Nanofibril Alignment in Flow Focusing: Measurements and

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Nanofibril Alignment in Flow Focusing: Measurements and Calculations Karl M.O. Håkansson, Fredrik Lundell, Lisa Prahl-Wittberg, and L. Daniel Söderberg J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b02972 • Publication Date (Web): 13 Jun 2016 Downloaded from http://pubs.acs.org on June 20, 2016

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The Journal of Physical Chemistry

Nanofibril Alignment in Flow Focusing: Measurements and Calculations Karl M. O. H˚ akansson,†,‡ Fredrik Lundell,∗,†,‡ Lisa Prahl Wittberg,†,‡ and L. Daniel Söderberg† Wallenberg Wood Science Center, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden, and Linné FLOW Centre, Department of Mechanics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden E-mail: [email protected] Phone: +46 8 790 68 75

∗

To whom correspondence should be addressed Wallenberg Wood Science Center ‡ FLOW, Department of Mechanics, KTH Royal Institute of Technology †

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Abstract The alignment of anisotropic super-molecular building blocks is crucial to control the properties of many novel materials. In this study, the alignment process of cellulose nanofibrils in a flow focusing channel has been investigated using small angle X-ray scattering (SAXS) and modeled using the Smoluchowski equation, which requires a known flow field as input. This flow field was investigated experimentally using micro particle tracking velocimetry and numerically applying the two-fluid level set method. The semi-dilute dispersion of cellulose nanofibrils was modeled as a continuous phase with a higher viscosity as compared to that of water. Furthermore, the implementation of the Smoluchowski equation also needed the rotational Brownian diffusion coefficient, which was experimentally determined in a shear viscosity measurement. The order of the nanofibrils was found to increase during the extension in the flow focusing channel, whereafter the rotational diffusion acted on the orientation distribution, driving the orientation of the fibrils towards isotropy. The main features of the alignment and de-alignment processes were well predicted by the numerical model, but the model over-predicted the alignment at higher rates of extension. The apparent rotational diffusion coefficient was seen to increase steeply as the degree of alignment was increased. Thus, the combination of SAXS measurements and modelling provides the necessary framework for quantified studies of hydrodynamic alignment followed by relaxation towards isotropy.

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Introduction Recently there has been an upswing regarding the interest in new concepts for cellulose-based materials. 1–3 Cellulose is the most abundant biopolymer on earth and the interest spurred is partly due to the recent developments towards large-scale production of strong and stiff (ultimate tensile strength more than 3 GPa; 4 Young’s modulus around 120 GPa 5 ), cellulose nanofibrils (CNF) from wood, as described by Päa¨kkö et al. 6 . The specific strength of CNF is comparable to glass fibres and the specific stiffness is similar to Kevlar. 3,4,7–10 Cellulose nanofibrils are around 20 nm thick and a micrometer long and were originally liberated from wood pulp using an energy expensive homogenization process. 11 However, new chemical and enzymatic pretreatment processes of the cellulose pulp have reduced the energy demand of the homogenization step by up to 98%, turning CNF into an industrially viable starting point for new materials. In engineering materials composed of anisotropic components, the structure on all scales, from nano to macro, influences the properties of the bulk material. 12 Structural control of materials to be produced industrially and in large quantities is very challenging, especially when detailed control of the smaller scales is desired. For materials containing slender particles, such as carbon or wood fibres, the flow conditions during the manufacturing processes can be used to control structural aspects, such as the orientation distribution of fibres in paper. 13 As an example, if the aim is anisotropic material properties, it is well known that extensional flows can be used to align slender particles. 14 However, the behaviour of a semidilute dispersion of nano-scale anisotropic particles in such a flow is not as intuitive due to more complex particle interactions (mechanical, chemical and electrostatic). In addition, Brownian diffusion must be considered. In the simplest approximation all such interactions are modeled by an adjustable single rotational diffusion coefficient. 15 The alignment of the fibrils in the final material is a critical aspect to exploit the full potential of CNF. Several studies have been dedicated to produce films and filaments with aligned CNF, for example Iwamoto et al. 16 used a wet-spinning technique by jetting a fibril3



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dispersion from a syringe into a coagulation bath whereafter the coagulated gel was dried. By changing the flow rate, the dry filament was shown to have different degrees of alignment. Sehaqui et al. 17 obtained aligned structures by preparing gel films and stretch them to different extents. The stretching induced a preferential orientation in the direction of the stretching. However, the films can only be stretched until breakage, which for the weak gel occurs well before full alignment is reached. A third method was introduced by H˚ akansson et al. 3 using the concept of flow focusing in order to produce filaments with aligned CNF. The flow focusing geometry, where a central flow is focused by outer (sheath) flows, has been used extensively in the last two decades with different aims: for example to mix, 18 induce chemical reactions, 19 produce droplets 20 or manufacture filaments. 21,22 In the filament manufacturing field, most efforts are spent on controlling the shape of the filaments. 21,23,24 However, in a process where the goal is to optimize the mechanical properties in a technical filament, the alignment of elongated polymers, fibrions or fibrils is of outmost importance. 12 It is well established that elongated particles align in the flow focusing. 25–27 H˚ akansson et al. 3 used ions in the outer flow that diffused into the dispersion after the alignment was achieved. The ions initiated a dispersion-gel transition 28 locking the aligned structure. Although it was shown that this concept can be used to produce a filament, there are several possibilities for further optimization. In particular, it would be valuable to increase the alignment of the fibrils in the dry filaments (35◦ was the highest mean alignment obtained). An improved understanding of the mechanisms affecting the alignment process is needed in order to achieve this. In the present work, aspects of velocity field and fibril orientation during assembly in flow focusing touched upon by H˚ akansson et al. 3 are studied in more detail. Velocity and orientation measurements are combined with calculations of the flow field and orientation of cellulose nanofibrils. The details of the orientational dynamics of the fibrils (alignment induced by extensional flow followed by relaxation towards isotropy) determine to what extent the nanostructure in the final material can be controlled. The aim of this study is to identify

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the physical processes that limits fibril alignment. In particular, it is investigated to what extent a model assuming fibril rotations from the flow according to Jeffery 14 and a rotational diffusion acting against any preferred alignment as derived by Doi 29 , Doi and Edwards 30 can predict the streamwise variation of fibril alignment. The theory for rotational diffusion has been verified for stiff polymers (around a fifth of the size of the cellulose nanofibrils) in experiments 31 and Monte Carlo simulations. 32 To validate the model, the alignment and de-alignment in the channel is measured in two series of synchrotron small angle X-ray scattering (SAXS) experiments, where one of the series were presented earlier. 3 These measurements are combined with fluid mechanical modelling and measurements of the flow field and fibril orientation. The model is shown to capture the alignment and de-alignment behaviour of the fibrils. However, at higher extension rates the assumption that the rotational diffusion is only weakly dependent on the orientation turns out to fail. Potential causes of this failure are discussed. In the next section, the Dispersion and flow cell will be described. This is followed by two sections on methods where the first presents velocity measurements (micro particle tracking velocimetry, µPTV) and flow field simulations and the second orientation measurements (small angle X-ray scattering, SAXS and polarized optical microscopy, POM) and modelling. The results and discussion follows and finally, the conclusions are given.

Dispersion and flow cell Similar suspensions of cellulose nanofibrils in water was used throughout the work and the flow setups, see figure 1, only needed minor adjustments in order to be compatible with the different measurement techniques.

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Figure 2: The black dots are shear viscosity measurements of the CNF dispersion at pH 7 and the red curve is a Carreau fit to the measurements. The critical shear rate, γ˙ crit , indicated with the black dashed line is a measure of Dr as explained in section .

is the relaxation time and n is the power index in the assumption:

ηeff (γ) ˙ = ηinf + (η0 − ηinf )[1 + (τ γ) ˙ 2]

n−1 2

.

(1)

The best fit yielded ηinf = 15.3 mPa s, η0 = 63.4 mPa s, τ = 39.8 s and n = 0.48. As can be seen in figure 2, the viscosity of the dispersion attains a practically constant shear viscosity for γ˙ < 1 s−1 and γ˙ > 100 s−1 . The Carreau model was chosen due to its ability to represent the viscosity data of the dispersion and the observation that the dispersion does not show any significant yield stress. The typical shear stress in the experiment is on the order of 10 s−1 . Rotational diffusion (to be strictly defined in section ) has been identified as a critical parameter for preparation of filaments with aligned fibrils. 3 The rotational diffusion, Dr , quantifies the rate at which a non-isotropic system relaxes towards isotropy. The diffusion

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was measured to be 0.04 rad2 s−1 . 3 This value was obtained by instantaneously stopping the flow and measure the decay of birefingence of the fibrils in the channel (the dispersion is more birefringent when the fibrils are aligned). This result should be viewed as a lower limit, since if the flow does not stop immediately, a lower value than the true value will be obtained. A second way of measuring Dr is by a rotary viscometer, 34 assuming that the critical shear rate, γcrit , at which the viscosity starts to decrease, is the inverse of the orientational relaxation time. A lower bound for the rotational diffusion is then obtained as Dr = γ˙ crit /6 where γ˙ crit is defined as the intersection of the first and the second slope, see figure 2. Here, the approximate value of Dr = 0.25 rad2 s−1 is extracted. Note that this is a mean value at different fibril alignments, and that the exact alignment at which this value is obtained is unknown.

Flow focusing device The term flow focusing usually refers to a channel system consisting of four channels, three inlets and one outlet, see figure 1a, which also defines the coordinate system (x, y, z). Fluids are transported in the three inlets towards a four-way intersection (focusing) where the core flow is focused by the two sheath flows at the intersection. In this experiment, a fibrilwater dispersion is focused by two water sheath flows (no gelling occurs and no filaments are produced since the topic of the present study is the orientational dynamics of the fibrils in dispersion). The flow focusing device used in this study is shown in figure 1b, where an h = 1 mm thick slit is cut out of a 1 mm thick steel plate. To encapsulate the channel and allow optical access, two 100 × 100 mm2 transparent plates were placed on both sides of the steel plate. These were made of poly(methyl methacrylate), PMMA, in the POM experiments and Kapton in the SAXS experiments. The Kapton was necessary in the high intensity X-ray environment, while the non-polarizing PMMA was suitable for the POM experiments. Furthermore, to keep the channel and windows in place outer layers made from aluminum 8


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Figure 3: Image sequence illustrating the µPTV procedure. Background image in the channel in a and three consecutive images in b-d, where three particles are marked with red circles.

plates with 20 holes (threaded in one of the plates) were used. The sandwiched construction was sealed with 20 screws. Two syringe pumps (WPI, Al-4000) were used to transfer the liquids (water and CNF dispersion) from syringes to the channels at constant volumetric flow rates. The flow rates were chosen based on those used in H˚ akansson et al. 3 and the core flow rate Q1 = Qcore = 6.5 mm3 s−1 was kept constant, while two sheath flow rates were used: Q2 = Qsheath = 7.5 and 15 mm3 s−1 . With the kinematic viscosity of water at 20◦ C equal to ν = 1 × 10−6 m2 s−1 the Reynolds number of the core becomes Recore = Qcore /hν = 6.5.

Velocity measurements and flow field simulations Even though the flow is laminar, it is non-trivial to determine the flow field, due to the non-zero Reynolds number, the 3D geometry of the channel, the viscosity ratio, interface between water and the fibril dispersion and the non-Newtonian behaviour of the dispersion. Concentrating on the rate of extension on the centreline of the channel, the flow velocity 9



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was measured with a micro particle tracking velocimetry, µPTV, technique and calculated with a two-fluid 2D simulation in COMSOL. The 2D simulation is fast and results in smooth velocity fields as compared to the experiments, hence, the experiments are used to validate the flow simulations. The calculated velocity along the centreling, wc (z) will be used in the particle orientation calculations described in section .

Micro Particle Tracking Velocimetry, µPTV The velocity of the core fluid in the channel was measured using Micro Particle Tracking Velocimetry, µPTV, which is a velocimetry technique based on identifying and tracking individual particles in the flow. 35 If the time between images is known the velocity can readily be calculated. The method assumes that the particles are small enough to follow the flow as passive tracer particles, and in this experiment polyamid seeding particles with 20 µm diameters obtained from Dantec dynamics were used. A low concentration of particles was mixed with the CNF dispersion and image sequences were captured at frequencies f = 50 and 100 Hz with an exposure time of 3 ms. The particles were identified by image analysis 36 and the straightforward PTV algorithm described below was used to track particles in three consecutive images. The algorithm stepped through all (n) particles with positions (zji , yji ) (j = 1, ..., n) in image i. First selecting particle p, and checking at all positions z i+1 > zpi and y i+1 = ypi ± 3.5 pixels in the next image (i + 1) in order to find the same particle. If a match was found, the difference in pixels is ∆z = zpi+1 − zpi . The last criterion was to search for a particle in a third image (i + 2) at 2∆z × 0.9 < z i+2 < 2∆z × 1.1 and y i+2 = ypi ± 3.5. The 10 % tolerance allows for accelerating particles. If the last criterion was fulfilled, the velocity was calculated from ∆z, the frequency of the image acquisition, f , and the number of pixels per length at the position zpi + ∆z/2 with the known pixel scale (155 pixel mm−1 ). A background image at the region of extensional flow is displayed in figure 3a and three consecutive example images are seen in figure 3b-d, where the background has been removed. 10


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Figure 5: Velocity error between the analytic and the simulated centreline velocity versus resolution in a and the level set interface parameter ǫ in b. The shapes and colours are three different grids with different resolutions.

the interface and the equation describing this is ∂χ + ∇ · χu + σ ∂t

∇χ ∇ χ(1 − χ) |∇χ|

− ǫ∇ · ∇χ = 0,

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where the thickness of the interface layer is proportional to teh interface parameter ǫ and the amount of re-initializations are determined by σ, which is set to be equal to the maximum speed in the simulation (small changes to this parameter did not change the solution). The interface parameter, ǫ, was adjusted along with the grid size in order to ensur the convergence of the simulations. In order to model the velocity field in a flow focusing device as inexpensively as possible, a 2D axisymmetric geometry (pipe) was used. The geometry is displayed in figure 4a were the top edge represents the axis of symmetry. The boundary conditions were set to fully developed laminar inflow at the lower and the left inlets while a zero pressure was set at the outlet (right edge). No-slip was implemented at all other boundaries and a small curvature (radius of 10−5 m) was implemented at the corners. The cross section of the pipe was √ matched to that of the flow focusing device, resulting in a diameter, dpipe = 2h/ π, where

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h = 1 mm. The sheath inlet was set to have the width equal to h and all mass flow rates were matched with the experiments. An inlet length of 1.5h was seen to be sufficient and the outlet channel was 5.5h long. The Navier-Stokes equations are possible to solve analytically for a fully developed twofluid core-annular flow in an infinitely long pipe. 38 The full black curve in figure 4b represents this solution, where the boundary between the two fluids is presented with the vertical black dashed line. The calculated velocity profiles at different positions is seen in figure 4b where each profile corresponds to one streamwise location as specified in the legend. The blue curve (z/h = 0) represent the position where the sheath flow channel starts and the subsequent numbers thus represents the number of channel widths h downstream of the start of the focusing. The velocity profile is seen to approach the analytical solution downstream of the focusing and the ratio between the analytical, wc,analytic , and simulated centreline velocity, wc,sim , is used to evaluate the convergence of the numerical solution. Figure 5a and 5b represent the velocity error (|1 − wc,sim /wc,analytic |) versus grid points and level set interface parameter, ǫ, respectively. The velocity on the centreline at z/h = 3.5 was used, where each colour represents one grid size and the difference between markers with the same colour is a variation of ǫ. From figure 5b it can be seen that it is the level set parameter that is controlling the convergence. The simulation results presented have ǫ = 4 µm and 262,000 grid points, resulting in an error less than 1 %. It should be noted that the mass flow error always is less than 0.015 %.

Orientation measurements and modelling The alignment of the nanofibrils in the flow focusing device was visualized with polarized light and quantified with synchrotron small angle X-ray scattering. It was also calculated with the Smoluchowski equation.

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Polarized Optical Microscopy, POM Since the dispersion consists of highly elongated particles (the fibrils) where all dimensions but one are significantly smaller than the wavelength of visible light, they are hard to detect. However, the fibril dispersion is birefringent and thus the variations in orientation distribution can be visualized using Polarized Optical Microscopy (POM). 39 The setup used in the experiments is illustrated in figure 1d, and consists of a light source (Schott, KL2500 LCD), two polarization filters and a camera (Basler, piA1900-32gm) mounted on a microscope (Nikon, SMZ 1500). A first polarization filter was placed between the light source and the channel and a second filter between the channel and the microscope. Without the channel between the two polarization filters, and the polarization directions of the two filters at 90◦ with respect to each other, no light will pass through to the camera. However, if a birefringent sample is put between the filters, the sample can shift the polarization of the light allowing it to pass through the second filter and thus to be detected by the camera. The fibril dispersion becomes more birefringent as the fibrils align and in order to highlight this alignment in the z-direction, the filters were rotated 45◦ with respect to the z-direction. POM images with the fibrils in the flow will be presented later and a higher light intensity means a higher degree of alignment.

In situ Small Angle X-ray Scattering (SAXS) Small Angle X-ray Scattering (SAXS) measurements were performed in situ in the flow device in order to quantify the nano-scale configuration of the fibrils in the dispersion. Diffraction is a standard method for determining structures at length scales shorter than the wavelength of visible light. 40 The experiments were performed in a transmission set-up at the P03 beamline 41 at the synchrotron PETRA III (DESY) in Hamburg, Germany. The set-up is sketched in figure 1d where an X-ray beam with wavelength λ = 0.957 ˚ A and beamsize 24 × 11 µm2 (Horiz. × Vert.) was allowed to pass through the centre of the channel at 14


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Figure 6: Original SAXS diffractogram in a, where the colour scale is the 10-logarithm of the photon count. Transformed diffractogram in b, where the image in a is ”unfolded” to have the Cartesian coordinates as the radial, q, and azimuthal, φ, directions in a. Dark blue parts indicate regions were no information is available, however, by symmetry arguments, full intensity distributions from the region between the two white lines in b can be obtained. In c, these intensity distributions are displayed, where the mean is shown as a thick red line on top of the individual distributions.

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different downstream positions. A single-photon counting detector (Pilatus 1M, Dectris) with pixel size 172 × 172 µm2 was placed at a distance of 8422 mm from the flow device. A typical scattering pattern (diffractogram) is shown in figure 6a (the colour scale is logarithmic and the horizontal and vertical blue bands are positions where no pixels are located in the detector). A beam stop is placed in the centre to stop the high intensity X-ray beam and the circular shadow is due to a vacuum tube running from approximately 300 mm after the flow device to approximately 100 mm before the detector. The vacuum tube is needed in order to minimize the interactions between particles in the air and scattered X-rays before they reach the detector. When the X-ray beam hits the fibrils, the fibrils scatter the X-rays resulting in constructive interference if the length, d, between the fibrils fulfill Bragg’s law: d = λ/[2 sin ζ/2], where ζ is the scattering angle, see figure 1d. Since the detector is two-dimensional, the orientation distribution of the fibrils in the plane of the detector, for a given d, is related to a scan along the azimuthal direction at the radial position defined by q = 4π sin(ζ/2)/λ, corresponding to that d. The 180 degree symmetry makes it possible to use data from other parts of the diffractogram at locations where data is missing due to the beam stop or lack of pixels. A background diffractogram, where the flow conditions are the same but pure water is fed into the core channel, is subtracted from all experiments. Extraction of the normalised orientation distributions, ΨSAXS (φ), is performed by transforming the diffractogram from the Cartesian coordinates to the azimuthal and radial coordinates φ and q (see figure 6a), which is displayed in figure 6b. The intensity scale is linear and each column is scaled with the maximum intensity. Now, each column represents the orientation distribution at a certain q, and in figure 6c the mean of every ∆q = 0.027 nm−1 between the two white lines at q = 0.39 and 1.08 nm−1 are shown. The mean of q = 0.39 − 1.08 nm−1 are shown with a red curve. It is clear that the orientation distribution is similar for all q in this range. Due to noise from diffuse scattering a zero-level for the distributions must be established. This zero-level is chosen based on the assumption that

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there are no fibrils with an orientation of 90◦ at the position of highest alignment (z/h = 2, Q2 /Q1 = 2.3) and that this noise is constant at all positions.

Fibril rotation and the Smoluchowski equation The Smoluchowski equation is used to compute the orientation distribution of the fibrils on the channel centreline. The Smoluchowski equation is a diffusion equation for the fibril orientation distribution Ψ, 15 with an extra forcing term originating from the fluid flow. Observations with polarised light indicate that the diameter of the fibril stream is constant far downstream (20h) of the focusing while the alignment has decayed. It can thus be argued that translational diffusion is negligible compared to rotational diffusivity and convection in this case. Under these assumptions the Smoluchowski equation for Ψ takes the form: DΨ = ℜDˆr · ℜΨ − ℜ · ωΨ, Dt

(3)

where, Ψ(r, p, t) is the orientation distribution, which is dependent on position, r, orientation described by the unit vector, p (see figure 1 c) and time, t. The rotational diffusion coefficient ˆ r (p), ω is the angular velocity of the fibril and ℜ is the rotational operator is denoted D ℜ=r×

∂ . ∂r

(4)

The orientation distribution along the centreline is assumed to be independent of θ (i.e. cylindrical geometry is used) and the coordinate system is defined in figures 1a and 1c. Assuming the fibrils to be rigid and inertia-free and the velocity in the z-direction on the centreline wc , the non-dimensional Smoluchowski equation becomes

wc∗

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∂Ψ ˙ − sin φφΨ sin φ ∂φ

(5)

where the superscript (∗) indicates a non-dimensionalized quantity, scaled with w and h. 17



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The orientation distribution, Ψ, is normalized according to Z

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˙ is obtained from analytically obtained The rate of change of the fibril orientation, φ, equations 14 assuming that inertial effects are negligible. The Reynolds number based on maximum strain, fibril length and water viscosity is on the order of 10−6 , validating this assumption. The rotation rate in axisymmetric flow is then given by ∂φ ∂w∗ φ˙ = ∗ = − ∗ ∂t ∂z

rp2 − 1 rp2 + 1

3 cos φ sin φ, 2

(8)

where rp is the aspect ratio of the fibrils. ˆ r , is strongly dependent on the concentration of The rotational diffusion coefficient, D fibrils, i.e. if the fibrils interact with other fibrils during their rotation. In this study, the system is in a semi-dilute state, which means that mechanical, hydrodynamical and possibly electrostatic interactions are important. The first two have been studied analytically for rigid polymers. 15,29,30 Starting from the dilute case, the rotational diffusion constant, Dr0 , was derived by Doi and Edwards 15 , using results from Kirkwood and Auer 42 , to assume the following form: Dr0 =

3kB T (2 ln(2rp ) − 1) . 16πηs a3

(9)

Here, kB is Boltzmann’s constant, T is the absolute temperature, a is the half length of the fibril and ηs is the viscosity of the solvent. For the semi-dilute case, the rotational diffusion constant increases due to particle-particle interactions and, based on the tube model, the

18


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following concentration dependent formula has been derived: 29,30

Dr = βDr0 (nl3 )−2 .

(10)

Here, n is the number of fibrils per unit volume and β is a constant ranging from unity to 103 that needs to be determined experimentally. 15 The final step is to account for the orientation of the fibrils. The reason for this is that if fibrils are aligned, they will not interact with the neighbouring fibrils. This effect is also investigated by Doi and Edwards 15 and the resulting rotational diffusion constant, Dˆr , is given as: Z −2 4 ′ ′ ′ ˆ Dr (p) =Dr dp |p × p |Ψs (p ) π Z πZ π Ψs (φ′ ) sin θ′ |(− cos θ sin θ′ sin φ′ )ˆ p1 + =Dr 8

(11)

0

0

′

′

′

′

′

′

(cos θ sin θ cos φ − sin θ cos θ )ˆ p2 + (sin θ sin θ sin φ )ˆ p3 | dθ dφ

′

−2

,

where Ψs is the orientation distribution of the surrounding fibrils, which in this work is assumed to be identical to Ψ. In this model, the fibrils are assumed to be both stiff and straight. Furthermore, electrostatic interactions due to the surface charge of the fibrils are not taken into consideration. Here, the measured Dr , see section , was used along with the orientation dependence above as the primary estimate of rotational diffusion. Moreover, calculations were also performed in order to study the effects of different rotational diffusions. During these calculations, β was varied while the following values where used: c = 0.003, l/2 = a = 1 µm, rp = 100, ηs = 1 mPa s, kb = 1.3 × 10−23 and T = 293 K. Equation 5 was discretised with finite differences in φ with periodic boundary conditions and integrated numerically in z using MATLAB.

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The order parameter, S In order to compare the fibril alignment and orientation distributions in the experiments with the computations by a single number, an order parameter, S, is used. The order parameter of van Gurp 43 is used and is calculated as the mean of the second order Legendre polynomial, P2 : S = hP2 (cos φ)i =

3 1 cos2 φ − 2 2

(12)

From this order parameter, the complete anisotropy tensor with respect to a given direction (in this case the streamwise direction) can be obtained. 15 It is thus a measure of the alignment of elongated particles in the z-direction, where S = 1 when all fibrils are aligned in the zdirection and S = 0 when the distribution is random. Expanding the mean results in:

S=

Z

π

Ψ(φ) 0

3 1 cos2 φ − 2 2

sin φ dφ

Z

π

dθ,

(13)

0

which is used to calculate S with Ψ normalized according to Z

π

Ψ(φ) sin φ dφ 0

Z

π

dθ = 1.

(14)

0

Results and discussion The flow and alignment of a semi-dilute cellulose nanofibril dispersion in a flow focusing device is investigated experimentally and modeled numerically. This section is divided into two parts, where the first part describes and discusses the velocity measurements and simulations, while the second part is focused on the alignment experiments and calculations.

Velocity measurements and simulations Figure 7a shows the calculated centreline velocity, wc , along the channel for four different viscosities (ηcore = 1, 15, 40, 65 mPa s) and the non-Newtonian Carreau viscosity model. 20


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w c /w c,s tart

4 3

3.5

b

ηcore = 1 ηcore = 15

3

ηcore = 40

2.5

w c /w c,s tart

5

a

ηcore = 65 Carreau

2

2 1.5 1

1

η c or e

EXP ηcore = 40

0.5

Carreau 0 −1

0

1

2 z /h

3

4

0

5

0

2

4

6

8

10

z /h

c 3.5

d

3

2

ηcore = 40

1.5

2.5 dw ∗ /dz ∗

w c /w c,s tart

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 1.5

Carreau

Q 2 /Q 1 = 2.3

1 0.5 0

1

EXP ηcore = 40

0.5

Q 2 /Q 1 = 1.15

−0.5

Carreau 0 −0.5

0

0.5

1

1.5 z /h

2

2.5

−1

3

0

2

4

6

8

10

z /h

Figure 7: Centreline velocity from the simulations for different viscosities of the core flow as function of downstream position are displayed in a. The gray dashed lines indicate the location of the focusing channels. In b (closeup in c), blue and green curves are simulated velocities using the Carreau model and ηcore = 40 mPa s, respectively. The red curves and black errorbars are results obtained by µPTV measurements of the core fluid in the flow focusing device. In c, the strain rate along the centreline using the Carreau model and ηcore = 40 mPa s are shown.

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The velocities have been normalized with the centreline velocity upstream of the focusing, wc,start = 0.0131 m s−1 . The velocity is seen to decrease significantly as the viscosity of the core fluid is increased from 1 to 15 mPa s, a result due to continuity of the shear stress over the jump in viscosity. Another effect of the viscosity increase is that the extension stretches over a longer distance downstream, from being significant in a region less than one channel height, ηcore = 1, to being significant for approximately two and a half channel heights when ηcore = 65 mPa s. Further increasing the viscosity only leads to minor changes, as do the addition of the non-Newtonian shear thinning (Carreau) model. It can also be observed that the inlet centreline velocity is slightly lower for the non-Newtonian case, due to the non-parabolic velocity profile. The velocity development in the channel was also measured by µPTV for two velocities, Qsheath = 7.5 and 15 mm3 s−1 , to validate the simulations. In figure 7b the velocity of the identified particles as function of downstream position is presented, along with the nonNewtonian (Carreau, green) and ηcore = 40 mPa s (blue) simulation cases. Since the focal length was larger than the channel depth, it was possible to determine the position in y but not in x. The red curve represents the mean of the experiment and the errorbars indicate the rms. The simulation with ηcore = 40 mPa s, is a good fit of the mean experimental data. Note that the mean velocity measured with the µPTV experiments represents the average in the x-direction of the velocity profile since the measurement will integrate throughout the x-direction of the channel. Hence, the measured velocity should be lower than the velocity obtained in the simulations, as can be observed in figure 7b. This discrepancy diminishes once the velocity profile is flat, which from figure 4b is seen to be the case for z/h > 0.5. The agreement between the measurements and Newtonian simulations in a cylindrical geometry in figure 7 shows that the flow model is adequate in spite of the strong assumptions regarding geometry and rheology. The strain rate along the centreline for the simulations with the Carreau model and ηcore = 40 mPa s are shown in figure 7d. The maximum strain rate is seen to be slightly higher with the Carreau model and it is possible that strong

22


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extensional viscosity, which is a typical feature of suspensions with non-spherical particles and has been observed for CNF suspensions, 44 has a stronger effect than the shear thinning.

Orientation measurements In the paper by H˚ akansson et al. 3 the alignment of cellulose nanofibrils as a function of downstream position was quantified by the order parameter S for one flow-rate ratio, see figure 8b; all SAXS data are averages along the X-ray beam. The images in figure 8c are SAXS diffractograms obtained at different downstream positions as indicated by the red dots in figure 8a-b. The red ellipses in figure 8c indicate levels of constant intensity, and the relation between the semi-principal axes is a footprint of the local mean fibril orientation at the given position. The ellipse initially becomes more circular from the first to the second diffractogram in 8c, whereafter the ellipse gradually becomes more elongated from the second to the fifth diffractogram. Figure 8a displays a POM image, where higher fibril orientation results in increased birefringence, giving a higher intensity in the image. The horizontal scale is the same in figures 8a and 8b and the location of the measurement points in 8b are displayed in 8a. Upstream of the focusing at z/h < 0 the orientation distribution of the fibrils is not isotropic, but a slight alignment in the flow direction (S ≈ 0.18) can be detected. This alignment is due to the shear close to the channel walls, inducing a rotational motion of elongated particles with the mean orientation in the flow direction. It is important to emphasize that the fibrils are periodically flipping in a shear flow and are not steady, in spite of the slight mean alignment. In figure 8a, the shear alignment is visualized by a higher light intensity close to the wall upstream of the focusing. The alignment observed in figure 8b is increasing as the flow is accelerated in the interval 0 < z/h < 2, (remember figure 7). Thereafter, when the velocity has assumed a constant value, the fibrils de-align due to Brownian diffusion.

23




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0.8

a

η

core

0.6

= 1 mPa s

Carreau ηcore = 40 mPa s

S

0.4 0.2 0 −2

0

2

4

6 z /h

8

10

12

14

12

14

0.8

b

Dr 0.6 0.4

S

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.2 0 −2

0

2

4

6 z /h

8

10

Figure 9: The order parameter from solving the Smoluchowski equation for three different flow fields is shown in a. The flow fields are solutions from simulations with different viscosity of the core, 1 and 40 times the viscosity of water along with the Carraeu model. b, Seven different numerical solutions to the Smoluchowski equation (solid curve) were the rotational diffusion coefficient, Dr , is varied. The red curve correspond to the measured case. The markers are the same experimental results as in figure 8.

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Orientation distribution calculations Figure 9 shows the same experimental points as in figure 8b along with the order parameter evolution (solid curves) from solutions of equation 5. The three different solutions seen in 9a correspond to three different flow simulations, namely: ηcore = 1 mPas, 40 mPas and the Carreau case, where Qcore = 6.5 mm3 s−1 , Qsheath = 7.5 mm3 s−1 . The rotational diffusion ˆ r is calculated from equation (11) based on the local Ψ and Dr = 0.25 rad2 s−1 . The D location of the maximum is shifted downstream as the viscosity, ηcore , is increased from 1 mPa s to 40 mPa s. It can also be seen that the Newtonian case with ηcore = 40 mPa s fits somewhat better than the non-Newtonian Carreau model, which is not surprising since the ηcore = 40 mPa s better represents the measured flow in the channel, see figure 7b. The model captures the overall features of the alignment and de-alignment process even though there are discrepancies compared to the experiments. Remember that the expressions for the rotational diffusion in the semi-dilute regime contain the fitting parameter β, which varies from 1 to 103 depending on the concentration, see equation 10. In figure 9b the calculated order as a function of downstream position for different values of β (and consequently the rotational diffusion coefficient, Dr ) are presented with the centreline velocity taken from the ηcore = 40 mPa s simulation. The exact values of the rotational diffusion are: Dr = 0.0025, 0.0076, 0.025, 0.076, 0.25 (red curve), 0.35 and 0.76 rad2 s−1 corresponding to β = 1, 3, 10, 30, 100 (red curve), 140 and 300. As Dr is increased, the fibrils de-align faster. The case when Dr = 0.35 rad2 s−1 fits better from z/h ≈ 2 to 10 than the measured case for Dr = 0.25 rad2 s−1 . However, it is clear that the discrepancies between measurements and calculations cannot be fully accounted for by adjusting β.

Increased rate of extension In figure 10 the measured order parameter at different downstream positions for two flow-rate ratios, Q2 /Q1 = 1.15 and 2.3, are presented, together with calculations with the measured 26


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Dr = 0.25 rad2 s−1 (β = 100). The calculation with the centreline velocity from the 2D, two-fluid simulation gives a good prediction of the actual alignment evolution for the milder extension. However, the predicted alignment for the Q2 /Q1 = 2.3 case overshoots the measurements significantly. Since the measured and predicted velocities agree fairly well, it will be argued that this discrepancy appears due to effects on the orientational dynamics of the fibrils that are not included in the model. At this point it should also be noted that the Carreau model results in even larger strain than the 40 mPa s case (see figure 7d). Since the strain drives the alignment, the overshoot would consequently be even larger if the centreline velocity from the Carreau model was used. In the next section it will be shown that the observed rotational diffusion is considerably higher for more aligned fibrils than for less aligned fibrils. Furthermore, this effect can be quantified as an orientational effect on Dr that is stronger than that given by equation (11), which is included in the calculation. Thus, in order to understand and control the alignment fully and find a description independent of fitting parameters, the Brownian rotational diffusion must be better understood, in particular the effect of increased orientational order in a semi-dilute dispersion consisting of surface charged, possibly entangling, elongated particles.

Observations regarding rotational diffusion, Dr It is clear that there is a resistance to further alignment in the process, and the resistance seems to increase as the alignment is increased. Now, three different measurements of the rotational diffusion, Dr , will be presented and compared. The two first are described in section , while a third measure of the rotational diffusion is carried out by fitting Dr in the above described numerical calculation (Smoluchowski equation) to the experimental measurements. This was performed after the extensional flow, since Dr is the only parameter affecting the development of the orientation distribution in this region (the velocity is constant). The SAXS measurements points are used in pairs. For each pair of consecutive SAXS 28


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0.5 0.4 Measured D r

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


SAXS Shear Visc Flow Stop

0.3 0.2 0.1 0 −0.1

0

0.1

0.2

0.3

0.4

S

Figure 11: Rotational diffusion coefficient, Dr , measured in three different experiments a different order parameter, S. The red dot is taken from H˚ akansson et al. 3 , the blue dashed line is from figure 2, where the order parameter is unknown but assumed to be small, and the black dots are results of fitting the Smoluchowski equation to the SAXS experiments.

measurement, the Dr providing the best description of the decay from the upstream to the downstream one is determined. This makes it possible to determine Dr for different values of S. Results from all three measurement techniques versus the order parameter are presented in figure 11. The red dot corresponds to the flow stopping experiment, the blue dashed line is taken from the viscosity measurement and the black dots are extracted from the SAXS experiments. The fibril alignment (order parameter) for the flow stopping and viscosity measurements are assumed to be low, here put at S = 0 and 0.02 < S < 0.18, respectively. The data indicates that the diffusion increases as the alignment increases and it seems as if the increase is stronger at higher degrees of alignment. By utilizing this measured behaviour of Dr , a better agreement between experiment and simulation at higher extensions should be possible. This would allow for the timescales of alignment, relaxation towards isotropy and diffusion of gelling inducing ions 3 to be tuned with more accuracy. 29



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It can also be hypothesised that interaction between fibrils contribute to the rotational diffusion at high strain rates as described theoretically 45,46 and studied experimentally. 47 This contribution to the rotational diffusion is typically modelled as a constant multiplied with the strain rate. Krochak et al. 47 determined the coefficient to be of the order of 0.004 in a concentration and aspect range similar to this study. In our case, the maximum strain rate is on the order of 10 s−1 and thus, the maximum contribution to rotational diffusion by interactions should be around 0.04, or considerably smaller than the rotary diffusion observed for the cellulose nanofibrils. When it comes to nanoparticles, rotational diffusion of metallic nanorods can be measured with several methods. 48,49 Our present results indicate that extension of such methods to aligned and well defined configurations is of great interest. This initial discussion of our observations in figure 11 assumes that the observed relaxation towards isotropy is due to Brownian dynamics only. As mentioned, this is an oversimplification. By adding another term to the Smoluchowski equation containing, for example, a potential, 50 the resistance to alignment at higher orientation could be described more realistically, which, in turn, could facilitate the prediction of the orientation in the channel at stronger elongations. It is clear that further studies are needed, both expermental, theoretical and numerical. Key aspects are effects of fibril interactions and non-straight, flexible fibrils.

Conclusions The present work has been inspired by the hydrodynamics of a promising process for filament production with the final goal to create a filament with fully aligned fibrils. 3 Specifically, a numerical model for fibril alignment is presented and the predictions are compared with experimental observations obtained by in-situ x-ray diffraction measurements. The experimental results show that increasing the extension rate increases the alignment of the fibrils, as expected. However, the fibrils do not align as much as predicted by the

30


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model for a stronger extensional flow. Either the assumption that the rotational Brownian diffusion is the only controlling parameter, or that the rotational diffusion is only weakly dependent on the overall fibril orientation, is not valid at higher degree of alignment. In particular, neither fibril/fibril interactions nor the configuration and dynamics of each fibril are incorporated in the present model. It is not known to what extent, or how, electrostatic, steric, hydrodynamic and other interactions cause clustering, ordering or de-ordering. Furthermore, studies of the configuration of individual fibrils are yet in their infancy 51 and more details can be expected to be revealed in the near future. Regarding the filament manufacturing process, the present results indicate that the extension rate necessary to fully align the fibrils may have to be several magnitudes larger than the extension rate required to start the alignment process. Or even worse, if the rotational diffusion is diverging as the alignment is increased, an infinite extension rate would be necessary to fully align the fibrils in the present setup. However, deeper understanding necessitates not only further knowledge of fibril interactions and configurations as indicated in the previous paragraph, but also even more detailed knowledge of how the fibril configuration is affected by the velocity gradients. Cleverly designed diffraction experiments are a key factor in this development. With increased knowledge of the involved mechanisms, one can also expect an improved ability to manipulate and control the nanofibril structure during assembly. The main contributions and observations leading up to the conclusions stated above are summarized below. • The computed velocity field of the dispersion agrees very well with µPTV measurements. • The fibril alignment is measured with SAXS and it is demonstrated that the model captures the evolution of the fibril alignment well at low alignments. However, at high alignments, the model over-predicted the alignment.

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• The plug flow downstream of the extensional flow allows for the rotational diffusion as a function of order parameter to be estimated from the streamwise decay of the order parameter. • These estimations show that the rate at which alignment decays increases rapidly (and more than given by equations 9–11) as the order is increased. Understanding and modelling the physical processes behind this is central to control the nanostructure in the dispersion (and eventually in the final material).

Acknowledgement This work was funded by the Knut and Alice Wallenberg foundation through the Wallenberg Wood Science Center. Innventia AB is acknowledged for providing the CNF. The simulations were performed on resources at PDC Centre for High Performance Computing (PDC-HPC). Kim Karlström and Göran R˚ adberg manufactured the experimental setups. Parts of this research was carried out at the light source PETRA III at DESY, a member of the Helmholtz Association (HGF). The authors thank Dr. S. V. Roth, Dr. S. Yu and Dr. G. Santoro for assistance in using beamline P03. Dr. A. Fall and Prof. L. W˚ agberg are greatly acknowledged for the many discussions and centrifugation the CNF. Dr. A. Fall and M. Kvick are thanked for the experimental help at PETRA III. Prof. F. Bark is acknowledged for reading and commenting on the manuscript.

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Nanofibril Alignment in Flow Focusing: Measurements and

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