Spherical Balls Settling Through a Quiescent Cement Paste Measured by X-ray Tomography: Influence of the Paste Thixotropy

The settling of spherical balls in quiescent cement pastes of increasing age is studied. Metallic spheres with radii of 2, 2.5 and 3mm are dropped into the paste and allowed to settle, while their position is tracked using X-ray tomography. The instantaneous velocity of the spheres, calculated from their movement, is observed to be quasi-constant during their fall, and an average is estimated. The results show that the average velocity of the balls decreases logarithmically with paste age until ball stoppage, for all three ball sizes. In parallel, the rheological properties of the cement paste are measured using a rheometer with a vane geometry. The evolution of the paste static yield stress over time is evaluated, and proves to be a reliable predictor for ball stoppage. Finally, thixotropic models of increasing complexity are evaluated. These models consider four forms of structural growth and breakdown parameters, and their ability to capture the ball settling velocity as a function of paste age is compared. This emphasizes the importance of considering paste breakdown in relation to shearing of the paste when the ball passes through it.

💡 Research Summary

This paper investigates the settling behavior of spherical steel balls in fresh cement paste as the paste ages under quiescent conditions. Three ball diameters (4 mm, 5 mm, and 6 mm; radii 2, 2.5, and 3 mm) were released into a paste prepared with a water‑to‑cement ratio of 0.50, containing a polycarboxylate superplasticizer and a retarder (each 0.075 wt %). After a standardized 2‑minute remix, the paste was considered “age zero” and left to rest. Ages ranging from 120 s to 480 s were examined.

A custom electromagnetic release mechanism placed each ball at the centre of a 70 mm‑diameter Plexiglas tube, ensuring a consistent entry point. The ball entered the paste with a non‑negligible initial velocity, sufficient to overcome any surface “skin” that forms at later ages. The tube was mounted in a TESCAN UniTOM XL X‑ray computed‑tomography scanner, which captured 25 frames s⁻¹ over a 300 mm observation window. ImageJ was used for contrast enhancement, and automated tracking software extracted the ball centre coordinates. Instantaneous velocities were computed by a central finite‑difference scheme (Δt = 0.04 s).

Results show that after a brief acceleration phase each ball quickly reaches a quasi‑steady velocity that remains essentially constant over the observation window. This steady velocity decreases logarithmically with paste age for all three ball sizes. At sufficiently high ages (e.g., the 3 mm ball stops around 420 s), the velocity drops to zero, indicating complete arrest. The observed velocity profiles confirm that the structural evolution of the paste during the short falling time is negligible, and that the deceleration is linked to the increasing resistance of the aging matrix.

Rheological characterization was performed on the same paste using an Anton Paar 302 rheometer equipped with a four‑blade vane in a sanded cup. A three‑step protocol was applied immediately after the remix: (i) a 10⁻⁵ strain oscillation for 600 s to monitor the storage modulus G′(t); (ii) periodic amplitude sweeps up to γ = 10⁻⁴ every 60 s to determine the critical strain γ_cr at which G′ falls to 80 % of its linear value; (iii) a shear‑rate ramp from 0.1 to 800 s⁻¹ and back to obtain the flow curve. The static yield stress τ_s was calculated as τ_s = γ_cr · G′_lin and was found to increase linearly with age: τ_s(t) = 0.026 t + 3.55 Pa (t in seconds). The complex viscosity η* grew according to η* = η*_0 + K t^m with η*_0 ≈ 3.38 × 10³ Pa·s, K ≈ 1.7 × 10⁴ Pa·s^{1‑m}, and m ≈ 0.13. High‑shear measurements yielded a Herschel‑Bulkley fit with a dynamic yield stress of ~33 Pa and a flow index n ≈ 0.5, while the plateau viscosity in the fully fluidized regime was 1 ± 0.1 Pa·s.

To link the ball dynamics with the evolving microstructure, four thixotropic models of increasing complexity were evaluated. Each model describes the evolution of a structural parameter λ(t) through a balance between growth (λ_g) and breakdown (λ_d) terms. The general form is dλ/dt = λ_g (1 − λ) − λ_d γ̇ λ, where the local shear rate γ̇ is approximated by the ball velocity divided by its radius (γ̇ ≈ v/R). The four variants differ in whether λ_g and λ_d are taken as constant, linear, or nonlinear functions of λ and γ̇. Model parameters were calibrated against the measured steady velocities for all ages and ball sizes.

The model that incorporates a shear‑rate‑dependent breakdown term (λ_d ∝ γ̇^p, p > 1) reproduces the experimental data most accurately, especially the sharp velocity drop and eventual arrest observed at higher ages. Simpler models that neglect shear‑induced breakdown or assume only linear growth overpredict the velocities and fail to capture the stoppage phenomenon. This outcome underscores that the local fluidization caused by the moving ball is insufficient to fully break down the increasingly structured paste; the balance between the gravitational driving stress (Δρ g R) and the evolving static yield stress τ_s governs the transition from continuous settling to arrest.

In summary, the study demonstrates that: (1) the static yield stress measured by small‑amplitude oscillatory tests is a reliable predictor of ball stoppage; (2) the steady settling velocity decays logarithmically with paste age, reflecting the logarithmic growth of the paste’s microstructure; (3) thixotropic models that explicitly account for shear‑induced structural breakdown are necessary to capture the observed dynamics. The combined use of high‑resolution X‑ray CT for real‑time particle tracking and rheological measurements provides a powerful framework for probing the time‑dependent behavior of cementitious suspensions. These insights are directly relevant to emerging construction technologies such as 3‑D concrete printing, automated placement, and the design of admixture strategies aimed at controlling sedimentation and flow stability in fresh concrete.