Rigid Dumbbell Research Articles

Lopsided rigid dumbbell rheology from Langevin equation: A graduate tutorial

Lopsided rigid dumbbell rheology from Langevin equation: A graduate tutorial

Assigning macromolecular meaning to nonlinear continuum rheology

The Oldroyd 8-constant continuum framework has yielded elegant analytical solutions for many polymer processing flows. However, continuum frameworks are silent on macromolecular structure. We can assign macromolecular meaning to the continuum constants by bridging continuum frameworks to the macromolecular theory of polymeric liquid dynamics. When the Oldroyd 8-constant framework has been bridged to rigid dumbbell theory (two-step), no higher order rheology was predicted (ν1=ν2=0). By higher order, we mean the nonlinear rheology. This troubled Bird (1972), motivating his modified Oldroyd 8-constant continuum framework, which does predict higher order rheology, to which meaning in rigid dumbbell theory is assigned. By two-step, we mean we get the three Jeffreys model constants from the macromolecular expression for the complex viscosity, and then solve five equations simultaneously for the five remaining constants. In this paper, in three steps, we bridge the Bird 8-constant framework to the more versatile rotarance theory (general rigid bead-rod theory). By three-step, we mean we get the three Jeffreys model constants from the macromolecular expression for the complex viscosity, and then solve three equations simultaneously for the next three, and finally solving two equation simultaneously for the remaining two higher order constants. By versatile, we mean accommodating any axisymmetric macromolecular structure (including the rigid dumbbell). We find the constants in the Bird 8-constant framework to be explicit functions of just one dimensionless macromolecular attribute: the ratio of the moment of inertia about the molecular axis, to the moment about either transverse axis. We thus assign macromolecular meaning to the higher order rheology. In passing, we also discover a new bridge to the Oldroyd 8-constant framework (three-step), which also assigns macromolecular meaning to the higher order rheology.

Cox–Merz rules from general rigid bead-rod theory

The value of this work is in its macromolecular explanations of both Cox–Merz rules, thus of when to expect them to work. For polymeric liquids and their solutions, the measured values of the steady shear viscosity and the magnitude of the complex viscosity often equate, within experimental error, when compared at common shear rate (in units of t−1) and angular frequency (in units of rad t−1). Called the first Cox–Merz rule, this remarkable empiricism, with one exception, has defied most macromolecular explanations. This one exception is the suspension of multi-bead rods and its special case of rigid dumbbells. The second Cox–Merz rule equates approximately the slope of the first derivative of steady shear viscosity with respect to shear rate with the real part of the complex viscosity when compared at common shear rate (in units of t−1) and angular frequency (in units of rad t−1). In this paper, we explain both Cox–Merz rules for all axisymmetric macromolecules, be they prolate or oblate, of almost any lopsidedness. Furthermore, through the lens of general rigid bead-rod theory, we define under what conditions these rules do not apply. Specifically, the first Cox–Merz rule fails when the macromolecules are too oblate.

Confined steady simple shear flow of polymeric liquids

In a confined simple shear flow, the macromolecules of a polymeric liquid reorient near the walls so that the measured viscosity decreases. For instance, in a small-amplitude oscillatory shear flow, the real part of the complex viscosity decreases with confinement, and macromolecular orientation explains this. These effects in oscillation have been explained analytically for a rigid dumbbell suspension and, for a confined small-amplitude oscillatory shear flow, the summation coefficients have been determined. By contrast, for the confined steady shear flow, the summation coefficients are undetermined. In this paper, we determine these coefficients and use them to evaluate the steady shear (i) viscosity and (ii) normal stress coefficients for a rigid dumbbell suspension. We find that the zero-shear viscosity and the zero-shear first normal stress coefficients decrease with confinement. We further find that the dimensionless (i) steady shear viscosity curve increases with confinement and (ii) first normal stress coefficient first decreases with light confinement and then increases with greater confinement. We confirm our theory, at low confinement, by comparing with our new measurements of the confined zero-shear viscosity of a polystyrene solution.

Reaction-Induced Molecular Dancing and Boosted Diffusion of Enzymes

A novel mechanism of reaction-induced active molecular motion, not involving any kind of self-propulsion, is proposed and analyzed. Because of the momentum exchange with the surrounding solvent, conformational transitions in mechano-chemical enzymes are accompanied by motions of their centers of mass. As we show, in combination with rotational diffusion, such repeated reciprocal motions generate an additional random walk - or molecular dancing - and hence boost translational diffusion of an enzyme. A systematic theory of this phenomenon is developed, using as an example a simple enzyme model of a rigid two-state dumbbell. To support the analysis, numerical simulations are performed. Our conclusion is that the phenomenon of molecular dancing could underlie the observations of reaction-induced diffusion enhancement in enzymes. Major experimental findings, such as the occurrence of leaps, the anti-chemotaxis, the linear dependence on the reaction turnover rate and on the rate of energy supply, become thus explained. Moreover, the dancing behavior is possible in other systems, natural and synthetic, too. In the future, interesting biotechnology applications may be developed using such effects.

Macromolecular microfluidic concentrators

Microfluidic design, fabrication, and experiments have developed rapidly, leading to lab-on-chip separation or fractionation. In this work, we design a continuous concentrator for macromolecular solutions. Our design relies on the analytical solutions for orientational diffusion under laminar pressure-driven slot flow through a microchannel [W. Stasiak and C. Cohen, “Dilute solutions of macromolecules in a rectilinear Poiseuille flow,” J. Chem. Phys. 78, 553 (1983)]. Using rigid dumbbell theory, we provide analytical solutions for the design of our microfluidic macromolecular hydrodynamic chromatography. We arrive at our design through the use of well-known confinement-driven composition profiles. Using a pair of razor-sharp blades, our design separates the slot flow into a symmetric core inner slot (concentrated) between two outer slots (diluted). We discover a minimum dimensionless blade leading edge separation for complete fractionation, and that this decreases with confinement and also decreases with dimensionless shear rate.

Stress Tensor of Single Rigid Dumbbell by Virtual Work Method

We derive the stress tensor of a rigid dumbbell by using the virtual work method. In the virtual work method, we virtually apply a small deformation to the system, and relate the change of the energy to the work done by the stress tensor. A rigid dumbbell consists of two particles connected by a rigid bond of which length is constant (the rigid constraint). The energy of the rigid dumbbell consists only on the kinetic energy. Also, only the deformations which do not violate the rigid constraint are allowed. Thus we need the dynamic equations which are consistent with the rigid constraint to apply the virtual deformation. We rewrite the dynamic equations for the underdamped SLLOD-type dynamic equations into the forms which are consistent with the rigid constraint. Then we apply the virtual deformation to a rigid dumbbell based on the obtained dynamic equations. We derive the stress tensor for the rigid dumbbell model from the change of the kinetic energy. Finally, we take the overdamped limit and derive the stress tensor and the dynamic equation for the overdamped rigid dumbbell. We show that the Green-Kubo type linear response formula can be reproduced by combining the stress tensor and the dynamic equation at the overdamped limit.

Hydrodynamic interaction within canonical macromolecular structures

In general rigid bead-rod theory, we deduce the rheological properties of a suspension of macromolecules from the orientation distribution that arises during flow. The most important feature governing this orientation is macromolecular architecture, and right behind this, enters hydrodynamic interaction. Until now, general rigid bead-rod theory has neglected hydrodynamic interactions, namely, the interferences of Stokes flow velocity profiles between nearby beads. The lopsidedness of the architecture affects orientability, and so do these heretofore unexplored interferences within the macromolecule. We here employ a new method for exploring how such hydrodynamic interactions affect the complex viscosity. This method has, with great effort, been used to examine hydrodynamic interactions in complex architectures, namely, multi-bead rods and backbone-branched polymers. However, it has yet to be applied to canonical forms. In this paper, we focus on the simplest of rigid architectures: (i) rigid dumbbell, (ii) tridumbbell, (iii) rigid rings, and (iv) planar stars. We call these forms canonical. We arrive at beautiful algebraic expressions for the complex viscosity for each canonical form. We find that for the dimensionless complex viscosity, for all canonical forms, hydrodynamic interactions just depend on the ratio of the bead diameter to the nearest bead separation, d/2L≡A. Furthermore, we find that for the dimensionless complex viscosity, for canonical forms (i) and (iii), hydrodynamic interactions shift the real part upward and minus the imaginary part downward. For canonical forms (ii), both parts are unaffected. For canonical forms (iv), the story depends interestingly on the number of beads. We advance the mathematics of fluids by establishing, for intramolecular hydrodynamic interactions, the foundational equations which future work must recover.

Spirographic motion in a vortex

The present study investigates the dynamics of an inertialess rigid dumbbell in a steady two-dimensional vortex. This system goes beyond the point-particle approximation but remains analytically solvable. For any vortex, the dynamics of the dumbbell follows from the existence of a constant of motion that is independent of the form of the vortex. In particular, if the fluid angular velocity decreases with the radial distance, the dumbbell performs spirographic trajectories around the vortex center, the shape of which is highly sensitive to the initial position and orientation of the dumbbell.

Hydrodynamic interaction and complex viscosity of multi-bead rods

One good way to explain the elasticity of a polymeric liquid is to just consider the orientation distribution of the macromolecules. When exploring how macromolecular architecture affects the elasticity of a polymeric liquid, we find the general rigid bead–rod theory to be both versatile and accurate. This theory sculpts macromolecules using beads and rods. Whereas beads represent points of Stokes flow resistances, the rods represent rigid separations. In this way, how the shape of the macromolecule affects its rheological behavior in suspension is determined. Until recently, general rigid bead–rod theory has neglected interferences of the Stokes flow velocity profiles between nearby beads. We call these hydrodynamic interactions, and we here employ our new method for exploring how these interactions affect the complex viscosity of suspensions of multi-bead rods. These multi-bead rods are also called shish-kebabs. We use the center-to-center distance between adjacent beads as the characteristic length. We proceed analytically, beginning with a geometric expression for the shish-kebab bead positions. Our analytical solution for the complex viscosity presents as one for N=3,5,6,7,8,…, one for N=4, and another for the rigid dumbbell, N=2. We find that for shish-kebabs, hydrodynamic interactions (i) increase zero-shear viscosity, (ii) increase zero-shear first normal stress coefficient, (iii) decrease the real part of the dimensionless complex viscosity, and (iv) increase minus the dimensionless imaginary part. We find that the combination of (iii) and (iv) explains crossovers of the parts of the complex viscosity. We further find that for a monodisperse polystyrene solution, the general rigid bead–rod theory with hydrodynamic interaction, for both parts of the complex viscosity, provides stunning improvement over without.

Chemically symmetric and asymmetric self-driven rigid dumbbells in a 2D polymer gel.

We employ computer simulations to unveil the translational and rotational dynamics of self-driven chemically symmetric and asymmetric rigid dumbbells in a two-dimensional polymer gel. Our results show that the activity or the self-propulsion always enhances the dynamics of the dumbbells. Making the self-propelled dumbbell chemically asymmetric leads to further enhancement in dynamics. Additionally, the direction of self-propulsion is a key factor for chemically asymmetric dumbbells, where self-propulsion towards the non-sticky half of the dumbbell results in faster translational and rotational dynamics compared to the case with the self-propulsion towards the sticky half of the dumbbell. Our analyses show that both the symmetric and asymmetric passive rigid dumbbells get trapped inside the mesh of the polymer gel, but the chemical asymmetry always facilitates the mesh to mesh motion of the dumbbell and it is even more pronounced when the dumbbell is self-propelled. This results in multiple peaks in the van Hove function with increasing self-propulsion. In a nutshell, we believe that our in silico study can guide researchers to design efficient artificial microswimmers possessing potential applications in site-specific delivery.

Confinement and complex viscosity

Whereas much is known about the complex viscosity of polymeric liquids, far less is understood about the behavior of this material function when macromolecules are confined. By confined, we mean that the gap along the velocity gradient is small enough to reorient the polymers. We examine classical analytical solutions [O. O. Park and G. G. Fuller, “Dynamics of rigid and flexible polymer chains in confined geometries. II. Time-dependent shear flow,” J. Non-Newtonian Fluid Mech. 18, 111–122 (1985)] for a confined rigid dumbbell suspension in small-amplitude oscillatory shear flow. We test these analytical solutions against the measured effects of confinement on both parts of the complex viscosity of a carbopol suspension and three polystyrene solutions. From these comparisons, we find that both parts of the complex viscosity decrease with confinement and that macromolecular orientation explains this. We find the persistence length of macromolecular confinement, Lp, to be independent of both λω and λγ̇0.

Analytical Model for Particle Capture in Nanopores Elucidates Competition among Electrophoresis, Electroosmosis, and Dielectrophoresis.

The interaction between nanoparticles dispersed in a fluid and nanopores is governed by the interplay of hydrodynamical, electrical, and chemical effects. We developed a theory for particle capture in nanopores and derived analytical expressions for the capture rate under the concurrent action of electrical forces, fluid advection, and Brownian motion. Our approach naturally splits the average capture time in two terms, an approaching time due to the migration of particles from the bulk to the pore mouth and an entrance time associated with a free-energy barrier at the pore entrance. Within this theoretical framework, we described the standard experimental condition where a particle concentration is driven into the pore by an applied voltage, with specific focus on different capture mechanisms: under pure electrophoretic force, in the presence of a competition between electrophoresis and electroosmosis, and finally under dielectrophoretic reorientation of dipolar particles. Our theory predicts that dielectrophoresis is able to induce capture for both positive and negative voltages. We performed a dedicated experiment involving a biological nanopore (α-hemolysin) and a rigid dipolar dumbbell (realized with a β-hairpin peptide) that confirms the theoretically proposed capture mechanism.

Zero-shear viscosity of Fraenkel dumbbell suspensions

Whereas rigid dumbbell suspensions predict, at least qualitatively, most of the viscoelastic material functions measured in the laboratory, Hookean dumbbells predict few of these. For instance, whereas rigid dumbbells predict a shear-thinning viscosity curve, as they should, Hookean dumbbells yield a constant for the steady shear viscosity. In this paper, we explore the addition of a Hookean spring to the end of a rigid rod, a dumbbell attributed to Fraenkel. In this way, we focus our attention on how macromolecular extensibility affects the configuration distribution in steady shear flow. We arrive at the exact solution to this configuration distribution in steady shear flow at low shear rate and then insert it into the Giesekus expression for the stress tensor to arrive at an exact solution for the zero-shear viscosity and the zero-shear values of the normal stress differences.

A charged finitely extensible dumbbell model: Explaining rheology of dilute polyelectrolyte solutions

A robust non-Newtonian fluid model of dilute polyelectrolyte solutions is derived from kinetic theory arguments. Polyelectrolyte molecules are modeled as finitely elongated nonlinear elastic dumbbells, where effective charges (interacting through a simple Coulomb force) are added to the beads in order to model the repulsion between the charged sections of polyelectrolyte chains. It is shown that the relative strength of this repulsion is regulated by the electric-to-elastic energy ratio, E, which is one of the key parameters of the model. In particular, E accounts for the intrinsic rigidity of polyelectrolyte molecules and can be used to explain the impact of solvent salinity on polyelectrolyte rheology. With two preaveraging approximations, the constitutive equations of the resulting fluid model are formulated in closed form. Material functions predicted by the model for steady shear flow, steady extensional flow, small-amplitude oscillatory shear flow, and start-up and cessation of steady shear flow are obtained and investigated using a combination of analytical and numerical methods. In particular, it is shown how these material functions depend on E. The two limiting cases of the model—uncharged dumbbells (E = 0) and rigid dumbbells (E → ∞)—are included in the analysis. It is found that despite its simplicity, the model predicts most of the experimentally observed rheological features of polyelectrolyte solutions.

Pattern method for higher harmonics of first normal stress difference from molecular orientation in oscillatory shear flow

This study examines the simplest relevant molecular model of a polymeric liquid in large-amplitude oscillatory shear (LAOS) flow: rigid dumbbells suspended in a Newtonian solvent. For such suspensions, the viscoelastic response of the polymeric liquid depends exclusively on the dynamics of dumbbell orientation. Previously, the explicit analytical expressions of the zeroth, second, and fourth harmonics of the alternating first normal stress difference response in LAOS have been derived. In this paper, we correct and extend these expressions by seeking an understanding of the next higher harmonic. Specifically, this paper continues a series of studies that shed light on molecular theory as a useful approach in investigating the response of polymeric liquids to oscillatory shear. Following the general method of Bird and Armstrong [“Time-dependent flows of dilute solutions of rodlike macromolecules,” J. Chem. Phys. 56, 3680 (1972)], we derive the expression of the first normal stress coefficient up to and including the sixth harmonic. Our analysis relies on the extension of the orientation distribution function to the sixth power of the shear rate. Our expression is the only one to have been derived from a molecular theory for a sixth harmonic and thus provides the first glimpse of the molecular origins of a first normal stress difference higher than the fourth.

Pattern method for higher harmonics from macromolecular orientation in oscillatory shear flow

For a suspension of rigid dumbbells, in any simple shear flow, we must first solve the diffusion equation for the orientation distribution function by a power series expansion in the shear rate. Our recent work has uncovered the pattern in the coefficients of this power series [L. M. Jbara and A. J. Giacomin, “Orientation distribution function pattern for rigid dumbbell suspensions in any simple shear flow,” Macromol. Theory Simul. 28, 1800046-1–1800046-16 (2019)]. Specifically, we have here used this pattern on large-amplitude oscillatory shear (LAOS) flow, for which we have extended the orientation distribution function to the 6th power of the shear rate. In this letter, we embed this extension into the Giesekus expression for the extra stress tensor to arrive at the alternant shear stress response, up to and including the seventh harmonic. We thus demonstrate that the pattern method for macromolecular orientation now allows our harmonic analysis to penetrate the shear stress response to oscillatory shear flow far more deeply than ever.

Small-angle light scattering in large-amplitude oscillatory shear

We explore wormlike micellar orientation during oscillatory shear using small-angle light scattering. Previous oscillatory-shear light scattering measurements focused on phase separation in polymeric solutions undergoing shear and none on wormlike micelles. We correlate light scattering videos of wormlike micelles undergoing oscillatory shear with molecular orientation. Specifically, we compare our orientation measurements with the predictions of rigid dumbbell theory. We find that “tulip” shaped scattering patterns caused by micellar orientation are only partially captured by the predicted scattering generated by rigid dumbbell theory. Additionally, we confirm that rigid dumbbell theory cannot describe the “butterfly” shaped scattering patterns arising from concentration fluctuations during micelle breakup. We successfully create a theory to describe both orientation and concentration fluctuation scattering by combining rigid rod Rayleigh-Debye scattering theory with flow induced Helfand-Fredrickson scattering theory.

Viscoelastic propulsion of a rotating dumbbell

Viscoelastic fluids impact the locomotion of swimming microorganisms and can be harnessed to devise new types of self-propelling devices. Here we report on experiments demonstrating the use of normal stress differences for propulsion. Rigid dumbbells are rotated by an external magnetic field along their axis of symmetry in a Boger fluid. When the dumbbell is asymmetric (snowman geometry), non-Newtonian normal stress differences lead to net propulsion in the direction of the smaller sphere. The use of a simple model allows to rationalise the experimental results and to predict the dependence of the snowman swimming speed on the size ratio between the two spheres.

Hydrodynamic interaction for rigid dumbbell suspensions in steady shear flow

From kinetic molecular theory, we can attribute the rheological behaviors of polymeric liquids to macromolecular orientation. The simplest model to capture the orientation of macromolecules is the rigid dumbbell. For a suspension of rigid dumbbells, subject to any shear flow, for instance, we must first solve the diffusion equation for the orientation distribution function. From this distribution, we then calculate the first and second normal stress differences. To get reasonable results for the normal stress differences in steady shear flow, one must account for hydrodynamic interaction between the dumbbell beads. However, for the power series expansions for these normal stress differences, three series arise. The coefficients for two of these series, (ck, dk), are not known, not even approximately, beyond the second power of the shear rate. Analytical work on many viscoelastic material functions in shear flow must be checked for consistency, in their steady shear flow limits, against these normal stress difference power series expansions. For instance, for large-amplitude oscillatory shear flow, we must recover the power series expansions in the limits of low frequency. In this work, for (ck, dk), we arrive at the exact expressions for the first 18 of these coefficients.