This study presents a computational investigation of self-assembly and rheological behavior of multichain amphiphilic block copolymers under varying chain length (N), architecture, composition (f), and shear rate (γ˙). Using Brownian dynamics simulations, we systematically examined bead-spring model multichain diblock and triblock copolymers with N = 12–48 beads, f ranging from 0 to 1.0, and γ˙ spanning 0–0.1 ns−1. These systems self-assemble into spherical, prolate, rodlike, and oblate/disk micelles, with transitions governed by each variable, where increasing shear rate drives nonmonotonic prolate-to-rod redistribution, aggregating at weak shear rate before fragmenting at high rates; longer chains strengthen the dominant rod population; and higher hydrophobic fraction shifts assemblies from rod toward sphere at f = 1. Triblock copolymers form bridged assemblies, where hydrophobic end-blocks anchor into distinct micellar cores while hydrophilic middle blocks span the intervening space, producing solution viscosities up to half an order of magnitude greater than counterpart diblock systems. Linear viscoelastic spectra show terminal power-law scaling (G′ ∼ ω2, G″ ∼ ω1) with a liquidlike response (G″ > G′) throughout the accessed window. The triblock relaxation is governed by chain-end escape. The bridge fraction rises with the chain length as a longer midblock spans neighboring cores but falls with hydrophobic fraction as the shorter midblock forces chains into loops. The bridge lifetime saturates with increasing chain length once bridging is established (N ≥ 24), while the loop lifetime shortens with chain length yet increases with hydrophobic fraction, providing qualitative design insights relevant to flow-responsive polymer self-assembling systems.
A fundamental challenge in direct ink writing (DIW) is identifying a quantitative rheological metric that robustly predicts 3D printability across diverse polymer/particle blends. By characterizing shear rate hysteresis through consecutive up- and down-ramps, we demonstrate that, under the rheological protocol employed in this study, a down-ramp yield stress exceeding approximately 30 Pa reliably predicts successful DIW printing of 20 mm-high slump cones. Beyond the polydimethylsiloxane matrices studied in earlier work, we systematically validate this framework across polyisoprene, polybutadiene, polyethylene glycol (PEG), and poly(ethylene glycol) diglycidyl ether, while expanding the dispersed phase beyond fumed silica to include carbon black and titania. The addition of oligomeric PEG was found to promote rapid thixotropic recovery of the down-ramp yield stress, thereby converting nonprintable suspensions into viable DIW inks for multiple polymer/particle combinations. Additionally, we address the impact of processing history, specifically mixing and degassing, on rheological reproducibility. These findings establish a predictive framework for DIW ink design rooted in the physics of yield stress recovery and demonstrate the transferability of this approach across a broad range of polymer-particle systems for additive manufacturing.
The dimensionless group quantifying viscosity change due to viscous heating has been known throughout the past century as the Nahme number, the Griffith number, the Nahme–Griffith number, and even misleadingly the Brinkman number. Until the time of this writing, there has been no published biographical record of Rudolf Nahme or Richard Griffith and no clear tracing of their namesake dimensionless groups through the literature. Here, we review these dimensionless groups, tell the stories of three key contributors—Rudolf Nahme, Richard Griffith, and Arthur Hagg—, and follow the development of the naming conventions to the present day. This review synthesizes an updated understanding of these numbers, including the distinction between stress and velocity forms, alternative measures of the temperature sensitivity of viscosity, and clarification of model-agnostic versus model-specific descriptions for thermorheological behavior.
Orthogonal superposition rheology is a powerful technique for probing the viscoelastic properties of complex materials under flow. However, its accuracy is heavily dependent on the precise calibration of four calibration parameters: the instrument’s transducer mass M, friction coefficient ζ, stiffness coefficient K, and geometry end-effects captured through the end-factor c⊥. This work demonstrates that the standard, decoupled calibration procedure, which utilizes static weights and Newtonian standards, can result in negative stiffness coefficients and nonminimum phase behavior, which are not representative for the experimental conditions when characterizing complex materials. The standard calibration also fails to account for the mathematical coupling between the friction coefficient and the orthogonal end-factor. In this paper, we demonstrate that, by exploiting the frequency-dependency of the moduli of a viscoelastic material, here different aqueous poly(ethylene)oxide solutions, the calibration offsets for the transducer mass, friction coefficient, and stiffness coefficient can be determined independently, together with the corresponding true orthogonal end-factor. Furthermore, the presence of the sample avoids the unstable behavior that occurs during the standard calibration methods. Validation of this novel calibration procedure across multiple rheometers with a polyisobutylene-CALTRAN solution shows that the proposed method performs equally to or better than traditional calibrations, particularly at high frequencies where instrument inertia dominates. This viscoelastic calibration procedure ensures a physically consistent calibration for the characterization of complex fluids.
Dripping-onto-substrate (DoS) rheometry is a well-established method for measuring the extensional rheology of low-viscosity liquids. However, clear guidelines on the capabilities and limitations of the technique are lacking. In the present work, we define operational limits for measuring a transient extensional viscosity directly from observations of the rate of filament thinning, as well as model-based bounds on calculating a viscosity eta and extensional relaxation time tau E of a liquid using DoS. Dilute solutions of polyethylene oxide and polyacrylamide are used to probe the lower limit of measurable tau E, demonstrating that values as low as 0.1 ms can be resolved, provided (a) the intrinsic Deborah number (based on the ratio of the relaxation time and the Rayleigh breakup time scale) is De >= O ( 0.1 ) and (b) an instrumental constraint related to spatial and temporal resolution is satisfied. This instrumental constraint is quantified through a new metric we define as the filament capture rate, a "figure of merit" (expressed in s(-1)) that can be used to quantify the number of data points within the elastocapillary regime that are available for extraction of tau(E). Finally, model-agnostic bounds for reliable measurements of the transient extensional viscosity eta(+)(E) using DoS rheometry are derived. These bounds prescribe experimental limitations due to inertia, gravity, and measurement resolution (spatial and temporal) that can assist rheologists in the design and interpretation of extensional viscosity measurements using DoS. Collectively, these results provide a quantitative roadmap for reliable DoS rheometry and affirm its use for measuring the transient extensional viscosity of liquid-like materials.
