Abstract
We revisit and improve the technique of piezo-operated sliding-plate rheometry in order to provide a versatile platform for measuring the linear viscoelastic properties of various soft matter systems at frequencies from 10 to 1.000 Hz. The sensitive loss angle measuring loop is validated explicitly against reference data from entangled amorphous polymer melts obtained with conventional rotational rheometers by means of time-temperature superposition (tTS). Frequency range limiting factors such as sample and tool inertia are discussed while errors are traced and theoretical correction is shown to be feasible when strong nonlinear behavior of the measuring cell is present. This gives confidence in measuring more complex systems where tTS does not apply. We also demonstrate the ability to probe the short-time dynamics of hard-sphere colloidal glasses. Important high-frequency features such as the behavior of the elastic modulus, G′, the moduli crossover frequency fc related to β-relaxation, and the associated limiting in-phase (with strain-rate), dynamic viscosity η∞′, are captured. This validates the suitability of this high-frequency rheometric technique to provide insights into interactions at nanometric particle separations.
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References
Baeza GP, Dessi C, Costanzo S, Zhao D, Gong S, Alegria A, Colby RH, Rubinstein M, Vlassopoulos D, Kumar SK (2016) Network dynamics in nanofilled polymers. Nat Commun 7:11368
Ballesta P, Petekidis G (2016) Creep and aging of hard-sphere glasses under constant stress. Phys Rev E 93:042613. https://doi.org/10.1103/PhysRevE.93.042613
Banchio AJ, Nägele G (2008) Short-time transport properties in dense suspensions: from neutral to charge-stabilized colloidal spheres. J Chem Phys 128(10):104903
Bard AJ (1980) LR Faulkner electrochemical methods. Wiley, New York
Bartolino R, Durand G (1977) Plasticity in a smectic-liquid crystal. Phys Rev Lett 39(21):1346–1349. https://doi.org/10.1103/PhysRevLett.39.1346
Bharadwaj NA, Ewoldt RH (2015) Single-point parallel disk correction for asymptotically nonlinear oscillatory shear. Rheol Acta 54(3):223–233
Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids. Volume 1: fluid mechanics. A Wiley-Interscience Publication, John Wiley & Sons
Booij H, Thoone G (1982) Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities. Rheol Acta 21(1):15–24
Bouzid M, Keshavarz B, Geri M, Divoux T, Del Gado E, McKinley GH (2018) Computing the linear viscoelastic properties of soft gels using an optimally windowed chirp protocol. J Rheol 62(4):1037–1050
Brack T, Bolisetty S, Dual J (2018) Simultaneous and continuous measurement of shear elasticity and viscosity of liquids at multiple discrete frequencies. Rheol Acta 57(5):415–428
Brady JF (1993) The rheological behavior of concentrated colloidal dispersions. J Chem Phys 99(1):567–581. https://doi.org/10.1063/1.465782
Brather A (1978) Numerisch einfache Beziehungen zwischen Verlust-und Speicherkomponente des dynamischen Schermoduls und der dynamischen Nachgiebigkeit. Rheol Acta 17(4):325–341
Bryant M, Keltie R (1986) A characterization of the linear and non-linear dynamic performance of a practical piezoelectric actuator part 1: measurements. Sensors Actuators 9(2):95–103
Bryant G, Williams SR, Qian L, Snook IK, Perez E, Pincet F (2002) How hard is a colloidal “hard-sphere” interaction? Phys Rev E Stat Nonlinear Soft Matter Phys 66(6 Pt 1):060501. https://doi.org/10.1103/PhysRevE.66.060501
Cheng Z, Zhu J, Chaikin PM, Phan SE, Russel WB (2002) Nature of the divergence in low shear viscosity of colloidal hard-sphere dispersions. Phys Rev E Stat Nonlinear Soft Matter Phys 65(4 Pt 1):041405. https://doi.org/10.1103/PhysRevE.65.041405
Colby RH (1989) Breakdown of time-temperature superposition in miscible polymer blends. Polymer 30(7):1275–1278
Colby RH, Fetters LJ, Graessley WW (1987) The melt viscosity-molecular weight relationship for linear polymers. Macromolecules 20(9):2226–2237
Colby RH, Fetters LJ, Funk WG, Graessley WW (1991) Effects of concentration and thermodynamic interaction on the viscoelastic properties of polymer solutions. Macromolecules 24(13):3873–3882
Collin D, Martinoty P (2003) Dynamic macroscopic heterogeneities in a flexible linear polymer melt. Physica A: Statistical Mechanics and its Applications 320:235–248
Cooper J, Henry F, Reiss EL (1966) Reflection of plane viscoelastic waves from plane boundaries. The Journal of the Acoustical Society of America 39(6):1133–1138
Dealy J, Giacomin A (1998) Sliding plate and sliding cylinder rheometers Rheological Measurement (pp. 237-259): Springer
Dealy J, Plazek D (2009) Time-temperature superposition—a users guide. Rheol Bull 78(2):16–31
Dhont JK, Wagner NJ (2001) Superposition rheology. Phys Rev E 63(2):021406
Doi M, Edwards SF (1988) The theory of polymer dynamics (Vol. 73): oxford university press
El Rifai OM, Youcef-Toumi K (2004) Modeling of piezoelectric tube actuators
Erwin BM, Rogers SA, Cloitre M, Vlassopoulos D (2010) Examining the validity of strain-rate frequency superposition when measuring the linear viscoelastic properties of soft materials. J Rheol 54(2):187–195
Ewing MW (1957) Elastic waves in layered media: McGraw-Hill
Ferry JD (1980) Viscoelastic properties of polymers: John Wiley & Sons
Franck A (2005) Understanding instrument inertia corrections in oscillation. TA Instrument
Fritz G, Maranzano B, Wagner N, Willenbacher N (2002) High frequency rheology of hard sphere colloidal dispersions measured with a torsional resonator. J Non-Newtonian Fluid Mech 102(2):149–156
Fritz G, Pechhold W, Willenbacher N, Wagner NJ (2003) Characterizing complex fluids with high frequency rheology using torsional resonators at multiple frequencies. J Rheol 47(2):303–319. https://doi.org/10.1122/1.1538608
Gallani J, Hilliou L, Martinoty P, Keller P (1994) Abnormal viscoelastic behavior of side-chain liquid-crystal polymers. Phys Rev Lett 72(13):2109–2112
Gautschi G (2002) Piezoelectric sensors Piezoelectric Sensorics (pp. 73-91): Springer
Ghiringhelli E, Roux D, Bleses D, Galliard H, Caton F (2012) Optimal fourier rheometry. Rheol Acta 51(5):413–420
Gold B, Pyckhout-Hintzen W, Wischnewski A, Radulescu A, Monkenbusch M, Allgaier J et al (2019) Direct assessment of tube dilation in entangled polymers. Phys Rev Lett 122(8):088001
Goldfarb M, Celanovic N (1997) Modeling piezoelectric stack actuators for control of micromanipulation. IEEE Control Syst 17(3):69–79
Gotze W, Sjogren L (1992) Relaxation processes in supercooled liquids. Rep Prog Phys 55(3):241–376
Gozen BA, Ozdoganlar OB (2012) A method for open-loop control of dynamic motions of piezo-stack actuators. Sensors Actuators A Phys 184:160–172
Graessley WW (2008) Polymeric liquids and networks: dynamics and rheology: Garland Science
Hall R, Kang B-G, Lee S, Chang T, Venerus DC, Hadjichristidis N, … Larson RG (2019) Determining the dilution exponent for entangled 1, 4-polybutadienes using blends of near-monodisperse star with unentangled, low molecular weight linear polymers. Macromolecules
Hecksher T, Torchinsky DH, Klieber C, Johnson JA, Dyre JC, Nelson KA (2017) Toward broadband mechanical spectroscopy. Proc Natl Acad Sci U S A 114(33):8710–8715. https://doi.org/10.1073/pnas.1707251114
Hopkins CC, de Bruyn JR (2016) Vibrating wire rheometry. J Non-Newtonian Fluid Mech 238:205–211
Huang Q, Mednova O, Rasmussen HK, Alvarez NJ, Skov AL, Almdal K, Hassager O (2013) Concentrated polymer solutions are different from melts: role of entanglement molecular weight. Macromolecules 46(12):5026–5035. https://doi.org/10.1021/ma4008434
Hudson R, Holder A, Hawkins K, Williams P, Curtis D (2017) An enhanced rheometer inertia correction procedure (ERIC) for the study of gelling systems using combined motor-transducer rheometers. Phys Fluids 29(12):121602
Ikeda A, Berthier L, Sollich P (2013) Disentangling glass and jamming physics in the rheology of soft materials. Soft Matter 9(32):7669. https://doi.org/10.1039/c3sm50503k
Kapnistos M, Vlassopoulos D, Roovers J, Leal L (2005) Linear rheology of architecturally complex macromolecules: comb polymers with linear backbones. Macromolecules 38(18):7852–7862
Kibble TW, Berkshire FH (2004) Classical mechanics: world scientific publishing company
Kim SA, Mangal R, Archer LA (2015) Relaxation dynamics of nanoparticle-tethered polymer chains. Macromolecules 48(17):6280–6293
Kirschenmann L, Pechhold W (2002) Piezoelectric rotary vibrator (PRV) – a new oscillating rheometer for linear viscoelasticity. Rheol Acta 41(4):362–368. https://doi.org/10.1007/s00397-002-0229-z
Koumakis N, Pamvouxoglou A, Poulos AS, Petekidis G (2012) Direct comparison of the rheology of model hard and soft particle glasses. Soft Matter 8(15):4271. https://doi.org/10.1039/c2sm07113d
Kramers HA (1927) La diffusion de la lumiere par les atomes. Paper presented at the Atti Cong. Intern. Fisica (Transactions of Volta Centenary Congress) Como
Kremer F, Schönhals A (2012) Broadband dielectric spectroscopy: Springer Science & Business Media
Kronig RDL (1926) On the theory of dispersion of x-rays. Josa 12(6):547–557
Läuger J, Stettin H (2016) Effects of instrument and fluid inertia in oscillatory shear in rotational rheometers. J Rheol 60(3):393–406. https://doi.org/10.1122/1.4944512
Likhtman AE, McLeish TC (2002) Quantitative theory for linear dynamics of linear entangled polymers. Macromolecules 35(16):6332–6343
Lionberger RA, Russel WB (1994) High frequency modulus of hard sphere colloids. J Rheol 38(6):1885–1908. https://doi.org/10.1122/1.550530
Lionberger RA, Russel W (2000) Microscopic theories of the rheology of stable colloidal dispersions. Adv Chem Phys 111:399–474
Liu C, He J, Ruymbeke E v, Keunings R, Bailly C (2006) Evaluation of different methods for the determination of the plateau modulus and the entanglement molecular weight. Polymer 47(13):4461–4479. https://doi.org/10.1016/j.polymer.2006.04.054
Mackay M, Cathey C (1991) A device to measure the dynamic shear properties of small samples. J Rheol 35(2):237–256
Macosko CW (1994) Rheology: principles, measurements, and applications: Wiley-vch
Mason TG (2000) Estimating the viscoelastic moduli of complex fluids using the generalized stokes–Einstein equation. Rheol Acta 39(4):371–378
Mason TG, Weitz DA (1995) Linear viscoelasticity of colloidal hard sphere suspensions near the glass transition. Phys Rev Lett 75(14):2770–2773. https://doi.org/10.1103/PhysRevLett.75.2770
Mattarelli M, Montagna M, Still T, Schneider D, Fytas G (2012) Vibration spectroscopy of weakly interacting mesoscopic colloids. Soft Matter 8(15):4235. https://doi.org/10.1039/c2sm07034k
McKenna GB (2006) Commentary on rheology of polymers in narrow gaps. Eur Phys J E Soft Matter 19(1):101–108; discussion 109-111. https://doi.org/10.1140/epje/e2006-00001-0
Mewis J, Haene P (1993) Prediction of rheological properties in polymer colloids. Paper presented at the Macromolecular Symposia
Mewis J, Wagner NJ (2012) Colloidal suspension rheology. Cambridge University Press, Cambridge
Morrison R (1967) Grounding and shielding techniques in instrumentation. Wiley, New York, p 1967
Müller G, Weber M, Rümpker G, Gajewski D (2007) Theory of elastic waves: Geoforschungszentrum
Nemirovsky Y, Nemirovsky A, Muralt P, Setter N (1996) Design of novel thin-film piezoelectric accelerometer. Sensors Actuators A Phys 56(3):239–249
Nommensen P, Duits MH, Van den Ende D, Mellema J (2000) Elastic modulus at high frequency of polymerically stabilized suspensions. Langmuir 16(4):1902–1909
Parot JM, Duperray B (2007) Applications of exact causality relationships to materials dynamic analysis. Mech Mater 39(5):419–433. https://doi.org/10.1016/j.mechmat.2006.07.004
Pham K, Petekidis G, Vlassopoulos D, Egelhaaf S, Poon W, Pusey P (2008) Yielding behavior of repulsion-and attraction-dominated colloidal glasses. J Rheol 52(2):649–676
Phillips R, Brady J, Bossis G (1988) Hydrodynamic transport properties of hard-sphere dispersions. I Suspensions of freely mobile particles. Phys Fluids 31(12):3462–3472
Plazek DJ (1996) 1995 Bingham Medal Address: oh, thermorheological simplicity, wherefore art thou? J Rheol 40(6):987–1014. https://doi.org/10.1122/1.550776
Poon WC, Weeks ER, Royall CP (2012) On measuring colloidal volume fractions. Soft Matter 8(1):21–30
Preumont A (2006) Mechatronics: Springer
Pritz T (2005) Unbounded complex modulus of viscoelastic materials and the Kramers–Kronig relations. J Sound Vib 279(3–5):687–697
Pusey P (1991) Colloidal suspensions in liquids, freezing, and the glass transition: les Houches
Pusey P (2008) Colloidal glasses. J Phys Condens Matter 20(49):494202
Rosedale J, Bates FS (1990) Rheology of ordered and disordered symmetric poly (ethylenepropylene)-poly (ethylethylene) diblock copolymers. Macromolecules 23(8):2329–2338
Roth M, D’Acunzi M, Vollmer D, Auernhammer GK (2010) Viscoelastic rheology of colloid-liquid crystal composites. J Chem Phys 132(12):124702
Rouleau L, Deü JF, Legay A, Le Lay F (2013) Application of Kramers–Kronig relations to time–temperature superposition for viscoelastic materials. Mech Mater 65:66–75. https://doi.org/10.1016/j.mechmat.2013.06.001
Royall CP, Poon WCK, Weeks ER (2013) In search of colloidal hard spheres. Soft Matter 9(1):17–27. https://doi.org/10.1039/c2sm26245b
Rubinstein M, Colby RH (2003) Polymer physics (Vol. 23). Oxford University Press, New York
Sánchez AM, Prieto R, Laso M, Riesgo T (2008) A piezoelectric minirheometer for measuring the viscosity of polymer microsamples. IEEE Trans Ind Electron 55(1):427–436
Schaertl W, Sillescu H (1994) Brownian dynamics of polydisperse colloidal hard spheres: equilibrium structures and random close packings. J Stat Phys 77(5–6):1007–1025
Schrag JL (1977) Deviation of velocity gradient profiles from the “gap loading” and “surface loading” limits in dynamic simple shear experiments. Trans Soc Rheol 21(3):399–413
Schrag J, Guess J, Thurston G (1965) Shear-wave interference observed by optical birefringence induced in a viscoelastic liquid. J Appl Phys 36(6):1996–2000
Schroyen B (2018) Bulk rheometry at high frequencies: a review of experimental approaches. In preparation
Schroyen B, Swan JW, Van Puyvelde P, Vermant J (2017) Quantifying the dispersion quality of partially aggregated colloidal dispersions by high frequency rheology. Soft Matter 13(43):7897–7906
Schroyen B, Hsu C-P, Isa L, Van Puyvelde P, Vermant J (2019) Stress contributions in colloidal suspensions: the smooth, the rough, and the hairy. Phys Rev Lett 122(21):218001
Shikata T, Pearson DS (1994) Viscoelastic behavior of concentrated spherical suspensions. J Rheol 38(3):601–616. https://doi.org/10.1122/1.550477
Sierou A, Brady JF (2001) Accelerated Stokesian dynamics simulations. J Fluid Mech 448. doi: https://doi.org/10.1017/s0022112001005912
Simon SL, Mckenna GB, Sindt O (2000) Modeling the evolution of the dynamic mechanical properties of a commercial epoxy during cure after gelation. J Appl Polym Sci 76(4):495–508
Sirohi J, Chopra I (2000) Fundamental understanding of piezoelectric strain sensors. J Intell Mater Syst Struct 11(4):246–257
Sternstein S (1983) Transient and dynamic characterization of viscoelastic solids: ACS publications
Stettin H (2016) Resonances in oscillatory rheometry. Appl Rheol 26(2):31–42
Szántó L, Vogt R, Meier J, Auhl D, Van Ruymbeke E, Friedrich C (2017) Entanglement relaxation time of polyethylene melts from high-frequency rheometry in the mega-hertz range a. J Rheol 61(5):1023–1033
Vaikuntanathan S, Jarzynski C (2009) Dissipation and lag in irreversible processes. EPL (Europhysics Letters) 87(6):60005
Van Ruymbeke E, Masubuchi Y, Watanabe H (2012) Effective value of the dynamic dilution exponent in bidisperse linear polymers: from 1 to 4/3. Macromolecules 45(4):2085–2098
Velankar S, Giles D (2007) How do I know if my phase angles are correct? Rheol. Bull, 76(8)
Vermant J, Moldenaers P, Mewis J, Ellis M, Garritano R (1997) Orthogonal superposition measurements using a rheometer equipped with a force rebalanced transducer. Rev Sci Instrum 68(11):4090–4096
Vleminckx G, Clasen C (2014) The dark side of microrheology: non-optical techniques. Curr Opin Colloid Interface Sci 19(6):503–513. https://doi.org/10.1016/j.cocis.2014.11.002
Walsh D, Zoller P (1995) Standard pressure volume temperature data for polymers: CRC press
Wang Y-Z, Wang G-H, Xiong X-M, Wang B, Zhang L-M, Zhang J-X (2010) Viscoelastic measurement of complex fluids using forced oscillating torsion resonator with continuously varying frequency capability. Rheol Acta 49(11–12):1117–1126. https://doi.org/10.1007/s00397-010-0484-3
Wang G, Chen G, Bai F (2015) High-speed and precision control of a piezoelectric positioner with hysteresis, resonance and disturbance compensation. Microsyst Technol 22(10):2499–2509. https://doi.org/10.1007/s00542-015-2638-9
Wen YH, Schaefer JL, Archer LA (2015) Dynamics and rheology of soft colloidal glasses. ACS Macro Lett 4(1):119–123. https://doi.org/10.1021/mz5006662
Willenbacher N, Oelschlaeger C (2007) Dynamics and structure of complex fluids from high frequency mechanical and optical rheometry. Curr Opin Colloid Interface Sci 12(1):43–49. https://doi.org/10.1016/j.cocis.2007.03.004
Winter HH (1997) Analysis of dynamic mechanical data: inversion into a relaxation time spectrum and consistency check. J Non-Newtonian Fluid Mech 68(2–3):225–239
Yamamoto J, Okano K (1991) Anomalous hydrodynamic behaviors of smectic liquid crystals at low frequencies. Jpn J Appl Phys 30(4R):754–763
Yamamoto J, Nakamura H, Okano K (1987) Apparatus for measurement of complex shear modulus of liquid crystals at low frequencies. Jpn J Appl Phys 26(S1):29
Acknowledgements
We thank A. B. Schofield for providing the PMMA particles, N. Hadjichristidis and H. Iatrou for providing and performing molecular characterization of the linear PBD samples, respectively, and D. Papazoglou, B. Schroyen, and J. Vermant for stimulating discussions. We acknowledge the contributions of D. Parisi, S. Costanzo, A. Mavromanolakis, and A. R. Jacob.
Funding
This study was financially supported by the EU (Horizon 2020 EUSMI GA731019).
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Athanasiou, T., Auernhammer, G.K., Vlassopoulos, D. et al. A high-frequency piezoelectric rheometer with validation of the loss angle measuring loop: application to polymer melts and colloidal glasses. Rheol Acta 58, 619–637 (2019). https://doi.org/10.1007/s00397-019-01163-x
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DOI: https://doi.org/10.1007/s00397-019-01163-x