Letter Cite This: ACS Macro Lett. 2018, 7, 1022−1027
pubs.acs.org/macroletters
Size-Dependent Orientational Dynamics of Brownian Nanorods Christophe Brouzet,*,†,‡ Nitesh Mittal,†,‡ L. Daniel Söderberg,†,‡ and Fredrik Lundell†,‡ †
Linné FLOW Centre, KTH Mechanics, KTH Royal Institute of Technology, Stockholm SE-100 44, Sweden Wallenberg Wood Science Center, KTH Royal Institute of Technology, Stockholm SE-100 44, Sweden
‡
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S Supporting Information *
ABSTRACT: Successful assembly of suspended nanoscale rodlike particles depends on fundamental phenomena controlling rotational and translational diffusion. Despite the significant developments in fluidic fabrication of nanostructured materials, the ability to quantify the dynamics in processing systems remains challenging. Here we demonstrate an experimental method for characterization of the orientation dynamics of nanorod suspensions in assembly flows using orientation relaxation. This relaxation, measured by birefringence and obtained after rapidly stopping the flow, is deconvoluted with an inverse Laplace transform to extract a length distribution of aligned nanorods. The methodology is illustrated using nanocelluloses as model systems, where the coupling of rotational diffusion coefficients to particle size distributions as well as flow-induced orientation mechanisms are elucidated.
D
length characterization techniques such as rheology measurements,29−31 electron microscopy,30,31 and X-ray scattering30−32 are predominantly limited to observing static systems or require significant investments in instrumentation. Other techniques such as dynamic light scattering33,34 and orientation relaxation methods based on flow35−39 or electric40−43 induced birefringence have been used extensively to study rotational diffusion. A straightforward dynamic characterization technique coupling rotational diffusion and length distribution during fabrication of nanostructured materials will thus contribute critically to the development of knowledge and technology for controlled assembly in colloidal systems. Herein, we describe a new experimental methodology that allows real-time assessment of orientation dynamics of nanorods under dynamic flow conditions. This approach is based on a flow-stop technique39 coupled with thorough analysis of the orientation relaxation.38,43 Once the flow has reached a steady state, it is rapidly stopped, and the relaxation of the particle orientation toward isotropy, depending on the diffusion of the system constituents, is then measured at different locations in the channel. Combining these measurements with a model of rotational diffusion,24−26 this methodology provides information about the typical size of the aligned particles before the flow is stopped. Therefore, the evolution of the length distribution of the aligned particles in the flow can readily be quantified. For the demonstration, we have utilized two types of nanocelluloses as model systems: cellulose nanocrystals (CNC), being almost monodisperse in length,
irected nanoparticle self-assembly is paramount to fabrication of novel materials.1 While microfluidics has emerged as a promising tool to accomplish this “bottom-up” approach,2,3 it demands scientific understanding concerning nanoparticle dynamics in flow systems. Particularly, controlled assembly of elongated nanoparticles (nanotubes, protein- and polymer-based nanofibrils) into high-performance structural components such as fibers or filaments have recently gained much attention.4−9 The mechanical performance of these macroscopic material components is given by their nanostructure, predominantly the orientation of the nanoparticles.5,6,9−14 Hydrodynamics can cause local nanoparticle alignment15,16 through shear and extensional flows, but our knowledge and models for interparticle interactions are not sufficient to readily describe the behavior of nanoparticles in flowing suspensions.17−21 Nanoscale assembly processes in flow systems are typically governed by Brownian diffusion, competing with hydrodynamic alignment mechanisms. Dealignment of elongated nanoparticles, here referred to as rotational diffusion, is detrimental to the performance of the macroscopic materials by altering the internal morphology.22,23 Extensive studies on rotational diffusion of monodisperse systems have been carried out24 and extended to the polydisperse case,25,26 which is typical for biological and chemical systems based on selfassembly of nanosized building blocks.6−9 However, quantification of rotational Brownian motion is not an easy task and fundamental concepts about motions of anisotropic macromolecules remain largely unexplored.27,28 Therefore, in-depth characterization of nanoparticle polydispersity and a thorough understanding of the physics behind rotational diffusion will pave the way for flow-based fabrication processes. Existing © XXXX American Chemical Society
Received: June 29, 2018 Accepted: July 26, 2018
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DOI: 10.1021/acsmacrolett.8b00487 ACS Macro Lett. 2018, 7, 1022−1027
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ACS Macro Letters
semidilute regime. The rotational diffusion coefficient of a rod in an isotropic semidilute system is
and cellulose nanofibrils (CNF), with a broad length distribution (polydisperse). The experimental setup used in this study consists of a flowfocusing channel,2 each branch having a square cross-section with h = 1 mm sides, as shown in Figure 1a (see Supporting
Dr =
β kBT ln(L /d) (cL3)2 3πηL3
(1)
where T is the temperature, kB is the Boltzmann constant, η is the solvent viscosity, and β is a numerical factor. β is expected be of order 1−10,33,37,38,49,50 but several experiments33,37,38 measured its magnitude in the range of 103−104. This discrepancy has led to refined theories,49,50 modeling more precisely the diffusion of the rods through the network, and can also be explained by the rods flexibility in some experiments.37,38 The birefringence relaxation toward isotropy is predicted to be exponential with a typical time scale of τ = 1/(6Dr).24 Figure 2 shows the birefringence relaxation of
Figure 1. Experimental setup and samples used. (a) Typical birefringence signal for the CNF suspension. The signal is normalized by its maximum value in the channel. The squares represent the different areas where the birefringence is averaged in space. (b) Length distributions of the CNC (dashed dotted line) and CNF (solid line) suspensions. The insets show TEM images with the scale bars representing 200 nm.
Figure 2. Normalized birefringence decays for the CNC suspension at different positions in the channel: x/h = −1 (blue), 0.75 (light blue), 1.25 (green), 1.75 (magenta), 4.75 (red), and 8 (black). The inset shows the same plot with a logarithmic scale for the vertical coordinates.
Information for a complete description). Fabrication of hierarchical superstructures by self-assembly is conceivable with a similar geometry.6,7,9,44−48 The orientation of the nanorods in the flow is visualized through birefringence using polarized optical microscopy.39 A typical CNF birefringence signal is shown in Figure 1a. Once the flow has been stopped, the birefringence decay B(t) is observed at different locations along the centerline of the channel marked by the black squares in Figure 1a. The sample preparation and characterization are described in the Supporting Information. Nanocelluloses have been chosen as model systems for large aspect ratio rigid rods. The length distributions of the corresponding samples are plotted in Figure 1b, together with representative transmission electron microscopy (TEM) images. To correlate the return-to-isotropy of the samples with a given particle length distribution, it is necessary to model the effect of Brownian motion. Doi and Edwards24 developed a model for monodisperse rods which depends on the length L, diameter d, and concentration c of the particles. For cL3 ≪ 1 (dilute regime), the rods are free to rotate without any interparticle interactions, while for 1 ≪ cL3 ≪ L/d (semidilute regime), the effects of particle interactions on the particle dynamics become significant due to volume exclusion24 and network formation.20,21 With a dry weight concentration of 41 g/L (cL3 ≈ 4.2), the CNC suspension belongs to the
the CNC suspension at different locations in the channel, both in a lin−lin and log−lin plot (inset). All the curves are normalized by their initial value, B0. They almost collapse and the inset indicates an exponential decay, as expected. This illustrates that independent of the initial alignment, only one time scale can be associated with the relaxation of the monodisperse CNC particle system, in agreement with Doi and Edwards theory.24 The diffusion model (eq 1), valid for the monodisperse CNC suspension, needs to be extended in order to describe the dynamics of the polydisperse CNF suspension, which is more realistic for general nanoparticle systems. A model for polydisperse systems has been proposed by Marrucci and Grizzuti.25,26 In such systems, the concentration distribution c̃ depends on the rod length L and the transition between dilute and semidilute regimes is defined by the entanglement length i L = jjj * k
∫0
+∞
y c (̃ L)LdLzzz {
−1/2
(2)
Rods shorter than L* are considered to be in the dilute regime, while rods longer than L* are in the semidilute regime. For the CNF suspension, L* ≈ 60 nm, reflecting that all CNF are in the semidilute regime (see Figure 1b). For a 1023
DOI: 10.1021/acsmacrolett.8b00487 ACS Macro Lett. 2018, 7, 1022−1027
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ACS Macro Letters polydisperse system close to isotropy, the diffusion coefficient for a rod of length L can be written as25,26(see Supporting Information) Dr (L) ≈
βkBTL 4 * ηL7
(3)
Equation 3 is similar to eq 1, with the concentration dependency hidden into the entanglement length L*. Close to isotropy, the dynamics of each rod takes place at its own time scale τL = 1/(6Dr(L)), depending only on its length L and the concentration distribution c̃. As the total birefringence is the sum of the contributions of each length component,38,43 the birefringence relaxation signal is given by B (t ) =
∫0
+∞
B0 (L)exp( −6Dr (L)t )dL
(4)
Here B0(L) is the contribution of the nanoparticles of length L to the total birefringence signal B0 before the flow is stopped. This leads to a relaxation toward isotropy with multiple time scales,37−39,43 visible for the CNF in Figure 3a (filled circles and inset). The difference with the exponential relaxation of the CNC, also plotted in Figure 3a (empty diamonds), is significant and highlights the strong coupling between the orientation relaxation of a system and its length distribution. This coupling allows us to extract information on the length distribution. Indeed, eq 4 is a Laplace transform of B0(L) and this quantity can be estimated by inverting B(t), using a method described by Rogers et al.38,43 (see details in the Supporting Information). The method is illustrated in Figure 3 with two examples, one for CNC (empty diamonds) and one for CNF (filled circles). Figure 3a shows the birefringence B/B0 as a function of time in a lin−log plot. The log-scale is necessary because a small change in length is amplified by the power 7 for time, as shown in eq 3. Each birefringence decay is first transformed into a discrete spectrum {B0j } of relaxation times {τj}, shown in Figure 3b. When B0j ≠ 0, the associated time scale τj contributes to the decay. For the CNC signal, Figure 3b exhibits a sharp peak, while the broad spectrum for the CNF signal spreads over more than 3 decades in time. This highlights that many time scales contribute to the CNF decay, as can be expected due to the polydispersity. The time scales {τj} are then converted to length scales L using eq 3 and τ(L) = 1/(6Dr(L)), while the discrete set {B0j } is turned into a continuous contribution B0(L). The numerical factor β given in eq 3 has to be adapted. For the CNF, it has been set to β = 1.5 × 103, which is the order of magnitude reported by other experiments.33,37,38 For the CNC, β = 0.5 matches well with the length distribution. The discrepancy between the two values reflects that CNC and CNF suspensions form threedimensional networks differently due to their aspect ratio, stiffness and morphology (see Supporting Information). Nevertheless, we finally obtain in Figure 3c an estimation of the different lengths contributing to the birefringence, that is, the lengths aligned by the flow before it was stopped. Note that the horizontal axis in Figure 3c is linear and within the range of the length distributions given in Figure 1b. By using systematically the method presented in Figure 3 at different channel locations, it is now possible to quantify the evolution of the length distribution of the aligned fibrils along the channel and therefore to assess the orientation dynamics of the suspension in the flow. Figure 4a shows normalized birefringence decays for CNF at different positions along the
Figure 3. Demonstration of the inverse Laplace transform methodology. (a) Normalized birefringence decays for CNC (empty diamonds) and CNF (filled circles), at x/h = 1.75. The fit of each curve using the inverse Laplace transform method is shown as dashed dotted line for CNC and solid line for CNF. The inset shows the normalized birefringence decay for CNF in a log−lin plot. (b and, respectively, c) Normalized contributions B0j /B0 (respectively, B0(L)/ B0) of the different time scales τj (respectively, length scales L) to the decay signals in panel (a) for CNC (dashed dotted line and empty diamonds) and CNF (solid line and filled circles).
channel, revealing different time scales in the dynamics. The contributions B0(L) obtained after the inverse Laplace transform of these decays are presented in Figure 4b,c for x/h ≤ 1.75, where fibrils are aligned by the extensional flow, and for x/h ≥ 1.75, where the alignment mechanism vanishes. The magenta curve (x/h = 1.75) is repeated in both figures for an easier comparison while the rescaled length distribution obtained from TEM images is shown by the thick solid line. The contributions B0(L) to the birefringence before the focusing is represented with blue triangles in Figure 4b. It is modified greatly after the focusing. First, it becomes narrower and the maximum shifts toward shorter particles (light blue triangles). Subsequently, longer particles contribute more (green stars and magenta dots), indicated by the maximum and the right flank of the distribution shifting toward longer 1024
DOI: 10.1021/acsmacrolett.8b00487 ACS Macro Lett. 2018, 7, 1022−1027
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Figure 4. Revealing the orientation dynamics within the flow. (a) Normalized birefringence decays for CNF. The symbols are experimental data and the solid lines are the inverse Laplace transform fit. The colors are for different positions in the channel: x/h = −1 (blue), 0.75 (light blue), 1.25 (green), 1.75 (magenta), 4.75 (red), and 8 (black). The two insets are schematics of the orientation of the short (blue) and long (red) fibrils before the stop, in the focusing region and far downstream. (b, c) Contributions B0(L) to the different length scales L upstream (x/h ≤ 1.75) and downstream (x/h ≥ 1.75), respectively. The colors used are the same as in panel (a). The thick solid line shows the rescaled length distribution obtained from TEM images and also plotted in Figure 1b.
comparison between the contributions B0(L) and the length distribution obtained with TEM images in Figure 4b,c. This highlights the fact that these fibrils are either too short, having a diffusion time scale smaller than the alignment time scale and therefore an inefficient alignment, or too long, causing a highly entangled network hindering alignment. As a consequence, a precise determination of the β value used in eq 3 is difficult to obtain and only a range β ≈ 1000−2500 can be specified. To summarize, combining flow-stop experiments and inverse Laplace transforms, our observations on the Brownian effects of nanorod suspensions provide exquisitely detailed information about the diffusive properties of anisotropic nanoobjects and the subtle interplay between alignment and diffusion mechanisms. Besides characterizing the nanoparticle dynamics under dynamic flow conditions, this technique is able to reveal that the orientation dynamics is strongly dependent on the nanoparticle length distribution and highlights the importance of the network formation due to the interparticle interactions, which is challenging to understand theoretically. To the best of our knowledge, this is the first of its kind methodology that allows such assessment and characterization of dynamic colloidal systems.
particles. This means that, during the alignment process, short fibrils are aligned first followed by alignment of longer particles. Further downstream, where no alignment mechanisms are present (Figure 4c), rotational diffusion reduces the contributions at all lengths, especially for short particles. These findings allow us to establish a comprehensive understanding of the orientation dynamics in microfluidic channels in general, and specifically in the extensional flows. Short particles are quickly oriented by the flow, while more time is needed to align long ones. Furthermore, short particles dealign faster and their motion is therefore difficult to control during the assembly. This scenario can be also deduced from Figure 4a, used as a diagnostic plot: aligned short fibrils in the focusing region introduce short time scales in the decay (light blue triangles), while aligned long fibrils further downstream exhibit longer time scales (red squares and black diamonds). These findings may have strong implications on the performance of the macroscopic nanocellulose structures fabricated via flow-based assembly.6,9,51,52 The typical time for the transition to colloidal glassy-state achieved with the flow focusing geometry of similar dimensions is estimated to be of the order of a second,6,9 that is, longer than the time found here for the dealignment of the shortest fibrils (see Figure 4a). Therefore, the hierarchical structures manufactured using similar approaches may be composed of long aligned particles embedded into an isotropic matrix of shorter particles. The orientation dynamics therefore exhibits a strong dependency on the length of the fibrils. This can be explained by the entanglement of the CNF which increases with their length, in the formed three-dimensional network. Indeed, during hydrodynamic alignment, all fibrils tend to rotate but the concentration required for the fabrication of hierarchical structures is typically sufficiently high for the fibrils to quickly collide and for their motion to be hindered. Such collisions tend to reorient the fibrils and slow down the alignment process.17 Thus, at a given total concentration small fibrils endure a less hindered motion, experience fewer collisions and a quicker alignment than the long ones, despite higher diffusion coefficients. When the alignment mechanisms are no longer effective, the differences in diffusion coefficients, due also to the entanglement, result in faster dealignment of the short fibrils. It should be noted that contributions from the minimal and maximal lengths of the fibrils in the polydisperse sample could not been accounted for, as shown by the
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.8b00487.
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Complete description of the experimental setup with schematic (Figure S1). Details about the nanocellulose samples (Figure S2). Full derivation of eq 3 from the works of Marrucci and Grizzuti25,26 (Figure S3). Details about the inverse Laplace transforms performed (PDF).
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Christophe Brouzet: 0000-0003-3131-3942 L. Daniel Söderberg: 0000-0003-3737-0091 Notes
The authors declare no competing financial interest. 1025
DOI: 10.1021/acsmacrolett.8b00487 ACS Macro Lett. 2018, 7, 1022−1027
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ACKNOWLEDGMENTS The Wallenberg Wood Science Center at KTH is acknowledged for providing the financial assistance. The authors are thankful to Dr. Assya Boujemaoui for providing the CNC suspension and to Dr. Michaela Salajkova for assistance with the TEM images. Dr. Tomas Rosén is acknowledged for experimental assistance and helpful discussions. The authors thank Prof. Howard A. Stone for valuable comments.
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DOI: 10.1021/acsmacrolett.8b00487 ACS Macro Lett. 2018, 7, 1022−1027
Letter
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DOI: 10.1021/acsmacrolett.8b00487 ACS Macro Lett. 2018, 7, 1022−1027
