A.
Ansón-Casaos
*^{a},
J. C.
Ciria
^{b},
O.
Sanahuja-Parejo
^{a},
S.
Víctor-Román
^{a},
J. M.
González-Domínguez
^{a},
E.
García-Bordejé
^{a},
A. M.
Benito
^{a} and
W. K.
Maser
^{a}
^{a}Instituto de Carboquímica, ICB-CSIC, Miguel Luesma Castán 4, 50018 Zaragoza, Spain. E-mail: alanson@icb.csic.es
^{b}Departamento de Informática e Ingeniería de Sistemas, Universidad de Zaragoza, María de Luna 1, 50018 Zaragoza, Spain
First published on 27th April 2020
Controlling the physicochemical properties of nanoparticles in fluids directly impacts on their liquid phase processing and applications in nanofluidics, thermal engineering, biomedicine and printed electronics. In this work, the temperature dependent viscosity of various aqueous nanofluids containing carbon nanotubes (CNTs) or graphene oxide (GO), i.e. 1D and 2D nanoparticles with extreme aspect ratios, is analyzed by empirical and predictive physical models. The focus is to understand how the nanoparticle shape, concentration, motion degrees and surface chemistry affect the viscosity of diluted dispersions. To this end, experimental results from capillary viscosimeters are first examined in terms of the energy of viscous flow and the maximum packing fraction applying the Maron–Pierce model. Next, a comparison of the experimental data with predictive physical models is carried out in terms of nanoparticle characteristics that affect the viscosity of the fluid, mostly their aspect ratio. The analysis of intrinsic viscosity data leads to a general understanding of motion modes for carbon nanoparticles, including those with extreme aspect ratios, in a flowing liquid. The resulting universal curve might be extended to the prediction of the viscosity for any kind of 1D and 2D nanoparticles in dilute suspensions.
According to the ideas of green chemistry, inks and nanofluids should be most preferably prepared using water as the base liquid. Both CNTs and graphene are hydrophobic and cannot be directly dispersed in water. Therefore, CNTs have to be stabilized in suspension through either the addition of surfactants,^{8,9} or suitable chemical functionalization.^{10} Besides, the oxidized form of graphene, namely graphene oxide (GO), is relatively stable in water due to the high ratios of oxygen functional groups.^{11}
The rheology of aqueous carbon nanofluids has been extensively studied for thermal transfer and lubricant applications.^{12–15} Experimental works include measurements on different kinds of multi-walled carbon nanotubes (MWCNTs),^{16–20} single-walled carbon nanotubes (SWCNTs),^{15,21–23} graphene nanoplatelets,^{24} and GO.^{25} Also, measurements on covalently functionalized CNTs (f-CNTs) have been reported,^{26,27} and it has been observed that GO reduction by chemical methods leads to a decrease in the viscosity of the nanofluid.^{25} Both CNT and GO dispersions in water are non-Newtonian liquids at relatively high concentrations, and typically form isotropic (nematic) phases.^{28,29} However, they approach the Newtonian behavior in the low concentration region.^{18–20,30,31}
From a fundamental viewpoint, CNTs and GO are nanoparticles with extreme aspect ratios, by far exceeding those of micrometric rods, cylinders and discs made of metals, ceramics and polymer materials.^{32–34} The viscosity of dilute CNT suspensions has been correlated with the CNT length,^{35,36} as well as with the bundling degree.^{37,38} Sedimentation studies by analytical ultracentrifugation reveal that CNTs in stable dispersions can be regarded as rigid cylinders.^{39,40} Also, analytical ultracentrifugation has been applied to the determination of lateral dimensions of GO, considering GO flakes as flat discs.^{41}
Most theoretical studies on the CNT viscosity are based on different expansions of classical two phase flow models,^{42} which are here the starting point for the discussion. Molecular dynamics simulations have been applied to the CNT viscosity,^{15} but are out of the scope of the present study. Also, recent mathematical approaches searching for improved algorithms to numerically fit experimental data are not discussed.^{43,44} Our focus lies on a general conceptual understanding by first principles theories. Since both CNTs and GO are nanostructures based on sheets of hexagonally bonded carbon atoms, the search for a common description of their properties might be natural.
We performed viscosity measurements on dilute dispersions of various CNT samples, f-CNTs and GO. All the GO samples and one type of f-CNT sample were directly dispersed in water. The other CNT and f-CNT samples needed the addition of surfactants. The complete set of measurements was first analyzed as a function of the temperature and the nanoparticle concentration, according to the empirical equation of Maron and Pierce.^{45} Next, size distributions for each sample were determined, and the viscosity was evaluated in terms of the aspect ratio (r_{p}) using predictive physical models. The r_{p} parameter is defined identically for both CNTs and GOs as the ratio between the particle dimension along a symmetry rotation axis and the perpendicular diameter. For CNTs, r_{p} is the ratio between the nanotube length and diameter, and for GOs, it is the ratio between the thickness and the average flake width. In this novel way, r_{p} ≫ 1 for CNTs, while r_{p} ≪ 1 for GO particles and we interpret the viscosity of 1D and 2D nanostructures only as a function of a unique common r_{p} parameter. Further analysis allows the discussion of the nanoparticle dynamics in the fluid according to the two employed predictive models.
Sample | Manufacturer | Synthesis method | t _{o} [h] |
---|---|---|---|
GO-2 | GCNN, ICB-CSIC | Modified hummers | 2 |
GO-4 | GCNN, ICB-CSIC | Modified hummers | 4 |
GO-16 | GCNN, ICB-CSIC | Modified hummers | Overnight |
GO-G | Graphenea (San Sebastián, Spain) | — | — |
Label^{a} | Manufacturer (commercial name) | Synthesis method | D [nm] | L [μm] | Carbon purity^{b} [%] |
---|---|---|---|---|---|
a The abbreviation in SWCNTs indicates the synthesis method. b From TGA residues. | |||||
MWCNT | Nanocyl (NC7000) | CCVD | 9.5 | 1.5 | 90 |
CVD-SWCNT | Chasm Advanced Materials (SG76) | CVD (CoMoCAT®) | 0.83 | 1 | ≥90 |
AD-SWCNT | Carbon Solutions Inc. (AP-SWNT) | Arc discharge | 1.4 | >1 | 60–70 |
Label | Treatment | Functional groups |
---|---|---|
c-HNO_{3}-MWCNT | 9.5 M HNO_{3}, 18 h reflux | –COOH, –CHO, –OH |
d-HNO_{3}-MWCNT | 1.5 M HNO_{3}, 2 h reflux | –COOH, –CHO, –OH |
F-MWCNT | Isopentyl nitrite, H_{2}N–C_{6}H_{5}–(CF_{2})_{7}–CF_{3}, 12 h, 60 °C | –C_{6}H_{5}–(CF_{2})_{7}–CF_{3} |
HSO_{3}-MWCNT | Isopentyl nitrite, H_{2}N–C_{6}H_{5}–SO_{3}H, 1 h, 80 °C | –C_{6}H_{5}–SO_{3}H |
Most of the CNT samples were dispersed with the help of a surfactant, either sodium dodecyl benzene sulfonate (SDBS) or Solsperse™ 27000, which is an acrylic commercial component for inks. One of the f-CNT samples, the HSO_{3}-MWCNT sample, and the GOs were directly dispersed in water. Dispersions of the nanomaterials were prepared at initial concentrations in the range of 0.4–4 mg mL^{−1} with the help of ultrasound. Most of the dispersions were centrifuged at a moderate speed of 4500 rpm to avoid large aggregates. Only the GO-16 and GO-G dispersions were not centrifuged since they did not contain large aggregates. Further details on the quality of the dispersions are provided in the ESI† (Section S2).
The viscosity (η) of a fluid is defined as the ratio between the shear stress (F) and the shear rate (G): η = F/G. For a suspension of solid particles in a liquid, several models aim to describe the dependence of η on factors such as temperature and particle concentration and shape. The relative viscosity (η_{r}), which is defined as the ratio between the dispersion viscosity (η) and the viscosity of the base fluid (η_{0}), is mainly controlled by the particle shape and size, as well as the volume fraction (ϕ). Einstein's pioneering work on dilute suspensions of hard spheres,^{46} for which η_{r} = 1 + 2.5ϕ, has been complemented by successive approaches taking into account the non-sphericity of the particles and the effect of higher concentrations. In some of those approaches, η_{r} is a polynomial function of ϕ, whose coefficients model the dependence on size and shape.^{47–50} A drawback of the mentioned models is their prediction of a finite viscosity regardless of the ϕ value, which is clearly unphysical. Viscosity must diverge as ϕ tends to the maximum packing fraction (ϕ_{m}), which depends on the particle geometry. This behaviour is predicted by models such as Mooney,^{51} Krieger–Dougherty,^{52} and Maron–Pierce.^{45} While Mooney and Krieger–Dougherty models have two fitting parameters, the Maron–Pierce model considers just one, as far as regards η_{r}. Fitting our experimental data with two parameters results in a slightly modified value of ϕ_{m}, but brings out no significant improvement in the standard deviation. Thus, the Maron–Pierce model is chosen here for simplicity.
The model of Maron and Pierce leads to the following equation for the absolute viscosity:^{45}
(1) |
(2) |
Therefore, the relative viscosity (η_{r}) is defined as
(3) |
For the present work, viscosity measurements were performed on dilute aqueous suspensions. The solid concentration (C) in the liquid was always kept below 4 mg mL^{−1}, so it can be reasonably assumed that dispersions behave as Newtonian fluids. Accordingly, viscosities were measured in Ubbelohde capillary viscosimeters, which provide very precise measurements in the low-viscosity regime. For instance, an Ubbelohde viscosimeter has been utilized for the assessment of the exfoliation degree of SWCNT dispersions.^{37} In Ubbelohde viscosimeters, a suspended column of the liquid flows inside the capillary under laminar regime, by gravity and in a unidirectional way, while the flowing time is registered. The direct measurement from Ubbelohde viscosimeters is the kinematic viscosity (v), which is related to the dynamic viscosity (η) through the fluid density (ρ): η = v·ρ. The complete set of v and ρ measurements is provided in the ESI† (Sections S3 and S4).
The kinematic viscosity of GO dispersions at temperatures between 298.15 and 318.15 K is in the range of 4.0 to 0.7 mm^{2} s^{−1}. For CNTs, the kinematic viscosity is between 2.8 and 0.6 mm^{2} s^{−1}, except in the Solsperse medium, where it reaches 6.1 mm^{2} s^{−1} at 298.15 K. The ρ values of all the measured CNT and GO dispersions are nearly that of pure water (Table S4, ESI†).
Before continuing with the analysis of viscosity at different nanoparticle concentrations, it was confirmed that the relative viscosity (η_{r}) does not substantially change with temperature.^{21} This condition, which indicates a Newtonian behavior, was clearly fulfilled by all the GO and CNT systems in water and SDBS. Deviations of 10% in the relative viscosity were found for MWCNT dispersions in Solsperse at the highest temperatures and concentrations. Solsperse leads to viscosities of >4 cP at room temperature, and thus the data interpretation has to be performed carefully. However, for most of the analyzed dispersions, it is evident that eqn (1) and (3) can be applied instead of the complete model for non-Newtonian suspensions (ESI,† Sections S5 and S6).
Fig. 1 Relative viscosity (η_{r}) as a function of concentration (C) of GO-16 in water at 5 temperatures. The fitting line was calculated with the Maron–Pierce model (eqn (1)). |
Fig. 2 Relative viscosity (η_{r}) as a function of concentration (C) of MWCNTs at 5 temperatures in (a) 0.5% SDBS and (b) 20% Solsperse. The fitting line was calculated with the Maron–Pierce model (eqn (1)). |
It has been commented above that ΔH_{1}* is expected to depend only on the liquid medium. Fig. 3 presents the calculated ΔH_{1}* values for all the samples. In fact, ΔH_{1}* is nearly constant in almost all the dispersions, and in pure water and the 0.5% SDBS medium as well. The values of ΔH_{1}*/R are typically around 1864 K with a deviation of ±54 K. Apparently, the SDBS surfactant does not lead to any substantial change with respect to pure water, and thus both media can be regarded as equivalent in terms of ΔH_{1}*. However, certain deviations from water exist, which should be consequently interpreted as induced changes in the molecular structure of the liquid.
Fig. 3 Enthalpy of activation for viscous flow (ΔH_{1}*) for all the CNT and GO systems that are considered in the present work. |
The addition of HSO_{3}-MWCNTs to pure water causes a strong decrease in ΔH_{1}*. The presence of benzene sulfonic acid groups on MWCNT walls allows the preparation of a stable dispersion in water, but also disrupts the liquid structure. The creation of hydrogen bonds between benzene sulfonic acid groups and water might be an explanation. Curiously, the effect is not observed in the presence of SDBS surfactant. The ΔH_{1}* value decreases by 5% after the addition of the CVD-SWCNT sample to the 0.5% SDBS medium. Even though the change is relatively small, it might be related to the extraordinarily high aspect ratio of CVD-SWCNTs (Table 2), which leads to a correction in the flowing energetics of the liquid. An increase of 6% in ΔH_{1}* was calculated for the GO-G dispersion in water. The change in ΔH_{1}* might reflect the presence of a stabilizing additive. A strong increase in ΔH_{1}* up to 2330 K occurs in the 20% Solsperse medium. Acrylic chains of Solsperse greatly influence the associated structure of water, hindering their molecular mobility. The effect is evident despite a decrease in the accuracy of viscosity measurements for the Solsperse medium.
In terms of the particle volume fraction (ϕ), the equation of Maron and Pierce (eqn (1)) is written as
(4) |
(5) |
The CNTs are considered as rigid cylinders, which is a reasonable assumption according to the literature.^{39,40} An average diameter (D) for each sample is taken from product specifications (Table 2). The presence of chemical groups on MWCNT walls is neglected in terms of D. Length distributions for each CNT and f-CNT sample were analyzed from scanning electron microscopy (SEM) images. The length of a number of CNTs (N = 290–480) was directly measured on SEM images (ESI,† Section S10). The length determination is assumed to be valid, although the morphology of the solid on the SEM substrate might be substantially different from that of free nanostructures in the liquid dispersion.^{54} According to the literature,^{38,55} the length histogram was fitted to a log-normal distribution (Section S10, ESI†). The calculated average and median lengths (L_{a} and L_{m} respectively) from log-normal distributions are listed in Table 4. The median length was used for the calculation of ρ_{p} in each case. In order to complete the geometrical description of MWCNTs, the number of layers was deduced from an experimental measurement of the specific surface area (SSA_{MWCNT} = 230 m^{2} g^{−1}) by nitrogen adsorption at 77 K.
Sample | L _{m} [nm] | L _{a} [nm] |
---|---|---|
MWCNT | 439 | 506 |
CVD-SWCNT | 426 | 515 |
AD-SWCNT | 315 | 384 |
c-HNO_{3}-MWCNT | 293 | 375 |
d-HNO_{3}-MWCNT | 463 | 585 |
F-MWCNT | 289 | 389 |
HSO_{3}-MWCNT | 339 | 453 |
Surface area distributions of GO flakes were also determined from SEM images. The calculated average and median surface areas (S_{a} and S_{m} respectively) of the four GO samples are listed in Table 5. Assuming an approximately elliptic shape, the complete description of flake dimensions requires determining its thickness. Equivalently, the number of layers (n_{L}) and the distance between layers (d_{LL}) in the flake were calculated from X-ray diffraction (XRD) patterns, applying the Bragg and Scherrer equations (ESI,† Section S11). The calculated d_{LL} and n_{L} dimensions are listed in Table 5. The flake thickness can be directly calculated as t_{f} = (n_{L} − 1)·d_{LL}.
Sample | S _{m} [μm^{2}] | S _{a} [μm^{2}] | n _{L} | d _{LL} [nm] |
---|---|---|---|---|
GO-2 | 0.33 | 0.61 | 7 | 0.821 |
GO-4 | 0.63 | 1.17 | 7 | 0.833 |
GO-16 | 0.86 | 1.56 | 9 | 0.833 |
GO-G | 3.84 | 7.27 | 5 | 0.879 |
Once the ρ_{p} value was calculated from geometric parameters for all the CNT and GO samples, a correction due to the presence of heteroatoms and impurities was considered. The correction was performed with a parameter that took into account the weight of atoms different from carbon in the structures. However, this additional parameter only causes relatively small changes in the calculated values, which do not alter the qualitative discussion of the results. Similarly, another correction factor was considered for the weight change undergoing if the inner cavity of CNTs was filled by the liquid medium. The filling factor of CNTs, which was not finally applied, does not change the qualitative discussion that is presented next.
Now, theoretical models to predict the relationship of ϕ_{m}vs. r_{p} are discussed. In the first approximation, both CNTs and GO are considered ellipsoidal particles with extreme aspect ratios. Ellipsoidal particles are defined by the semi-axes a, b and c. The CNTs may be described as elongated (prolate, a ≫ b = c) ellipsoids, while GO flakes look like flattened (oblate, a ≪ b = c) ellipsoids. The coefficient ϕ_{m} can be related to the extra power or rate of work (Ẇ_{p}) required to shear a volume of the nanofluid due to the presence of the particle:^{56}
(6) |
In its movement inside the fluid, the particle is free to rotate along its axis of symmetry (in the a direction), which is a natural easy movement. The ϕ_{m} coefficient has a maximum when all the ellipsoids are oriented in the flow direction. However, a gradient of velocity in the fluid may exist that additionally forces the particle to tumble end over end. In fact, the velocity of a fluid in a capillary is maximum at the center and minimum at the capillary wall. The end-over-end rotation is much more energy consuming than the simple rotation along the symmetry axis. Moreover, the particle requires a lot of extra space to rotate, which hinders particle packing and leads to a minimum ϕ_{m} coefficient (Fig. 4). A number of intermediate situations can be considered depending on the angle between the symmetry axis of the spheroid and the flow direction.
Fig. 4 Schematic explanation of GO and CNT nanoparticle dynamics, according to their viscous behavior in a velocity gradient. |
Therefore, the theoretical value of ϕ_{m} for spheroidal particles can be calculated as a function of r_{p} for different orientations of the particle in the flowing liquid. GO flakes are considered as oblate spheroids (r_{p} ≪ 1) and CNTs as prolate spheroids (r_{p} ≫ 1). Three hypothetical orientations of the spheroidal particle are here taken into account, giving three ϕ_{m} functions: maximum, minimum and average. The average ϕ_{m} coefficient is calculated accounting for all the possible orientations of the particle, which are assumed to be equally probable.
(7) |
In the dilute regime, eqn (7) can be approximated by the first term of a series expansion:
η ≈ η_{0}(1 + χϕ) | (8) |
(9) |
Fig. 5 Intrinsic viscosity (χ) as a function of the aspect ratio (r_{p}). Experimental data are superimposed to trend lines that were calculated for spheroidal particles^{56} and for cylinders.^{58} The dashed line indicates the correction arising from the statistical dispersion in the CNT length. |
The integral equation of Ẇ_{p} can be solved analytically for spheroids, but not for cylinders. Therefore, Aragon and Flamik proposed a solution by numerical methods, more specifically applying the boundary element method.^{58} Cylindrical particles are characterized by their aspect ratio r_{p} = L/D. The model of Aragon and Flamik assumes r_{p} > 1 and takes into account both rotational and translational movements of cylindrical particles in the fluid. They considered three possible configurations for cylinder edges: open, closed by spherical caps, and closed by a circle in the rectangular configuration. The results for the three configurations converge at high enough r_{p} values. The final equation of Aragon and Flamik for χ totally agrees with a solution that was given by Mansfield and Douglas applying another numerical method, namely, the path integration method.^{59}
Experimental values of χ for CNTs match quite well the numerical equation for cylindrical particles (Fig. 5). The numerical model of cylinders suitably predicts the behavior of CNTs in the dilute regime. Since the χ vs. r_{p} branch (solid line in Fig. 5) was first calculated under the assumption that all nanotubes have an identical length L_{m}, a question remains about the reliability of such a simplifying approximation. Therefore, another calculation was performed taking into account the range of L_{m} values that comes out from considering a log-normal distribution (dashed line in Fig. 5). The expected deviation in terms of length variability is not qualitatively relevant. Regarding the applicability of numerical methods to solve Eqn 6 for the case of GO as a flat cylinder with r_{p} ≪ 1, the solution is not trivial and falls out of the scope of the present work.
From the theoretical χ = f(r_{p}) function, the variation of χ with the geometrical parameters of length, diameter and thickness can be calculated for CNT and GO dispersions (Fig. 6). Experimental data of GO in water and MWCNTs in SDBS agree well with the theoretical function. Chemical functionalization of MWCNTs, which causes certain changes in the nanotube length, has a minor influence on viscosity. The theoretical χ value for SWCNT dispersions is overestimated with respect to experimental data. The deviation might be associated with a certain degree of bundling and, indirectly, with the presence of impurities in the starting SWCNT powder material.
Fig. 6 Experimental and theoretical intrinsic viscosity (χ) as a function of particle shape parameters: (a) experimental data for CNT dispersions as a function of their length (L) and theoretical trends for cylinders,^{58} with diameters D = 0.83, 1.49 and 9.5 nm; (b) experimental data for GO dispersions as a function of their flake diameter (b + c) and theoretical trends for oblate ellipsoids,^{56} whose minor axis (2·a) is equal to the flake thickness t_{f} = 3.53, 5.00 and 6.18 nm. Theoretical calculations correspond to the distributions of multi-disperse particles whose length (CNT) or surface (GO) follows a log-normal distribution. |
A step forward in the assessment of the theoretical model is the calculation of the relative viscosity (η_{r}) as a function of geometrical parameters and the particle concentration. According to the Maron–Pierce model, η_{r} is related to χ by:
(10) |
Finally, it is noteworthy that the absolute viscosity (η) depends on the particle size,^{13} while the intrinsic viscosity (χ value) does not theoretically depend on size. Regarding the intrinsic viscosity, let us recall the universal value χ = 5/2 deduced by Einstein for spheres. Moreover, the more general expressions reported for spheroids and rods depend only on the shape and the aspect ratio.^{56–59} Thus, the relationship χ = f(r_{p}) can be regarded as a universal prediction of the viscosity for 1D and 2D nanoparticle dispersions in the dilute regime.
The GO-2, GO-4 and GO-16 samples were synthesized from graphite flakes by a modified Hummers’ method including the following stages:^{11} (i) treatment in NaNO_{3}/H_{2}SO_{4}/KMnO_{4} at 35 °C for 2, 4 or 16 h, respectively; (ii) cooling in 30% H_{2}O_{2}; (iii) filtration, washing with diluted HCl and drying; and (iv) resuspension in water at 0.5 mg mL^{−1} and exfoliation in an ultrasound bath for 2 h. The GO-G sample was purchased from Graphenea (San Sebastian, Spain).
The SDBS surfactant was purchased from Sigma Aldrich and acrylic Solsperse™ 27000 was provided by Lubrizol Advanced Materials (Sant Cugat del Vallés, Barcelona, Spain). For the preparation of CNT dispersions, SDBS was dissolved at 0.5 mass/vol% and Solsperse was diluted at 20 vol% in water. The solutions were added to CNT powders and the mixtures were treated with a 400 W Hielscher ultrasonic probe for 1 h at 60% amplitude and a frequency of 0.5 cycles. Dispersions were centrifuged at 4500 rpm for 30 min and the sediment was discarded.
Footnote |
† Electronic supplementary information (ESI) available: Elemental analysis of powder samples; analysis of dispersion quality; kinematic viscosity; density of the CNT and GO dispersions; dependence of relative viscosity with temperature; the Maron and Pierce equations for non-Newtonian suspensions; fitting parameters of eqn (1); accuracy of viscosity measurements; calculation of the particle density; electron microscopy images and the log-normal distribution; XRD experiments for the determination of n_{L} and d_{LL} in GO; chemical functionalization of MWCNTs. See DOI: 10.1039/d0cp00468e |
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