Bayesian estimations of orientation distribution functions from small-angle scattering enable direct prediction of mechanical stress in anisotropic materials

Properties of soft materials are influenced by their anisotropic structuring under nonequilibrium fields. Although anisotropic structure-property relationships have been extensively explored theoretically, comparison to experiments requires determination of the microstructural orientation probability distribution function (OPDF) of microstructural elements. Small angle scattering (SAS) measurements encode information about the OPDF, but tools to navigate this connection are incomplete. Here, we develop and validate an explicit framework to link arbitrary OPDFs to SAS measurements. Specifically, we propose, validate, and apply a method, maximum a posteriori scattering inference (MAPSI), whereby the OPDF may be obtained from SAS measurements using a Bayesian estimation method. Using this method, we obtain estimates of the full 3D OPDF for two model semidilute fd-virus (rodlike) dispersions at concentrations that are approximately equal to and twice the overlap concentration. From the OPDF, we calculate its second and fourth moments and compare these to predictions for a dilute suspension of rigid rods and to a recent theory for semidilute suspensions. Finally, we use both the theoretical and measured moments to calculate the stress, both for dilute and semidilute suspensions. These predictions are not only compared to each other, but also to measured values of the shear stress, and point to new insights into the behavior of suspensions of highly elongated particles in the transition between dilute and semidilute behavior. We also use this new framework to provide perspective on the connection between scalar parameterizations of scattering and the OPDF that have frequently been used in the past. The new tools developed in this work provide an unprecedented path toward experimental validation of dynamical theories of rodlike colloids and polymers, and for measurement of nonequilibrium structures and stresses of other complex fluids and soft materials with SAS.

Figure 1

The orientational coordinate system for a rod relative to the simple shear flow field indicated on the coordinate axis. The angles (ϕ, θ) provide information about particle orientation in the flow ($q_u$), gradient ($q_{\nabla u}$), and vorticity ($q_{\omega }$) directions. For the $1-2$ and $1-3$ sample environments, the relationship between the flow field and beam coordinate systems for the two sample environments used experimentally in this work are provided. In a real experiment, the direction of incident radiation ($q_z$) will not change, only the orientation of the flow relative to the incident radiation using different flow cells. Not pictured is the detector for the $2-3$ plane, where incident radiation is along the flow direction.

Figure 2

1D equilibrium SANS measurements and fits for the fd-virus dispersions considered in this work. The measured intensities are corrected for incoherent scattering and volume fraction extracted from fits of the measurements. Measurements are included for the concentrations (0.1 and 0.2 vol%) and measurement planes (1–2 and $1-3$) as the open points of varying color where the error bars indicate the standard deviation of the intensity from the SANS measurement. The cylinder model considered for the fd-virus is included as the black line.

Figure 3

Representative 3D OPDFs (i) simulated from dilute theory and extracted from MAPSI using theory-predicted scattering in the ($ii$) $1-2, 1-3$, and $2-3$ planes, (iii) $1-2$ and $1-3$ planes, (iv) $1-2$ plane and (v) $1-3$ plane. For the $1-3$ scattering, the incident radiation is in the $\nabla u$ (2) direction; for the $1-2$ scattering, the incident radiation is in the ω (3) direction; and for the $2-3$ scattering, the incident radiation is in the $u$ (1) direction. The representative results are included for ${Pe}_r=\left(\mathrm{a}\right)2$, (b) 7.5, and (c) 30. The color on the unit sphere represents the value of the OPDF in a particular direction in the $u$, ∇u, and ω reference frame specified in the bottom left corner. The black lines on the unit sphere indicate the edges of the hat functions used to discretize the OPDF.

Figure 4

Components of the second moment tensor, S, calculated from simulations of the dilute rod theory (black lines) and calculated from OPDFs extracted from MAPSI using theory predicted scattering (points) at varying ${Pe}_r$. Components of the moment tensor extracted using MAPSI include results using data from different scattering projections included: $1-2$ plane only (up-pointing red triangles with right side filled), $1-3$ plane only (down-pointing blue triangles with left side filled), using both $1-2$ and $1-3$ plane scattering (purple filled diamonds), using $1-2+1-3+2-3$ plane scattering (filled black circles) and using $1-2+1-3+2-3$ plane scattering with an extended $q$-range (filled gray circles). The error bars indicate the standard deviation of the value from posterior sampling where the standard deviation was taken to be $I\left(\mathbf{q}\right)/3$.

Figure 5

Components of the fourth moment tensor, ${\mathrm{S}}^{\left(4\right)}$, calculated from simulations of the dilute rod theory (black lines) and calculated from OPDFs extracted from MAPSI using theory predicted scattering (points) at varying ${Pe}_r$. The component of the moment tensor is indicated in the legend, where the four numbers correspond to the $i, j, k$ and $l$ indices for the moment. Components of the moment tensor extracted using MAPSI include results using data from different scattering projections included: $1-2$ plane only (up-pointing red triangles with right side filled), $1-3$ plane only (down-pointing blue triangles with left side filled), using both $1-2$ and $1-3$ plane scattering (purple filled diamonds), using $1-2+1-3+2-3$ plane scattering (filled black circles) and using $1-2+1-3+2-3$ plane scattering with an extended $q$-range (filled gray circles). The error bars indicate the standard deviation of the value from posterior sampling where the standard deviation was taken to be $I\left(\mathbf{q}\right)/3$.

Figure 6

A comparison of OPDFs from (i) dilute theory to those inferred by MAPSI using (ii) dilute theory predictions of SANS in the $1-3$ and $1-2$ planes and (iii, iv) experimentally measured SANS of the fd-virus dispersions at 0.1 and 0.2 vol%. For the $1-3$ measurement, the incident radiation is in the $\nabla u$ (2) direction, while for the $1-2$ measurement, the incident radiation is in the ω (3) direction. The representative results are included for ${Pe}_r$ of (a) 0.94, (b) 1.9, (c) 7.5, (d) 15, (e) 30 ($\dot{\gamma }=16$, 32, 128, 256, and $512{\mathrm{s}}^{-1}$, respectively). The color on the unit sphere represents the value of the OPDF in a particular direction in the $u,\nabla u$, and ω reference frame specified in the bottom left corner. The black lines on the unit sphere indicate the edges of the hat functions used to discretize the OPDF.

Figure 7

Components of the second moment tensor, S, calculated from simulations of the dilute rod theory (black lines), simulations of the modified D-B model (colored lines), and calculated from OPDFs extracted from MAPSI using experimentally measured scattering (colored points) at varying ${Pe}_r$. Concentrations of 0.1 vol% (blue) and 0.2 vol% (red) corresponding $\varphi /{\varphi }^{*}\approx 1$ and 2 are included. The solid black points are moments extracted from MAPSI and dilute theory predictions of the scattering in the $1-2$ and $1-3$ planes (Fig. 4). The error bars indicate the standard deviation of the value from posterior sampling.

Figure 8

Components of the fourth moment tensor, ${\mathrm{S}}^{\left(4\right)}$ computed for the OPDFs extracted from MAPSI (points) compared to the dilute theory predictions (black lines) with varying ${Pe}_r$. The component of the moment tensor is indicated in the legend, where the four numbers correspond to the $i, j, k$, and $l$ indices for the moment. Moments from SANS experiments with the fd-virus dispersion are indicated with the open circles while moments from SANS predictions from the dilute theory are indicated with the closed diamonds. The error bars indicate the standard deviation of the value from posterior sampling.

Figure 9

Rheological quantities (a) the particle contribution to the viscosity, (b) the first normal stress difference, and (c) the second normal stress difference for the fd-virus dispersions in simple shear flow. The quantities are normalized by the solvent viscosity and the volume fraction of particles. The open symbols represent values determined using S and ${\mathrm{S}}^{\left(4\right)}$ from MAPSI with Batchelor's expression for the stress tensor. The filled symbols are results from rheological measurements of the viscosity. The colors indicate the dispersion concentration including the dilute theory predictions of the scattering (black) and the measured 0.1 vol% (blue) and 0.2 vol% (red) fd-virus dispersions. Predictions from the dilute suspension theory are included as the solid black line. Predictions from the modified D-B model are included as the solid lines with colors corresponding to the concentrations of the measured points. The error bars represent the standard deviation of the measured quantity.

Figure 10

Representative results of the $1-3$ plane rheo-SANS experiments and theory predictions for ${Pe}_r$ of approximately (a) 2, (b) 7.5, and (c) 30 corresponding to $\dot{\gamma }=32$, 128, and $512{\mathrm{s}}^{-1}$, respectively. Included are the (i) dilute theory predicted orientation distribution function (OPDF) relative to the real-space flow, gradient, and vorticity directions ($u$, ∇u, and ω, respectively), (ii) theoretically predicted SANS patterns, (iii) experimental SANS patterns, and (iv) difference between these two patterns. The intensities and differences for the 2D patterns are on logarithmic scales while the OPDFs are presented on a linear scale. The low-$q$ cutoff and gap between detectors is indicated on the 2D patterns in black.

Figure 11

Representative results of the $1-2$ plane flow-SANS experiments and theory predictions for ${Pe}_r$ of approximately (a) 2, (b) 7.5, and (c) 30 corresponding to $\dot{\gamma }=32$, 128, and $512{\mathrm{s}}^{-1}$, respectively. Included are the (i) dilute theory predicted orientation distribution function (OPDF) relative to the real-space flow, gradient, and vorticity directions ($u$, ∇u, and ω, respectively), (ii) theoretically predicted SANS patterns, (iii) experimental SANS patterns, and (iv) difference between these two patterns. The intensities and differences for the 2D patterns are on logarithmic scales while the OPDFs are presented on a linear scale. The low-$q$ cutoff and gap between detectors is indicated on the 2D patterns in black.

Figure 12

(a) Averaged scattering intensity, $I$($\theta$,q*), as a function of annular angle $\theta$ within a $q$-range of 0.032 and $0.046{\AA }^{–1}$ for the $1-3$ plane. (b) Averaged scattering intensity, $I$($\varphi$,q*), as a function of annular angle $\varphi$ within a $q$-range of 0.032 and $0.046{\AA }^{–1}$ for the $1-2$ plane. For (a) and (b), the colored points correspond to experimentally measured intensity for ${Pe}_r$ from 0.06 to 30 (purple to red), which corresponds to shear rates from 1 to $512{\mathrm{s}}^{-1}$. The solid colored lines correspond to the theoretically predicted average scattering intensity at conditions matching the experiments. The dotted colored lines are the in plane OPDF from the theory (i.e., ϕ = 0 or $\theta =\frac{\pi }{2}$ for the $1-3$ and $1-2$ planes, respectively). The value of α was chosen for each shear rate to have the value of $\alpha N\left(\theta -\pi \right)$ equal $I\left(\theta ,q^{*}\right)$ or $\alpha N\left(\varphi -\pi \right)$ equal $I\left(\varphi ,q^{*}\right)$ at the minimum intensity. (c) and (d) Alignment factor ($A_f\left(q^{*}\right)$) as a function of shear rate for the experimentally measured scattering (solid points) and scattering predicted from the dilute rod theory (solid line). Included on the same plot (dotted line) is $S_m=S_{11}-S_{33}$ or $S_m=S_{11}-S_{22}$ for the $1-3$ and $1-2$ planes, respectively, from the OPDF simulation.