Capillary Flow-MRI: Quantifying Micron-Scale Cooperativity in Complex Dispersions

Strongly confined flow of particulate fluids is encountered in applications ranging from three-dimensional (3D) printing to the spreading of foods and cosmetics into thin layers. When flowing in constrictions with gap sizes, w, within 102 times the mean size of particles or aggregates, d, structured fluids experience enhanced bulk velocities and inhomogeneous viscosities, as a result of so-called cooperative, or nonlocal, particle interactions. Correctly predicting cooperative flow for a wide range of complex fluids requires high-resolution flow imaging modalities applicable in situ to even optically opaque fluids. To this goal, we here developed a pressure-driven high-field magnetic resonance imaging (MRI) velocimetry platform, comprising a pressure controller connected to a capillary. Wall properties and diameter could be modified respectively as hydrophobic/hydrophilic, or within w ∼ 100–540 μm. By achieving a high spatial resolution of 9 μm, flow cooperativity length scales, ξ, down to 15 μm in Carbopol with d ∼ 2 μm could be quantified by means of established physical models with an accuracy of 13%. The same approach was adopted for a heterogeneous fat crystal dispersion (FCD) with d and ξ values up to an order of magnitude higher than those for Carbopol. We found that for strongly confined flow of Carbopol in the 100 μm capillary, ξ is independent of flow conditions. For the FCD, ξ increases with gap size and applied pressures over 0.25–1 bar. In both samples, nonlocal interactions span domains up to about 5–8 particles but, at the highest confinement degree explored, ∼8% for FCD, domains of only ∼2 particles contribute to cooperative flow. The developed flow-MRI platform is easily scalable to ultrahigh field MRI conditions for chemically resolved velocimetric measurements of, e.g., complex fluids with anisotropic particles undergoing alignment. Future potential applications of the platform encompass imaging extrusion under confinement during the 3D printing of complex dispersions or in in vitro vascular and perfusion studies.


Table of content
Table S1.Parameters obtained from fitting of the 1D autocorrelation functions of 0.5% Carbopol and 15% FCD with Eq. 1, with their respective fitting errors.Parameter was set manually and was not a fitting parameter.The characteristic size of the microstructure, was   calculated from .

S1. Platform design: radii of capillaries
Figure S2.Intensity profiles vs the position in the x-direction, extracted from 2D axial µCT scans of all capillaries with hydrophilic (left column) or hydrophobic walls (right column).The diameters of the capillaries are marked within each profile in units of µm, as measured with Avizo software, using the ruler function.The error associated with the measurement is 3.7 µm for all profiles and corresponds to the pixel size.
Table S1.Parameters obtained from fitting of the 1D autocorrelation functions of 0.5% Carbopol and 15% FCD with Eq. 1 with their respective fitting errors.Parameter was set manually and was not a fitting parameter.The characteristic size of the microstructure, was   calculated from . 

Figure S1 .Figure S2 .
Figure S1.a) Scheme of the rotational rheo-MRI setup, with drive shaft length marked as , and the flow geometry fixed in the sensitive   region of the probe, marked with a red rectangle.b, c) Commonly used geometries in rotational rheo-MRI, namely CC and CP, with gap size, .d, e) Illustrative velocity profiles of a Newtonian fluid flowing in a CC and CP respectively. Figure S2.Diameters of capillaries used in the flow-MRI platform, determined from the intensity profiles vs the position in the x-direction, extracted from 2D axial µCT scans of all capillaries with hydrophilic or hydrophobic walls.The diameters of the capillaries were determined with Avizo software, using the ruler function.The error associated with the measurement is 3.7 µm for all profiles and corresponds to the pixel size.

Figure S3 .Figure S4 .
Figure S3.Local flow curves of silicone oil calculated from the measured velocity profiles (symbols) in capillaries with diameters ranging from 100 to 540 μm, across all tested values.The solid line shows the global flow behavior measured with a rheometer.  Figure S4.Micrographs of 0.5% Carbopol and 15% FCD, used in the calculation of the respective autocorrelation functions.The open symbols in the plots represent the radial average of the full 2D function shown in the inset and the solid line is the fit of the exponential decay function described with Eq. 1.

𝑟 0 Figure S5 .Figure S6 .
Figure S5.Global flow curves (open circles), fitted with the Herschel-Bulkley model (solid line) of 0.5% Carbopol, described by the equation of the form 4.07 + 4.4 and 15% FCD, described by the equation of the form 3.46 + 0.9 . 0.5  0.7 Figure S6.Global flow curve (solid line) of 0.5% Carbopol compared with the local flow curves calculated from the 1 H MRI velocity profiles measured in 540 μm hydrophilic capillary, 100 μm hydrophilic capillary, and 100 μm hydrophobic capillary.

Figure S7 .
Figure S7.Global flow curve (solid line) of 15% FCD compared with the local flow curves calculated from the 1 H MRI velocity profiles measured in 540 μm hydrophilic capillary, and 250 μm hydrophilic capillary.

Figure S8 .
Figure S8.Radial averages of the 2D density images obtained from the velocimetry measurements of the 15% FCD in a 540 µm capillary and 250 µm capillary under the applied pressures of 0.25 bar, 0.5 bar, 0.75 bar, and 1 bar.Lines of different colors represent the subsequent repeats.

Figure S1 .
Figure S1.a) Scheme of the rotational rheo-MRI setup, with drive shaft length marked as , and the flow geometry fixed in the sensitive   region of the probe, marked with a red rectangle.b, c) Commonly used geometries in rotational rheo-MRI, namely CC and CP, with gap size, .Red shaded rectangles represent the typical location and orientation of a slice where a 1D velocity profile is measured.d, e)  Illustrative velocity profiles of a Newtonian fluid flowing in a CC and CP respectively.

Figure S5 .Figure S6 .
Figure S5.Global flow curves (open circles), fitted with the Herschel-Bulkley model (solid line) of a) 0.5% Carbopol, described by the equation of the form 4.07 + 4.4 and b) 15% FCD, described by the equation of the form 3.46 + 0.9 . 0.5  0.7 Figure S6.Global flow curve (solid line) of 0.5% Carbopol compared with the local flow curves calculated from the 1 H MRI velocity profiles

Figure S7 .
Figure S7.Global flow curve (solid line) of 15% FCD compared with the local flow curves calculated from the 1 H MRI velocity profiles measured in a) 540 μm hydrophilic capillary, and b) 250 μm hydrophilic capillary.

Figure S8 .
Figure S8.Radial averages of the 2D density images obtained from the velocimetry measurements of the 15% FCD in a 540 µm capillary (top row) and 250 µm capillary (bottom row) under the applied pressures of 0.25 bar (a and e), 0.5 bar (b and f), 0.75 bar (c and g) and 1 bar (d and h).Lines of different colors represent the subsequent repeats.