} [nm] | σ | r_{n,DLS} [nm] | r_{v,DLS} [nm] | r_{h,ACS} [nm] | σ | τ_{Brown} [s] | |
CF_12 | 8.2 | 0.13 | 10.7 | 13.3 | 11.6 | 0.31 | 4.3 × 10^{−6} |
CF@SiO_{2}_25 | 18.3 | 0.07 | 25.6 | 28.2 | 24.8 | 0.24 | 4.2 × 10^{−5} |
CF@SiO_{2}_29 | 21.7 | 0.07 | 28.4 | 33.6 | 29.2 | 0.35 | 6.8 × 10^{−5} |
CF@SiO_{2}_37 | 25.3 | 0.07 | 32.2 | 37.5 | 37.0 | 0.46 | 1.4 × 10^{−4} |
CF@SiO_{2}_42 | 25.1 | 0.09 | 38.8 | 46.4 | 42.3 | 0.36 | 2.1 × 10^{−4} |
In addition to the basic architecture of the tracers, it is important to collect relevant information on their surface properties, their hydrodynamic behavior and their magnetic properties.
The surface properties of the tracers are analyzed by attenuated total reflection infrared (ATR-FT IR) spectroscopy (see Fig. S1, ESI†). For SiO_{2}-coated particles, dominant Si–O–Si stretching modes are found. All particles are negatively charged on the surface at pH 6 (the pH of the investigated polymer solutions is 6–7) with deprotonated acid groups on the magnetic cores, which are stabilized with polyacrylic acid, and deprotonated hydroxy groups on the surface of the silica particles. The ζ potential is shown for a wide pH range for the magnetic cores and exemplarily at pH 6 for the silica particles used for the measurements in Fig. S2 (ESI†). All batches show at pH 6 a ζ potential of approximately 50 mV in water which ensures good stability of the particles in aqueous solutions at that pH.
The hydrodynamic properties of the tracers are investigated independently based on the translational and rotational diffusion of the particles. Dynamic light scattering (DLS) is used to determine the number- and volume-averaged hydrodynamic radius r_{h,DLS} of the particles (see Table 2) based on translational diffusion. Accordingly, the hydrodynamic radii are slightly larger than r_{g} as expected, representing the impact of the solvation shell and confirming that mainly single-dispersed particles are present.
The quasi-static magnetic properties of the samples are analyzed by vibrating sample magnetometry (VSM) measurements in the dried state as well as in aqueous solution and are shown in Fig. 2 and Table 3.
Probes | CF_12 | CF@SiO_{2}_25 | CF@SiO_{2}_29 | CF@SiO_{2}_37 | CF@SiO_{2}_42 |
---|---|---|---|---|---|
M_{S} [A m^{2} kg^{−1}] | 74.1 | 11.1 | 10.6 | 10.5 | 10.2 |
μ_{mag} [m%] | 98.1 | 14.6 | 14.1 | 13.9 | 13.5 |
H_{c} [kA m^{−1}] | 13.8 | 10.8 | 13.5 | 10.2 | 16.0 |
M_{R}/M_{S} | 0.30 | 0.25 | 0.28 | 0.27 | 0.29 |
χ_{ini}/M_{S} [m A^{−1}] | 3.2 × 10^{−5} | 3.0 × 10^{−5} | 2.9 × 10^{−5} | 2.7 × 10^{−5} | 2.3 × 10^{−5} |
m [A m^{2}] | 3.6 × 10^{−19} | 3.0 × 10^{−19} | 2.8 × 10^{−19} | 2.6 × 10^{−19} | 2.2 × 10^{−19} |
In the dried state, all particle batches show at ambient temperature a hysteretic magnetization curve, and the saturation magnetization M_{S} is according to the expected range by taking into account the volume and diamagnetism of the SiO_{2} shell. The normalized curves are of similar shape, and the coercive field H_{c} and normalized remanence M_{R}/M_{S} do not significantly depend on the shell thickness.
In aqueous dispersion, all particle batches show pseudo-superparamagnetic behavior, characterized by Langevin-type behavior and the absence of hysteresis. With increasing particle size, the normalized initial susceptibility of the virgin curve, χ_{ini}/M_{S}, shows a decrease which is attributed to the diamagnetic contribution of the SiO_{2} shell.
With respect to rotational diffusion-based hydrodynamic properties, dynamic magnetometry based AC susceptometry is employed. These experiments also serve as an important reference for the MPN experiments discussed in the following chapters. The AC susceptograms of the tracers are shown in Fig. 3. All tracer particles show a typical Debye-type relaxation behavior in water, with a sigmoidal shape of the real part of the susceptibility χ′, and a maximum for the imaginary part χ′′. The fit of the single Debye relaxation process including a particle size distribution is exemplarily shown in Fig. S3 (ESI†). With increasing diameter of the tracer particles, the transition is shifted to lower frequencies as expected. All susceptograms can be fitted according to a single Debye relaxation process in a Newtonian fluid with a minor distribution of relaxation times (due to the minor size distribution of the tracer particles), the average characteristic relaxation time τ_{B} is estimated, and by employing the viscosity of water at 25 °C, η_{0} = 0.89 mPas,^{30} the volume-weighted hydrodynamic radius r_{h} of the respective tracer particles and its distribution are determined. A summary of the obtained hydrodynamic sizes of the probe particles in water is given in Table 2.
According to the results discussed above, all particle batches show spherical shape, narrow size distribution, and well-defined hydrodynamic and magnetic properties and are predominantly magnetically blocked, suggesting that the particles are well suited as magnetic tracer particles for Magnetic Particle Nanorheology based on dynamic magnetometry.
The structure and dynamics of polymer solutions are characterized by various parameters. First of all, the concentration regime of the polymer solution is important, and thus information on the overlap volume fraction ϕ*, the entanglement volume fraction ϕ_{e} and the critical molar mass M_{e} is required. With M_{e} = 5870 g mol^{−1} is accessible from the literature,^{41} we can expect that P4k solutions are not entangled independent of the polymer volume fraction, since the critical molar mass of PEG with M_{e} is not reached.
ϕ* and ϕ_{e} of the aqueous PEG solutions are molar-mass-dependent and obtained from static macroscopic rheological measurements. The zero shear viscosity η_{0} is extracted from rheometric steady-state flow curves by extrapolating the viscosity η to σ → 0. In Fig. 4, the dependence of η_{0} on the volume fraction ϕ and molar mass M_{w} of the polymer component is shown. For each polymer solution series, there are two concentration regimes with different scaling exponents m_{1} and m_{2}, corresponding to the semi-diluted regime and to the concentrated regime, respectively. From the cross-section of the linear fits in the double-log plot, the critical polymer volume fraction ϕ_{c} is obtained and found to be close to the literature values for the overlap volume fraction ϕ*, or the entanglement volume fraction ϕ_{e} (Table 4).^{24,31}
Sample | ϕ_{theo.}* [v%] | ϕ_{e,theo} [v%] | ϕ_{c,exp} [v%] | m_{1} (ϕ < ϕ_{c,exp}) | m_{2} (ϕ < ϕ_{c,exp}) |
---|---|---|---|---|---|
a Apparent, since theoretically not defined for this molar mass. | |||||
Theo. | — | — | — | m_{1,theo} = 1.000 | m_{2,theo} = 3.927 |
P4k | 7.59 | n. a. | 13.88^{a} | 0.65 ± 0.08 | 2.26 ± 0.11 |
P20k | 2.22 | 14.35 | 11.94 | 0.92 ± 0.11 | 3.66 ± 0.23 |
P35k | 1.45 | 9.36 | 9.46 | 1.09 ± 0.15 | 3.57 ± 0.13 |
With increasing molar mass M_{w}, ϕ_{c} decreases as expected, and the scaling of η_{0} with ϕ is in good agreement with the theoretical predictions. The deviations present can be ascribed to the finite molar mass distribution of the polymer component.
When considering the relevance of the tracer size, it becomes further important to estimate the characteristic length scales within the solutions based on theoretical predictions. Of relevance are the radius of gyration R_{g}, the correlation length ξ, and, for entangled solutions, the tube diameter a.
The radius of gyration R_{g}, defined via the mean square distance of the building blocks to the center of gravity of the molecule, is weakly dependent on the polymer volume fraction ϕ (eqn (5)).^{24}
(5) |
(6) |
The correlation length ξ is defined as the average distance between one monomer unit of a polymer chain to the nearest monomer unit of another polymer chain, while a is related to the topological confinement of a polymer chain by the surrounding chains to a tube-like region.^{24} Both parameters are independent of M_{n}, but depend on the polymer volume fraction ϕ. (eqn (7) and (8)).^{24,25}
ξ(ϕ) = bϕ^{−ν/(3υ− 1)} | (7) |
a(ϕ) = a(1)ϕ^{−ν/(3υ−1)} | (8) |
The dependence of R_{F} on M is shown in Fig. 5(left) and indicates its relation to the characteristic tracer particle size r_{h}. R_{F} increases with M, indicating that by varying the molar mass M_{n} of the polymer and the size r_{h} of the tracer particles, it is possible to selectively study different regimes of r_{h} with respect to R_{F}. The correlation length ξ and the imaginary tube diameter a are shown in dependence on the polymer volume fraction ϕ in Fig. 5(right). Both parameters decrease with increasing ϕ, with a being always larger than ξ. By relating a and ξ to the probe particle size r_{h}, one obtains critical concentrations where ξ(ϕ_{ξ}) = r_{h} or a(ϕ_{a}) = r_{h}, as listed in Table S1 (ESI†). Based on these parameters, the particles can be classified into small, intermediate and large particles as suggested by Rubinstein, as shown in Fig. 12.^{25} Upon systematic variation of the polymer molar mass M_{n}, the polymer volume fraction ϕ and the probe particle size r_{h}, a whole bandwidth of polymer concentration regimes and relations of particle size to polymer size is accessible and its specific influence on the nanorheological results can be studied. In the investigated polymer fraction and molar mass regime, the particles CF_12 undergo all of the three different regimes (large particles: d_{h} < a; intermediate particles: a < d_{h} < ξ; small particles: ξ < d_{h}), whereas the two larger particles are expected to mainly behave as intermediate and large particles.
Accordingly, the polymer solutions under investigation are well-suited as model systems, and thus in line with recent experimental studies, the system is particularly suited to allow the prediction of characteristic time- and length scales based on the theoretical findings and experimental results of others.
In Fig. 6, the AC susceptograms based on three different tracer particles and solutions of three different polymers are shown. Generally, the spectra are shifted to lower frequencies with increasing polymer concentration ϕ, in accordance with an increase in viscosity η (see 1.2). Similarly, the spectra shift to lower frequencies with increasing particle size r_{h}, in accordance with an increased rotational relaxation time τ_{B}.
The spectra based on the two larger tracer types mainly show a symmetric shift with a conserved curve shape, indicating that even at high volume fractions the particles experience a single Debye relaxation process in these solutions, similar to the one observed in water. In contrast, the spectra of CF_12 are more complex. They show a broadening of the peak and partly even bimodal behavior with increasing volume fraction ϕ.
Analyzing the susceptograms by means of the Gemant-Bishop-diMarzio model allows the detailed investigation of the influence of the intrinsic parameters of the polymer solutions on the rotational particle diffusivity. For this purpose, we calculate the mechanical loss modulus G′′(f) according to eqn (2), and compare it to the results of the macrorheological frequency sweeps (see Fig. 7a and Fig. S4–S6, ESI†) with emphasis on the influence of different parameters such as the polymer volume fraction ϕ, the molar mass of the polymer M, and the particle size r_{h}.
Fig. 7 (a) Loss moduli determined by macrorheology (full symbols) and nanorheology (empty symbols) in dependence on frequency f for P35k aqueous solutions for probe particles CF_12. The volume concentration is varied between 1.8–35.2 v% (dark red: ϕ = 1.8 v%; red: ϕ = 4.5 v%; orange: ϕ = 9.0 v%; light orange: ϕ = 13.5 v%, yellow: ϕ = 18.1 v%, light green: ϕ = 22.8 v%; olive green: ϕ = 27.5 v%; turquoise: ϕ = 32.3 v%; light blue: ϕ = 35.2 v%) and (b) schematic representation of loss moduli curve course in dependence on frequency for different relations of probe particle size to structural units of the polymer based on the theory set up by Rubinstein for translational motion of particles in complex and structured solutions.^{25} (blue: large particles (a < d_{h}), red: intermediate particles (ξ < d_{h} < a), and green: small particles (d_{h} < ξ)). Here, τ_{ξ} is the relaxation time of the polymer connected to the correlation length, τ_{d} is the relaxation time of a polymer segment with the same size as the particle size, τ_{e} represents the relaxation time of an entanglement strand and τ_{rep} is the relaxation time of a whole polymer chain. |
While the macroscopic results for all samples show a Newtonian behavior for all samples in the investigated frequency range, generally the correspondence between macroscopically and nanoscopically derived data is the best for large particles, low polymer fractions and low molar mass of the polymer component. For increasing M_{n} and polymer fraction, the smaller particles show deviations between the methods, and a frequency-dependent behavior is observed for the smaller tracer particles. In the following, these observations are analyzed in more detail.
First of all, it is important to point out that the absolute values in the overlapping frequency region of macroscopic and nanoscopic measurements show a good agreement for the two larger (SiO_{2}-coated) particle batches, in particular at low molecular mass M and polymer volume fraction ϕ. In contrast to this, with decreasing particle size, the increasing volume fraction of the polymer ϕ and molar mass M the discrepancies between the two measurement methods increase. For the P35k solutions investigated with CF_12 particles, for the two lowest polymer volume fractions a good correspondence between the nanoscopic and macroscopic results is found. Here, the particles are expected to behave as ‘small probe particles’ (compare to Fig. 5 and 7b) with a slope of 1, resulting in a constant viscosity and this is experimentally found. For higher polymer fractions a deviation in the frequency-dependent behavior as well as in the extrapolated apparent viscosity at low frequencies is found. This can be explained since the particles go through the transition from ‘intermediate’ to ‘large particles’ with a frequency dependent scaling behavior.
A theoretical description of the scaling of G′′ with f for different relations of probe particle size to structural units is based on the theory set up by Rubinstein^{25} for translational motion of particles in polymer solutions. According to this approach, a different scaling behavior is expected for the so-called ‘large particles’ (larger than the tube diameter of the solution a), for ‘intermediate-sized particles’ with a radius or diameter between the correlation length and the tube diameter, and for ‘small particles’ that are smaller than the correlation length ξ, as schematically shown in Fig. 7b.
Accordingly, in the series discussed here, we expect a scaling (G′′ vs. f) of 1 over the full frequency range for volume fractions below ϕ_{ξ} = 4.5%, since the particles dynamics are mostly affected by an effective viscosity of the polymer solution. Since r_{h} < R_{F}, a reduction in effective viscosity as compared to the macroscopic behavior is present, as the particles mainly diffuse the solvent-rich domains of the solution. Hence, the apparent viscosity as obtained from these experiments is lower than the macroscopic zero-shear viscosity η_{0} and closer to the viscosity of the solvent η_{s}. For entangled solutions between ϕ_{ξ} < ϕ < ϕ_{a}, we expect a behavior as ‘intermediate-sized’ tracer, and this is in accordance with the observed subdiffusive behavior at intermediate frequencies for solutions between ϕ = 4.5% and 22.8%.
According to the theory, the subdiffusive regime is found at time scales between τ_{ξ} (relaxation time of the polymer connected to the correlation length) and τ_{d} (relaxation time of a polymer segment with the same size as the probe particle size). In the polymer volume fraction regime where ϕ > ϕ_{a}, the tracers are expected to behave as ‘large particles. In entangled solutions and at long observation times (t < τ_{rep}, reptation time of a polymer chain), the diffusion of ‘large’ particles is determined by bulk viscosity of the polymer system, and a slope of 1 is predicted for G′′(f). At shorter times (τ_{rep} > t > τ_{e}, the relaxation time of an entanglement strand), the tracer particles have to wait for the polymer mesh to relax, and strongly subdiffusive behavior or even a plateau is predicted. Below τ_{e}, the particle diffuses within its mesh with a behavior similar to intermediate particles. For CF_12 tracer particles in P35k solutions, this is clearly confirmed for ϕ > 22.8%.
It is thus experimentally shown that the nanorheological behavior as extracted from MPN is in accordance with the predicted behavior and that for the investigated polymer solutions, the smallest tracer particles CF_12 allow the observation of the transition from small via intermediate-sized to large tracers. The volume fractions of the transitions observed are close to the expected volume fractions ϕ_{ξ} = 4.5% and ϕ_{a} = 21.9% as calculated for the CF_12 particles.
Accordingly, this classification seems feasible for the detailed discussion of the size-related results, and is later used to establish a phase diagram of the entire system (see Section 2.6).
In the series with ϕ = 4.5 v%, the tracer particles are expected to be in the ‘intermediate’ size regime, as for CF@SiO_{2}_29, ϕ_{ξ} = 1.4 v% and ϕ_{a} = 7 v%. Experimentally, the results confirm a nearly Newtonian-like behavior for f > 100 Hz at a viscosity close to the macroscopic one, however, also a reduced effective viscosity at lower frequencies. The deviations at low frequencies can result from the fact that in the correction procedure for the evaluation of the data using the GDB model a distribution of hydrodynamic size and viscosity resulting from the measurements in Newtonian fluids is assumed and the direct transfer of these parameters to complex fluids might account for the errors. This latter effect is not very exposed due to the fact that the tracer radius is larger than or close to R_{f}, and it should be pointed out that none of these solutions is entangled. The deviations at high frequencies may be caused by the application of a frequency-independent value of χ_{N}′ at χ_{∞} for the correction of the data.
It will be helpful to compare this behavior to higher concentrated, entangled and non-entangled solutions.
Accordingly, the results of the series employed with the same tracer particles (CF@SiO_{2}_29) at a polymer volume fraction of ϕ = 22.8 v% show a more complex behavior. Again, for the P4k solution there is a good agreement between the macroscopic and the nanoscopic measurements. This is expected, since the molar mass of P4k is below the critical value for the occurrence of entanglements M_{e}, and thus the solution in all regimes behaves principally similar to a simple Newtonian fluid for all length scales considerably larger than R_{F} (or R_{g}) and ξ.
The nanoscopic measurements of the higher molar mass polymers at ϕ = 22.8 v% show a strong decrease of the apparent viscosity η′ at higher frequencies. This behavior is attributed to the diffusion of a tracer particle within its mesh of the entangled networks of the P20k and P35k solutions. According to the theoretical predictions,^{25} the respective transition is expected at time scales between τ_{e} (or τ_{d}) and τ_{rep}, which is addressed further in Section 2.6.
For the entangled solutions at ϕ = 22.8 v%, a good agreement between macroscopic and nanoscopic measurements is found for the two silica coated particle batches in the frequency overlap region, while a plateau is observed in the intermediate frequency range. CF_12 tracers show additionally a deviation in the effective viscosity to lower values in the lowest accessible frequency range. Such a plateau is in accordance with the predictions and is expected between τ_{e} (or τ_{d}) and τ_{rep} (see Section 2.6).
For all investigated systems, a decrease in the normalized rotational diffusion is found with increasing reduced volume fraction ϕ/ϕ_{a}. Depending on the tracer particle size and the molar mass of the polymer, up to three distinct regimes of scaling with the reduced polymer fraction are observed.
For all particle batches, we observe a decrease of D_{r} with ϕ^{−1.5} for volume fractions between ϕ_{ξ} and ϕ_{a}, and the results are thus principally in agreement with theoretical predictions.^{25} Deviations are observed for P4k, where at this low molar mass, ϕ_{a} is not defined.
At volume fractions larger than ϕ_{a}, for the two larger particle batches, a good agreement between the macroscopic and nanoscopic experiments is shown, as indicated by the scaling behavior and cross-over frequency of the two regimes identified. For the smallest tracer particles CF_12 though, a particularly sharp transition as compared to the larger tracers is observed at ϕ_{a}, and a sharp decrease of D_{r} at increasing volume fraction is found, in accordance with an arrest of the particle within the entanglement mesh. This behavior is in good accordance with the predictions from Rubinstein's theory, predicting a scaling behavior of 0 for ϕ < ϕ_{ξ}, of −1.52 for ϕ_{ξ} < ϕ < ϕ_{a}, with a sharp step of D_{r} at ϕ = ϕ_{a} for the case of small particles. For intermediate-scaled particles, we obtain a scaling exponent of about −3.0 from these results, in accordance with Fig. S10 (ESI†).
By analyzing the frequency dependence of the loss moduli G′′ in detail, all experiments can be classified into ‘small’, ‘intermediate-sized’ or ‘large’ tracer behavior. Therefore, the scaling behavior of the different curve segments is analyzed and compared to the theoretical expectations (e.g., Fig. 7b). From this, also the inherent characteristic time for reptation, τ_{rep}, is extracted for experiments on entangled solutions with ‘large’ tracer particles (compare Fig. 11) which is discussed later in this section.
For a comprehensive overview on the results obtained for the entire investigated system, a phase diagram is constructed (Fig. 12), in which the hydrodynamic particle radius r_{h} normalized over the radius of gyration at infinite dilution R_{F} is plotted against the reduced polymer volume fraction ϕ/ϕ*. From theoretical calculations ξ, a, R_{F} are calculated in dependence on ϕ/ϕ* and included in the phase diagram.
The results can be summarized as follows: for three series of experiments, the tracers are considerably larger than the radius of gyration at infinite dilution, R_{F}: CF@SiO_{2}_42 tracer particles in P4k and P20k, as well as CF@SiO_{2}_29 tracers in P4k. In these experiments it is observed that the behavior is nearly Newtonian for most concentrations (indicated by the green color of the data point in the phase diagram), and that subdiffusive behavior is observed at intermediate time scales, in case the concentration becomes large enough (ϕ ∼ ϕ_{a} or ϕ_{ξ}). This is indicated by red symbols in the phase diagram.
The increasing polymer molar mass and/or decreasing particle size results in r_{h} ∼ R_{F}. This is the case for CF@SiO_{2}_42 in P35k, CF@SiO_{2}_29 in P20k, and for CF_12 in P4k. Principally, we observe a similar behavior, however, in the entangled or overlapped solutions with ϕ > ϕ*, the predicted plateau-behavior expected for ‘large’ tracers can be observed (indicated by blue symbols in the diagram).
Finally, the tracer size is clearly lower than R_{F} for the series of CF@SiO_{2}_29 in P35k, as well as for CF_12 in P20k and in P35k. In these series, the predictions on all three regions (‘small’, green symbols; ‘intermediate’, red symbols; and ‘large’ tracers, blue) are excellently met with the predictions from theory. On the other hand, when the particles are slightly larger than the polymer size, still, the different regimes are found, but here the border between the different regimes seems to be dominated by the entanglement volume fraction ϕ_{e}/ϕ*. The picture that arises here is, that below ϕ_{e}/ϕ* the particles diffuse nearly Newtonian, whereas above ϕ_{e}/ϕ* the response of the particles is mainly dominated by the entanglements of the system.^{42}
Accordingly, the other important result is that R_{F} (or R_{g}) is the relevant cross-over quantity in order to classify macro- to nanoscopic rheological behavior. However, it should be stated that for r_{h} > R_{F}, still time-related processes are detected, and for the scaling with tracer size in this regime a good theoretical description is not available to date.
As a final step in our evaluation, we confirm the suitability of this size-dependent method to obtain realistic information on the solution dynamics. For this purpose, we exemplarily extract the reptation time τ_{rep} as a well-defined property from the curves of sufficiently concentrated and entangled solutions. The data are plotted in Fig. 11 and compared to the theoretical prediction. Taking into account the available data points we determine an experimental scaling with ϕ^{2.35}, whereas the theoretical expected scaling is ϕ^{1.65}. This requires more theoretical and experimental consideration in upcoming studies.
Taking further into account that Rubinstein's theory is set up for the passive translation of tracer particles in similar systems, the agreement with our data, that is exclusively on the rotational relaxation of magnetically deflected tracer particles, is not fully unexpected, but still surprising.
Based on the experimental and theoretical considerations of others,^{27,38,43–45} e.g. the scaling behavior of η_{app} vs. ϕ is expected to be weaker for rotation than for translation, and might further be affected by a depletion layer.^{38,43,45,46}
Even if the latter was not the focus of the present study, we hope that our results foster the development of a more developed theoretical framework for rotational diffusion in complex fluids and might give rise to upcoming detailed theoretical description focusing on rotation. In view of the versatility and broad applicability of the method, along with the actual development of similar or related approaches, this would be very attractive.
For this purpose, we use spherical, magnetically blocked tracer particles of different hydrodynamic size, and determine the frequency dependent rheological properties, such as viscosities η, loss moduli G′′ and diffusion properties of the probes, e.g. the rotational diffusion coefficient, from the real and imaginary part of the AC susceptometry spectra. The results are compared and complemented by macroscopic measurements as well as theoretical approaches to obtain a complete picture on the dynamic behavior of this system.
Our results show that it is possible to classify the solutions and their combination with the tracers into three important regimes, by considering the tracer size and the polymer-derived length scales in the solution, as long as the tracers are in the size range of the gyration radius R_{g}. Our experimental findings are consistent and show good semi-quantitative agreement with theoretical expectations for the passive, translational diffusion of particles in such solutions, as predicted by Rubinstein and coworkers. In particular, for tracer particles smaller than the radius of gyration of the polymer solutions, the predictions for ‘large’, intermediate-sized’ or ‘small’ tracer behavior are excellently met, while partly pronounced deviations from macrorheological measurements are found. If the tracer particles are larger than R_{F}, the nanorheological results are closer to the expectations from macrorheology, and are strongly affected if entanglements are present.
This detailed study on the size-dependent rheological properties of polymer solutions can serve as a basis for the advancement of the present rheological descriptions, and is suitable to serve as a basis for further investigations of more complex polymer systems such as dynamic networks,^{32} covalently crosslinked gels or biological systems. In particular, MNP can be used to specifically address particle–matrix interactions and to determine for example mesh sizes in less known systems.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp04083h |
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