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BREAKDOWN OF THE STOKES-EINSTEIN RELATION FOR THE ROTATIONAL DIFFUSIVITY OF POLYMER GRAFTED NANOPARTICLES IN POLYMER MELTS Lorena Maldonado-Camargo, and Carlos Rinaldi Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b02359 • Publication Date (Web): 11 Oct 2016 Downloaded from http://pubs.acs.org on October 13, 2016
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BREAKDOWN OF THE STOKES-EINSTEIN RELATION FOR THE ROTATIONAL DIFFUSIVITY OF POLYMER GRAFTED NANOPARTICLES IN POLYMER MELTS Lorena Maldonado-Camargo1 and Carlos Rinaldi, 1,2* 1
Department of Chemical Engineering, University of Florida, P.O. Box 116005, Gainesville, FL 32603 2
J. Crayton Pruitt Family Department of Biomedical Engineering, P.O. Box 116131, Gainesville, FL 32611
*Author to whom correspondence should be addressed, J. Crayton Pruitt Family Department of Biomedical Engineering, P.O. Box 116131, Gainesville, FL 32611 Fax: (352) 273-9221. Tel.: (352) 294-5588. E-mail:
[email protected] We report observations of breakdown of the Stokes-Einstein relation for the rotational diffusivity of polymer-grafted spherical nanoparticles in polymer melts. The rotational diffusivity of magnetic nanoparticles coated with poly(ethylene glycol) dispersed in poly(ethylene glycol) melts was determined through dynamic magnetic susceptibility measurements of the collective rotation of the magnetic nanoparticles due to imposed time-varying magnetic torques. These measurements clearly demonstrate the existence of a critical molecular weight for the melt polymer, below which the StokesEinstein relation accurately describes the rotational diffusivity of the polymer-grafted nanoparticles and above which the Stokes-Einstein relation ceases to apply. This critical molecular weight was found to correspond to a chain contour length that approximates the hydrodynamic diameter of the nanoparticles. Keywords: rotational diffusivity; magnetic nanoparticles; Stokes-Einstein; dynamic magnetic susceptibility Introduction Increasingly, advancing applications of nanoparticles requires quantitative understanding of their dynamics in complex fluids. For example, nanoparticles intended for biomedical applications must navigate what is often a complex, crowded, and confined environment consisting of concentrated mixtures of macromolecules with sizes in the nanoscale and larger cellular and tissue structures. On the other hand, many attractive applications of nanoparticles require their dispersion in a polymer matrix, where they influence mechanical/structural properties or grant new functional properties to the composite. Fundamental, quantitative understanding of the translational and rotational motion of nanoparticles in such complex environments is crucial for engineering nanoparticles with optimal properties, as well as understanding their transport and fate.
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Commonly, it is assumed that the translational and rotational diffusion of small particles through simple viscous (Newtonian) fluids are described by the Stokes-Einstein (SE) relations and its modification by Debye1
DT − SE =
k BT 3πη Dh
;
DR − SE =
k BT πη Dh3
(1)
where DT − SE and DR − SE are the translational and rotational diffusion coefficients of the particles, η is the viscosity of the fluid, Dh is the hydrodynamic diameter of the particles, T the temperature of the system, and k B is the Boltzmann constant. The applicability of the SE relations in describing the random thermal motion of particles in simple fluids and under conditions where the particles are much larger than other fluid constituents is well established and forms the basis of techniques to characterize the properties of the particles (e.g., dynamic light scattering and nanoparticle tracking analysis) or of the fluid (e.g., as in microrheology, using the generalized SE relation). In contrast, recent theoretical and experimental work has shown that the translational diffusion of nanoparticles in complex fluids, such as polymer melts, can deviate from the classical SE relation.2-4 For example, the translational dynamics of nanoparticles in polymer melts could be influenced by melt properties such as polymer static length, spacing between entanglements, correlation length and radius of gyration, by the internal dynamics of the polymer chains, or by the interactions between the melt polymer and the particle surface, causing the SE relation to fail.5-8 Breakdown of the SE relation for the translational diffusivity of nanoparticles when the particle’s characteristic length approximates characteristic lengths of the polymer melt was suggested by Brochard-Wyart and de Gennes4 and more recently extended by Cai et al.9 Experimentally, the work by Mackay and Tuteja3 is a landmark in the field as it was the first experimental observation of the breakdown of the SE relation for the translational diffusivity in nanoparticle systems. Using oleic acid coated cadmium selenide quantum dots with a small hydrodynamic diameter (~5 nm), smaller than the Edwards’ tube diameter for the molecular weight of the polystyrene used in the experiments, they saw that the experimental diffusivity of the quantum dots was larger than the one predicted by the SE relationship by about 2 orders of magnitude. Subsequent studies have confirmed this finding and further tested the predictions of the theory by Brochard-Wyart and de Gennes by characterizing the extent of SE breakdown as a function of melt molecular weight and nanoparticle size over limited ranges.10, 11 In spite of the several studies of breakdown of the SE relation for translation of nanoparticles in complex fluids, the phenomenon of breakdown of the SE relation for rotation of nanoparticles remains unexplored. In this work, we directly measure the rotational diffusivity of polymer-grafted sub-40 nm diameter nanoparticles in polymer melts of a wide range of molecular weights using dynamic magnetic ACS Paragon Plus Environment
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susceptibility (DMS) measurements and identify a critical molecular weight above which the SE relation is no longer applicable. The magnetic nanoparticles (MNPs) used in these experiments were grafted with the same polymer as used in the melt phase. This is a departure from prior studies of the breakdown of the SE relation for the translational diffusivity, where the surface chemistry of the nanoparticles was not "matched" to that of the polymer melts. Nanoparticles intended for dispersion in polymers are often coated with a brush of polymer to improve their dispersion12-15 and our observations are a first example of breakdown of the SE relation for polymer-grafted nanoparticles. Estimating Nanoparticle Rotational Diffusivity through Dynamic Magnetic Susceptibility Measurements DMS measurements are a unique method to assess the rotational dynamics of magnetic nanoparticles in simple and complex fluids. This is achieved by inducing rotation of the nanoparticles using time-dependent magnetic torques. Because the measurement principle is magnetic and highly sensitive to changes in rotational diffusivity, the technique can be applied to small (50-200 µl) and optically-complex samples. Previously, this technique has been used to detect the presence of MNPs aggregates in polymer melts and to monitor adsorption of proteins to MNPs by changes in the rotational diffusitivity of the nanoparticles.16, 17 DMS measurements of suitably synthesized MNPs could also be used to sense porperties in polymer melts such as phase transitions of polymer matrices and nanoscale viscosity of small solvent fluids.18-20 Finally, DMS measurements have been used to evaluate the mobility of nanoparticles in polymer solutions and hydrogels.21, 22 During a DMS measurement, a small amplitude alternating magnetic field (AMF) is applied to the sample and the frequency-dependent magnetization of a suspension of MNPs is measured. For the small field amplitudes used in a DMS measurement, the magnetic response of the nanoparticles is linear with the field and the model developed by Debye for non-interacting dipoles in an alternating electric field can be adapted to describe the dynamic response of the nanoparticles to the AMF.23-25 According to Debye’s theory, the dynamic magnetic susceptibility χˆ of the particles in a quiescent, dilute suspension is described by
χˆ = χ ′ − j χ ′′ ; χ ′ = χ ∞ +
χ0 − χ∞ ( χ − χ ∞ ) Ωτ ; χ ′′ = 0 2 2 1+ Ω τ 1+ Ω 2τ 2
(2)
where, χ ′ and χ ′′ are the real and imaginary components of the dynamic magnetic susceptibility, Ω is the AMF frequency, χ 0 is the initial susceptibility, χ ∞ is the infinite frequency susceptibility, and τ is the magnetic relaxation time of the particles in the suspension. According to Eq. (2), as the frequency
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increases the in-phase component
( χ ′)
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monotonically decreases while the out-of-phase component
( χ ′′ ) has a maximum at the condition where Ωτ = 1 . Under a small amplitude magnetic field, MNPs tend to align their magnetic dipoles in the direction of the applied field.26 For the cobalt ferrite nanoparticles used here the magnetic relaxation time is related to the rotational diffusivity DR of the nanoparticles
τ=
π Dh3η 1 = 2DR 2k BT
(3)
where Dh is the hydrodynamic diameter of the particles, η the viscosity of the surrounding fluid, T is the absolute temperature, and 𝑘! is Boltzmann’s constant. Hence, an estimate of the rotational diffusivity of the particles can be determined using the frequency corresponding to the peak in the outof-phase susceptibility from
DR− p =
Ω peak 2
=
1 2τ
(4)
which is accurate for nanoparticles with narrow size distributions. The effects of particle size polydispersity can be accounted for by weigthing the Debye model using a volume-weighted lognormal hydrodynamic diameter distribution n ( Dh ) 27, 28 3 πη Dhpv Dh 1 χ ′′ = ∫ nv ( y ) dy ; y = ; τ hpv = = 2 2 6 Dhpv 2k BT 2DR−r 1+ Ω τ hpv y 0 ∞
χ 0,hpv Ωτ hpv y 3
(5)
where
⎛ ln 2 ( Dh Dhpv ) ⎞ 1 ⎟ nv ( Dh ) = exp ⎜ − ⎜ 2ln 2 σ g ⎟ 2π Dh ln σ g ⎝ ⎠
(6)
Dhpv is the volume weighted mean diameter and ln σ g is the geometric deviation. In Eq. (5) χ 0,hpv , τ hpv are the volume weighted average initial susceptibility and relaxation time, respectively. The DMS spectra can be fitted to Eq. (5), weighted using Eq. (6), through a non-linear regression in order to determine properties such as the size distribution of the nanoparticles in a fluid of known viscosity, or the viscosity of the suspending medium by using nanoparticles of known size distribution. Of relevance to the present work, Eq. (5) and Eq. (6) can be used to determine the effective relaxation time τ hpv , and therefore the rotational diffusivity DR-r of the nanoparticles. For particles with narrow hydrodynamic size distributions the rotational diffusivities determined by both methods are expected to be quantitatively similar.
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Results and Discussion In the experiments reported here, poly(ethylene glycol) (PEG) was chosen as both the melt polymer and the polymer coating grafted onto the nanoparticles, due to its relative commercial availability in a wide range of molecular weights, its relevance for biomedical applications, and our past experience with coating magnetic nanoparticles with covalently-grafted PEG brushes.29,
30
PEG melts (Polymer
Standards Services) had molecular weights (Mw) ranging from 1 to 55 kDa (see Table S1 in the Supporting Information). All polymers used had narrow molar mass distributions, with most having a polydispersity index (PDI) below 1.2 (see Table S1). Furthermore, all polymer melts displayed apparent Newtonian behavior at the temperatures studied here (see Fig. S1). As expected, the viscosity of the PEG melts increased with increasing molecular weight and decreased with temperature. The magnetic nanoparticles consisted of cobalt ferrite cores with narrow size distribution synthetized by a thermal decomposition procedure and grafted with PEG-silane.30-33 Four nanoparticle samples were prepared, with different core diameters and grafted PEG molecular weights. The core diameter was determined by transmission electron microscopy (TEM), whereas hydrodynamic diameters were determined by dynamic light scattering, as illustrated in Fig. 1 and summarized in Table S2. ,There was evidence in the DLS measurements of some aggregates of ~200 nm in diameter, however, the primary distribution was narrow and the particles were found to be colloidally stable. To confirm that the nanoparticles responded to alternating magnetic fields according to the Debye model weighted by the lognormal size distribution and that their magnetic relaxation time was given by their rotational diffusivity, their DMS spectra was measured in water. Fig S2 in the Supporting Information illustrates these measurements, while Table S2 summarizes the hydrodynamic diameters obtained by fitting the DMS spectra to Eq. (5) using the viscosity of water. The close agreement between the hydrodynamic diameters obtained from DLS and DMS measurements indicates that the nanoparticles respond to alternating magnetic fields by physical particle rotation, in what we call predominant Brownian relaxation.18-20 Fig 2a shows representative DMS spectra for the sample ICO-16 PEG 2kDa dispersed in the 2 kDa PEG melt at various temperatures. The observation that the in-phase and out-of-phase susceptibilities cross at the peak of the out-of-phase susceptibility indicates Debye behavior in the PEG melt. Since the SE relation predicts that the rotational diffusivity of a particle is linearly proportional to the viscosity of the sample, as the molecular weight of the free polymer increases one would expect the peak frequency of the out-of-phase susceptibility to decrease. This is illustrates in Fig 2b, which shows out-of-phase susceptibility spectra calculated according to Eq. (5) using the melt viscosities determined by rheometry (Fig S1). In contrast, experimental spectra for the out-of-phase susceptibility (Fig 2c) show that the measured peak frequency decreases with increasing melt molecular weight up to about 4 kDa and then ACS Paragon Plus Environment
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remains essentially constant, independent of the molecular weight of the melt polymer. Importantly, for all melt molecular weights the DMS spectra continue to possess the shape predicted by the Debye model for nanoparticles with a single relaxation time. This suggests that fitting of the spectra to Eq. (5) can be used to estimate the nanoparticle’s rotational diffusivity in the melt, for comparison with the rotational diffusivity estimated from the SE relationship. Fig 3 illustrates several comparisons between the rotational diffusivity determined from DMS measurements and calculated according to the SE relationship. Fig 3a compares the experimental and calculated rotational diffusivities for sample ICO-16 PEG 2kDa at 353 K. Similar comparisons for the other three samples are shown in Fig S3. From these results it is evident that the rotational diffusivity of the nanoparticles in the PEG melts is accurately predicted by the SE relation for low PEG melt molecular weights. However, as the molecular weight of the melt increases above a critical value the nanoparticles are found to diffuse faster than predicted by the SE relation. These observations seem to indicate that breakdown of the SE relation occurs at a PEG melt molecular weight that varies with nanoparticle sample. The contour length corresponding to each melt molecular weight, calculated based on the Kuhn length34 of PEG and the molecular weigth of the melt polymer, is shown in Fig. 3a as a secondary horizontal axis. According to these calculations, the melt polymer chain length associated with breakdown of the SE relation for the ICO-16 PEG 2kDa nanoparticles corresponds to 20-40 nm. This range is close to the hydrodynamic diameter of the nanoparticles, suggesting that breakdown occurs when the melt polymer contour length is greater than the characteristic length scale of the nanoparticles. The molecular weight for deviation from the SE relationship did not appear to change with temperature of the measurements, in the experimental range of 333-353 K. This is demonstrated in Fig 3b for ICO-16 PEG 2kDa, where the ratio of rotational diffusivity determined from DMS measurements to rotational diffusivity calculated according to the SE relationship is shown for three measurements temperatures as a function of the contour length of the melt polymer. Finally, Fig 3c shows the ratio of rotational diffusivity determined from DMS measurements to rotational diffusivity calculated according to the SE relationship as a function of the ratio of melt polymer contour length to hydrodynamic diameter of the nanoparticles for all four nanoparticle samples. Interestingly, the data for all four samples appears to fall along a master curve. This plot shows that, for all samples, breakdown of the SE relationship appears to happen when the melt polymer contour length exceeds the diameter of the nanoparticles. It is relevant to consider how the characteristic length scales of the nanoparticles used here (namely their core and hydrodynamic diameters) compare with important length scales for the polymer melts. For all samples, the nanoparticle hydrodynamic diameters are much larger than the radius of gyration of the polymers in the melts (see Table S3 in the Supporting Information). It is also much larger than the ACS Paragon Plus Environment
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(
)
Edward’s tube diameter for PEG35 dT = 3.73nm ≪ d h = 32 nm . Thus, it is unlikely that the observed breakdown of the SE relationship for the rotational diffusivity can be explained by the theory of Brochard-Wyart and de Gennes.4 One possible explanation for the deviation from the predictions of the SE relationship for the rotational diffusivity might be changes in the conformation of the relatively thick grafted PEG coating as the molecular weight of the melt polymer increases. To assess this possibility we calculated the effective hydrodynamic diameter the nanoparticles would need to have in order for the DMS rotational diffusivity to correspond to that calculated using the SE relation and the melt viscosity determined in the rheometer. The results of these calculations are summarized in Table S4. These calculations indicate that the effective hydrodynamic diameter of the nanoparticles dispersed in the highest molecular weight melts would have to be significantly smaller than the diameter of the organic core, as determined by TEM. Thus, while we cannot rule out compression of the grafted polymer coating as one of the contributors to the observed deviations from the predictions of the SE relationship, it is unlikely that this mechanism alone can explain our observations. Another potential explanation for the observed breakdown of the SE relationship might be the occurrence of hydrodynamic slip at the surface of the nanoparticles, which may depend on the relative molecular weights of the melt and graft polymers. Autophobic dewetting of melt polymer from grafted polymer has been reported to occur on flat surfaces and on nanoparticles,7-11 and this phenomenon could in principle lead to slip at the surface. To explore this possibility we used a model developed by Ganesan et al.12 for the rotational diffusivity of nanoparticles in polymer melts the estimate the extent of hydrodynamic slip required to explain the observed deviation from the predictions of the SE relationship. The results of these calculations are summarized in Table S5. These calculations indicate that the slip length required for agreement between the experimental rotational diffusivity and the model of Ganesan et al.12 is a function of molecular weight of the melt polymer and would have to be much larger than the hydrodynamic diameter of the nanoparticles. Indeed, for the highest melt molecular weights tested the slip length would have to be greater than 1µm, which we find implausible. Finally, Ganesan et al.12 argued that for a small particle in a polymer melt the hydrodynamic resistance to translation and rotation is determined by a phenomenological nonlocal viscosity that is determined by segments of the polymer chain of size commensurate with the wave vector for the velocity gradients induced by particle motion. The nonlocal viscosity of Ganesan et al.12 decreases with increasing ratio of melt polymer radius of gyration to hydrodynamic size of the particles. Similar arguments were used by Berg et al.,36, 37 to explain observations of breakdown of the SE relation for the rotational diffusivity of anthracene in polymer melts. According to these authors, for small melt polymer chains the SE relation describes the rotation of small objects as long as the polymer chains remain rigid, ACS Paragon Plus Environment
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i.e. particle rotation is coupled with whole polymer chain motion at the time scale of the measurements. However, as the melt polymer chain length increases further, resistance to rotation of small objects will be determined by local motions of polymer chain segments, of length similar to the size of the object. This explanation can be traced back to predictions from the Rouse model,38 in which the polymer is a chain of segments with a certain degree of flexibility. Thus, according to Ganesan et al.12 one would expect to see deviations in the effective viscosity experienced by the nanoparticles when the ratio of melt polymer radius of gyration to nanoparticle hydrodynamic radius exceeds unity, whereas according to Berg et al.,36, 37 one would expect this deviation to occur when the ratio of melt polymer contour length to nanoparticle hydrodynamic diameter exceeds unity. To test these two alternatives we calculated the effective viscosity such that the SE relationship agreed with the DMS rotational diffusivities, using the hydrodynamic diameters of the nanoparticles in the 2 kDa PEG melt. The results of the calculations are summarized in Table S6 and illustrated in Fig S4, where it is seen that deviation from unity in the ratio of effective to bulk viscosity occurs when the contour length of the melt polymer exceeds the hydrodynamic diameter of the nanoparticles, in apparent agreement with the mechanism proposed by Berg et al.,36, 37 for breakdown of the SE relationship for rotational diffusion of anthracene. In summary, the analyses described above seem to indicate that breakdown of the SE relationship for the rotational diffusivity of polymer-grafted nanoparticles in polymer melts occurs when the contour length of the melt polymer exceeds the hydrodynamic diameter of the nanoparticles. While we cannot conclusively attribute this observation to a particular mechanism, we can rule out several potential mechanisms. First, because of the large size of the particles, it is unlikely that the underlying mechanism is that proposed by Brochard-Wyart and de Gennes,4 which has been used to explain breakdown of the SE relationship for the translational diffusivity of nanoparticles smaller than the Edward’s tube diameter in polymer melts. Second, it would seem that compression of the grafted polymer coating alone cannot explain the observed deviations from the predictions of the SE relationship, as the required effective diameter of the nanoparticles would have to be smaller than the diameter of the inorganic core. Third, the existence of hydrodynamic slip at the particle/melt interface as the responsible mechanism seems implausible as it would require slip lengths of over 1 µm for the highest molecular weight melts. We note, however, that while compression of the polymer coating and hydrodynamic slip appear unlikely explanations on their own, their combination is a possibility. Finally, the observed scaling with respect to the ratio of melt polymer contour length to nanoparticle hydrodynamic diameter for deviation from the predictions of the SE relationship seems to indicate that the effective viscosity opposing rotation of the nanoparticles is determined by local motions of the polymer chain segments of size commesurate with the hydrodynamic diameter of the nanoparticles. However, further work is needed to fully attribute our observations to this mechanism. ACS Paragon Plus Environment
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Conclusions Breakdown of the SE relationship for the rotational diffusivity of polymer-grafted nanoparticles dispersed in polymer melts is reported for the first time. The rotational diffusivity of the polymergrafted nanoparticles was accurately predicted by the SE relationship for low melt polymer molecular weights, for which the melt polymer contour lengths were smaller than the hydrodynamic diameter of the nanoparticles. At high melt polymer molecular weights, such that the contour length of the polymer exceeded the hydrodynamic diameter of the nanoparticles, the nanoparticle’s rotational diffusion was found to be faster than predicted from the SE relation based on the melt viscosity measured using a traditional rheometer. Consideration of various potential mechanisms indicates that one potential explanation for the observed deviation from the predictions of the SE relationship for the rotational diffusivity is that the hydrodynamic resistance to nanoparticle rotation is determined by an effective viscosity arising from the motion of polymer chain segments of length commensurate with the size of the nanoparticles. Acknowledgements. This work was supported by the US National Science Foundation (CBET143993). The authors are grateful to The Polymer Chemistry Characterization Lab for access to GPC and DSC instrumentation. Supporting Information Available: Detailed methods of nanoparticle synthesis and characterization; PEG molecular weight distributions and melt viscosities; summary of nanoparticle size characterization; DMS spectra of nanoparticles in water; additional results for rotational diffusivities of nanoparticles in PEG melts; and results for calculations of effective diameters, slip lengths, and effective viscosities. This material is available free of charge via the Internet at http://pubs.acs.org References 1. Squires, T. M.; Mason, T. G. Annu Rev Fluid Mech 2010, 42, 413-438. 2. Omari, R. A.; Aneese, A. M.; Grabowski, C. A.; Mukhopadhyay, A. J Phys Chem B 2009, 113, 8449-8452. 3. Tuteja, A.; Mackay, M. E.; Narayanan, S.; Asokan, S.; Wong, M. S. Nano Lett 2007, 7, 12761281. 4. Wyart, F. B.; de Gennes, P. G. Eur Phys J E 2000, 1, 93-97. 5. Egorov, S. A. J Chem Phys 2011, 134. 6. Liu, J.; Gardel, M. L.; Kroy, K.; Frey, E.; Hoffman, B. D.; Crocker, J. C.; Bausch, A. R.; Weitz, D. A. Phys Rev Lett 2006, 96. 7. Valentine, M. T.; Perlman, Z. E.; Gardel, M. L.; Shin, J. H.; Matsudaira, P.; Mitchison, T. J.; Weitz, D. A. Biophys J 2004, 86, 4004-4014. 8. Yamamoto, U.; Schweizer, K. S. Macromolecules 2015, 48, 152-163. 9. Cai, L. H.; Panyukov, S.; Rubinstein, M. Macromolecules 2011, 44, 7853-7863. ACS Paragon Plus Environment
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10. Grabowski, C. A.; Adhikary, B.; Mukhopadhyay, A. Appl Phys Lett 2009, 94. 11. Grabowski, C. A.; Mukhopadhyay, A. Macromolecules 2014, 47, 7238-7242. 12. Currie, E. P. K.; Norde, W.; Stuart, M. A. C. Adv Colloid Interfac 2003, 100, 205-265. 13. de Gennes, P. G. Macromolecules 1980, 13, 1069-1075. 14. Green, P. F. Soft Matter 2011, 7, 7914-7926. 15. Kumar, S. K.; Jouault, N.; Benicewicz, B.; Neely, T. Macromolecules 2013, 46, 3199-3214. 16. Bohorquez, A. C.; Rinaldi, C. Part Part Syst Char 2014, 31, 561-570. 17. Sierra-Bermudez, S.; Maldonado-Camargo, L. P.; Orange, F.; Guinel, M. J. F.; Rinaldi, C. J Magn Magn Mater 2015, 378, 64-72. 18. Barrera, C.; Florian-Algarin, V.; Acevedo, A.; Rinaldi, C. Soft Matter 2010, 6, 3662-3668. 19. Calero-DdelC, V. L.; Santiago-Quinonez, D. I.; Rinaldi, C. Soft Matter 2011, 7, 4497-4503. 20. Herrera, A. P.; Barrera, C.; Zayas, Y.; Rinaldi, C. J Colloid Interf Sci 2010, 342, 540-549. 21. Frickel, N.; Messing, R.; Schmidt, A. M. J Mater Chem 2011, 21, 8466-8474. 22. Roeben, E.; Roeder, L.; Teusch, S.; Effertz, M.; Deiters, U. K.; Schmidt, A. M. Colloid Polym Sci 2014, 292, 2013-2023. 23. Fannin, P. C. J Alloy Compd 2004, 369, 43-51. 24. Mazo, R. M., Brownian Motion: Fluctuations, Dynamics, and Applications. Clarendon Press: 2002. 25. Shliomis, M. I.; Stepanov, V. I. J Magn Magn Mater 1993, 122, 176-181. 26. Rosensweig, R. E., Ferrohydrodynamics. Dover Publications: 1997. 27. Nutting, J.; Antony, J.; Meyer, D.; Sharma, A.; Qiang, Y. J Appl Phys 2006, 99. 28. Payet, B.; Vincent, D.; Delaunay, L.; Noyel, G. J Magn Magn Mater 1998, 186, 168-174. 29. Barrera, C.; Herrera, A. P.; Rinaldi, C. J Colloid Interf Sci 2009, 329, 107-113. 30. Barrera, C.; Herrera, A. P.; Bezares, N.; Fachini, E.; Olayo-Valles, R.; Hinestroza, J. P.; Rinaldi, C. J Colloid Interf Sci 2012, 377, 40-50. 31. Bao, N. Z.; Shen, L. M.; An, W.; Padhan, P.; Turner, C. H.; Gupta, A. Chem Mater 2009, 21, 3458-3468. 32. Herrera, A. P.; Polo-Corrales, L.; Chavez, E.; Cabarcas-Bolivar, J.; Uwakweh, O. N. C.; Rinaldi, C. J Magn Magn Mater 2013, 328, 41-52. 33. Park, J.; An, K. J.; Hwang, Y. S.; Park, J. G.; Noh, H. J.; Kim, J. Y.; Park, J. H.; Hwang, N. M.; Hyeon, T. Nat Mater 2004, 3, 891-895. 34. Rubinstein, M.; Colby, R. H., Polymer Physics. OUP Oxford: 2003. 35. Fetters, L. J.; Lohse, D. J.; Graessley, W. W. J Polym Sci Pol Phys 1999, 37, 1023-1033. 36. Sluch, M. I.; Somoza, M. M.; Berg, M. A. J Phys Chem B 2002, 106, 7385-7397. 37. Somoza, M. M.; Sluch, M. I.; Berg, M. A. Macromolecules 2003, 36, 2721-2732. 38. Ferry, J. D.; Landel, R. F.; Williams, M. L. J Appl Phys 1955, 26, 359-362.
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Figure 1. Characterization of core and hydrodynamic diameter distributions for PEG-grafted cobaltferrite magnetic nanoparticles. a) ICO-16 PEG 2kDa, b) ICO-12 PEG 2kDa, c) ICO-12 PEG 5kDa, d) ICO-21 PEG 5kDa.
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Figure 2. a) Representative DMS spectra of ICO-16 PEG 2 kDa nanoparticles dispersed in 2.0 kDa PEG melts at three different temperatures. Solid lines represent fit to the Debye model weighted by the lognormal hydrodynamic diameter distribution. b) Predicted out-of-phase susceptibility spectra for ICO16 PEG 2 kDa nanoparticles in polymer melts of different molecular weights at 343 K, calculated using the Debye model weighted by the lognormal hydrodynamic diameter distribution and using the melt viscosity determined by rheometry. c) Experimental out-of-phase susceptibility spectra for ICO-16 PEG 2 kDa nanoparticles in polymer melts of different molecular weights at 343 K.
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Figure 2. a) Experimental rotational diffusivity ( DR ) and rotational diffusivity predicted by the SE relationship ( DR−SE ) for ICO-16 PEG 2kDa in PEG melts of different molecular weights at 353 K. b) Ratio of experimental to SE rotational diffusivisities for ICO-16 PEG 2kDa as a function of the melt polymer contour length for measurements at 333 K, 343 K, and 353 K. c) Ratio of experimental to SE rotational diffusivisities as a function of ratio of the melt polymer contour length to the nanoparticle’s hydrodynamic diameter for particles with different inorganic core diameters and grafted polymer molecular weights.
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TOC Graphic 34x27mm (300 x 300 DPI)
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Figure 1 152x107mm (300 x 300 DPI)
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Figure 2 156x120mm (300 x 300 DPI)
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Figure 3 124x95mm (300 x 300 DPI)
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