Coil-stretch Transition Research Articles

Scaling variation in the pinch-off of colloid-polymer mixtures

Hypothesis: The scaling laws of drop pinch-off are known to be affected by drop compositions including dissolved polymers and non-Brownian particles. When the size of the particles is comparable to the characteristic length scale of the polymer network, these particles may interact strongly with the polymer environment, leading to new types of scaling behaviors not reported before.Experiments: Using high-speed imaging, we experimentally studied the time evolution of the neck diameter hmin of drops composed of silica nanoparticles dispersed in PEO solution when extruded from a nozzle.Findings: After initial Newtonian necking with hmin ∼ t2/3, the subsequent stage may exhibit scaling variation, characterized by either exponential or power-law decay, depending on the nanoparticle volume fraction ϕ. The exponential decay hmin ∼ e-t/τ signifies the coil-stretch transition in typical viscoelastic suspensions. We conducted an analysis of the power-law scenario hmin ∼ tα at high ϕ, categorizing the entire process into three distinct regimes based on the exponents α. The dependences of critical thicknesses at transition points and exponents on polymer concentration offer initial insights into the potential transition from heterogeneous to homogeneous thinning in the mixture. This novel scaling variation bears implications for accurately predicting and controlling droplet fragmentation in industrial applications.

The influence of low-temperature shear on crystallization of shish-kebab of unimodal/ bimodal C4 short-chain branching polyethylene

The shish crystal, a crucial structure in the formation of shish-kebab crystals, has drawn widespread attention due to its highly oriented chain arrangement, significantly enhancing the mechanical properties of products. This study utilizes polyethylene samples with longer C4 short branches, employing in-situ synchrotron SAXS/WAXD to investigate the influence of short branch length and content on the formation of shish-kebab crystals in unimodal and bimodal polyethylene under low-temperature shear conditions (with supercooling degrees of 13.5 °C and 10.5 °C). The results reveal that the presence of C4 short-chain branching affects system crystallization, but low-temperature shear induces stronger mechanical action on the melt, resulting in better-oriented lamellar crystals compared to high-temperature shear (superheat degrees of 22.5 °C and 19.5 °C), achieving higher orientation. This achieves the control of shish-kebab crystal generation through the stretched network mechanism. Both types of samples with different C4 short-chain branching contents generate shish-kebab crystals during the shear stage, with relatively higher short-chain branching content inducing more shish crystal formation but with less regular structure. The bimodal polyethylene exhibit higher proportions and absolute amounts of final shish crystals compared to the unimodal polyethylene, indicating that high molecular weight portions are conducive to shish crystal generation under shear. The result demonstrates the feasibility of generating shish-kebab crystals during the low-temperature shear process of PE in the presence of C4 branches using the stretched network mechanism under shear flow conditions. Moreover, controlling the degree of entanglement of molecular chains by adjusting shear temperature, branch content, and branch length can modulate two formation mechanisms of shish crystals—coil-stretch transition mechanism at high temperature and stretched network mechanism at low-temperature.

Elastic particle model for coil-stretch transition of dilute polymers in an elongational flow

The phenomenon of the ‘coil-stretch’ (C-S) transition, wherein a long-chain polymer initially in a coiled state undergoes a sudden configuration change to become fully stretched under steady elongational flows, has been widely recognized. This transition can display intricate hysteresis behaviours under specific critical conditions, giving rise to unique rheological characteristics in dilute polymer solutions. Historically, microscopic stochastic models and Brownian dynamics simulations have shed light on the underlying mechanisms of the transition by uncovering bistable configurations of polymer chains. Following the initial work by Cerf (J. Chem. Phys., vol. 20, 1952, pp. 395–402), we introduce a continuum model in this study to investigate the C-S transition in a constant uniaxial elongational flow. Our approach involves approximating the unfolding process of the polymer chain as an axisymmetric deformation of an elastic particle. We make the assumption that the particle possesses uniform material properties, which can be represented by a nonlinear, strain-hardening constitutive equation to replicate the finite extensibility of the polymer chain. Subsequently, we analytically solve for the steady-state deformation using a polarization method. By employing this reduced model, we effectively capture the C-S transition and establish its specific correlations with material and geometric properties. The hysteresis phenomena can be comprehended through a force-balance analysis, which involves comparing the externally applied viscous forces with the intrinsic elastic responsive forces. We demonstrate that our model, while simple, unveils rich elastohydrodynamics of the C-S transition.

Single droplet formation by controlling the viscoelasticity of polymer solutions during inkjet printing

Inkjet printing has emerged as a potential solution processing method for large-area patterned films. During inkjet printing, a single droplet without satellite droplet is required for high-quality film. Herein, we propose a strategy for obtaining a single droplet by adjusting the reduced concentration (c/c*, where c* is the critical overlap concentration) in the range of 1.0–1.5. Droplet formation can be categorized into three distinct regimes: (1) c/c* < 1.0, satellite droplet; (2) c/c* = 1.0–1.5, single droplet; (3) c/c* > 2.0, no droplet. Furthermore, an inertial-capillary balance led to the 2/3-power scaling of the minimum radius with time for the solutions of c/c* < 1.0. However, for the solutions of c/c* = 1.0–1.5, the ligament radius decreased exponentially with time. Moreover, the Weissenberg number was higher than the critical value of 0.5, indicating that the polymer chains underwent coil-stretch transition. The viscoelastic-capillary balance dominated instead of the inertial-capillary balance. The resulting viscoelastic resistance reduced the length of the ligament and increased the velocity difference between the satellite and main droplets. Consequently, a single droplet was formed. In addition, the law can be successfully generalized to various molecular weights, molecular structures and solvents.

Measurement and control of extensional viscosity in barrier coating dispersions

This study aimed to understand the effect of various rheological additives on the extensional viscosity of barrier coating dispersions, as well as to understand the role extensional viscosity plays in stabilizing a liquid curtain. The apparent extensional viscosity was measured using two devices that create accelerating flows: a capillary viscometer and an orifice rheometer. Additives tested include several polyvinyl alcohols, a high molecular weight polyethylene oxide, and carboxymethylcellulose. Extensional viscosity plays a significant role in stabilizing a liquid curtain, as it slows down hole expansion and prevents impurities and disturbances from causing holes in the first place. Some of the additives could substantially increase the extensional viscosity of the dispersions without increasing the shear viscosity outside the typical range of processability for a curtain coater. Some of the additives exhibited coil-stretch transition, meaning they start increasing extensional viscosity above a certain extension rate. Polymers with low chain lengths exhibited finite extensibility, which indicates the polymer chain has fully extended and cannot provide further extensional viscosity, even though the extending force is increased. Polymeric additives with stiff or branched chains significantly raised shear viscosity without increasing extensional viscosity. Both methods could reliably measure extensional viscosity in curtain coating barrier dispersions.

DNA fragmentation in a steady shear flow.

We have determined the susceptibility of T4 DNA (166 kilobase pairs, kbp) to fragmentation under steady shear in a cone-and-plate rheometer. After shearing for at least 30 min at a shear rate of , corresponding to a Reynolds number of and a Weissenberg number of , % of the sample is broken into a polydisperse mixture with a number-averaged molecular weight of kbp and a polydispersity index of , as measured by pulsed-field gel electrophoresis (with a 95% confidence interval). The molecular weight distributions observed here from a shear flow are similar to those produced by a (dominantly extensional) sink flow of DNA and are qualitatively different than the midpoint scission observed in simple extensional flow. Given the inability of shear flow to produce a sharp coil-stretch transition, the data presented here support a model where polymers can be fragmented in flow without complete extension. These results further indicate that DNA fragmentation by shear is unlikely to be a significant issue in microfluidic devices, and anomalous molecular weight observations in experiments are due to DNA processing prior to observation in the device.

Measured energy injection, transfer, and dissipation rates in the bulk of dilute polymeric turbulent flow: The concentration and Weissenberg number effects

We present an experimental study of the effects of polymer additives on the energy cascade in the bulk of turbulent von Kármán swirling flow and its dependences on the polymer concentration (ϕ) and Weissenberg number (Wi). The turbulent flow is viscous, driven by the rotation of a pair of smooth disks. The velocity measurements show that the flow is more coherent in the presence of the polymers. It is found that there is a critical concentration ϕc below which the energy injection, transfer, and dissipation rates are rapidly suppressed to about 20% of the Newtonian case with increasing ϕ (referred to as the rapid suppression regime) and above which the suppression is saturated at that level (referred to as saturated suppression regime). We found that the Wi dependence of the energy cascade resembles that of concentration dependence with one additional regime referred to as no suppression regime when Wi is very small. The sharp transition from the no suppression regime to the rapid suppression regime implies the occurrence of the coil–stretch transition of the polymers. Although the energy dissipation rate is greatly suppressed in the presence of polymers, the functional form of its probability density function is the same as that of the Newtonian case, suggesting that the mechanism governing the energy dissipation is universal for both the Newtonian and dilute polymeric turbulence. Our experiments show for the first time the similarity between the effects of Wi and ϕ in dilute polymeric bulk turbulence.

Elucidating the role of network topology dynamics on the coil-stretch transition hysteresis in extensional flow of entangled polymer melts

Dissipative particle dynamics (DPD) simulations are performed on coarse-grained replicas of linear, monodisperse entangled polyethylene melts C1000H2002 and C3000H6002 undergoing both steady-state and transient planar elongational flow (PEF). The fidelity of the DPD simulations is verified by direct comparison of flow topological and rheological properties of a 334-particle chain liquid against the united-atom C1000H2002 liquid, simulated using nonequilibrium molecular dynamics (NEMD). These DPD simulations demonstrate that a flow-induced coil-stretch transition (CST) and its associated hysteresis caused by configurational microphase separation, as observed in previous NEMD simulations of PEF, can be replicated using a more computationally efficient coarse-grained system. Results indicate that the breadth of the CST hysteresis loop is enlarged for the longer molecule liquid relative to the shorter one. Furthermore, relaxation simulations reveal that reducing the applied flow Deborah number (De) from a high value corresponding to a homogeneous phase of highly stretched molecules to a De within the biphasic region results in a two-stage relaxation process. There is a fast initial stratification of the kinetically trapped highly stretched chains into regions of highly extended and less extended chains, which displays similar behavior to a system undergoing a spinodal decomposition caused by spatial configurational free energy fluctuations. After a short induction period of apparently random duration, the less extended chain regions experience a stochastic nucleation event that induces configurational relaxation to domains composed of coiled molecules over a much longer time scale, leaving the more highly extended chains in surrounding sheetlike domains. The time scales of these two relaxation processes are of the same order of magnitude as the Rouse and disengagement times of the equilibrium liquids.

Self-induced hydrodynamic coil-stretch transition of active polymers.

We analyze the conformational dynamics and statistical properties of an active polymer model. The polymer is described as a freely jointed bead-rod chain subject to stochastic active force dipoles that act on the suspending solvent where they drive long-ranged fluid flows. Using Langevin simulations of isolated chains in unconfined domains, we show how the coupling of active flows with polymer conformations leads to emergent dynamics. Systems with contractile dipoles behave similarly to passive Brownian chains with enhanced fluctuations due to dipolar flows. In systems with extensile dipoles, however, our simulations uncover an active coil-stretch transition whereby the polymer spontaneously unfolds and stretches out in its own self-induced hydrodynamic flow, and we characterize this transition in terms of a dimensionless activity parameter comparing active dipolar forces to thermal fluctuations. We discuss our findings in the context of the classic coil-stretch transition of passive polymers in extensional flows and complement our simulations with a simple kinetic model for an active trimer.

Crooks Fluctuation Theorem for Single Polymer Dynamics in Time-Dependent Flows: Understanding Viscoelastic Hysteresis.

Nonequilibrium work relations have fundamentally advanced our understanding of molecular processes. In recent years, fluctuation theorems have been extensively applied to understand transitions between equilibrium steady-states, commonly described by simple control parameters such as molecular extension of a protein or polymer chain stretched by an external force in a quiescent fluid. Despite recent progress, far less is understood regarding the application of fluctuation theorems to processes involving nonequilibrium steady-states such as those described by polymer stretching dynamics in nonequilibrium fluid flows. In this work, we apply the Crooks fluctuation theorem to understand the nonequilibrium thermodynamics of dilute polymer solutions in flow. We directly determine the nonequilibrium free energy for single polymer molecules in flow using a combination of single molecule experiments and Brownian dynamics simulations. We further develop a time-dependent extensional flow protocol that allows for probing viscoelastic hysteresis over a wide range of flow strengths. Using this framework, we define quantities that uniquely characterize the coil-stretch transition for polymer chains in flow. Overall, generalized fluctuation theorems provide a powerful framework to understand polymer dynamics under far-from-equilibrium conditions.

Tollmien-Schlichting route to elastoinertial turbulence in channel flow

Direct simulations of two-dimensional channel flow of a viscoelastic fluid have revealed the existence of a family of Tollmien-Schlichting (TS) attractors that is nonlinearly self-sustained by viscoelasticity [Shekar et al., J.Fluid Mech. 893, A3 (2020)]. Here, we describe the evolution of this branch in parameter space and its connections to the Newtonian TS attractor and to elastoinertial turbulence (EIT). At Reynolds number $Re=3000$, there is a solution branch with TS-wave structure but which is not connected to the Newtonian solution branch. At fixed Weissenberg number, $Wi$ and increasing Reynolds number from 3000-10000, this attractor goes from displaying a striation of weak polymer stretch localized at the critical layer to an extended sheet of very large polymer stretch. We show that this transition is directly tied to the strength of the TS critical layer fluctuations and can be attributed to a coil-stretch transition when the local Weissenberg number at the hyperbolic stagnation point of the Kelvin cat's eye structure of the TS wave exceeds $\frac{1}{2}$. At $Re=10000$, unlike $3000$, the Newtonian TS attractor evolves continuously into the EIT state as $Wi$ is increased from zero to about $13$. We describe how the structure of the flow and stress fields changes, highlighting in particular a "sheet-shedding" process by which the individual sheets associated with the critical layer structure break up to form the layered multisheet structure characteristic of EIT.

Shear-induced crystallization of unimodal/bimodal polyethylene at high temperatures affected by C4 short-branching

Generation of shish-kebab crystals of unimodal and bimodal polyethylenes (PEs) with different C4 short-branching sheared at high overmelt degree (the increment higher than the crystallization temperature) and a low shearing rate was investigated by using in-situ SAXS/WAXD techniques. Both PEs exhibiting SAXS streaks during the shearing indicate the shish crystals generated by shearing. The WAXD analysis indicates an orientation degree as high as 0.75–0.95 for both PEs. During the succeeding isothermal crystallization at 121 °C, kebab crystals and partially oriented stacked crystals in lamellae form, which elevate the crystallinity index but decrease the orientation degree. The shish crystal content of bimodal PE is higher than that of unimodal PE because the high molecular weight (MW) part facilitates the generation of shish crystals. The increase of the branching length is more favorable for the generation of shish crystals (comparing with C2). However, the high content of C4 short-chain branching makes shear-induced crystals less ordered. The generation mechanism of both PEs' shish crystals at high shearing temperatures can be ascribed to the coil-stretch transition (CST) mechanism.

Pinch-off of viscoelastic particulate suspensions

The formation of drops of a complex fluid, for instance including dissolved polymers and/or solid particles, has practical implications in several industrial and biophysical processes. In this Letter, we experimentally investigate the generation of drops of a viscoelastic suspension, made of non-Brownian spherical particles dispersed in a dilute polymer solution. Using high-speed imaging, we characterize the different stages of the detachment. Our experiments show that the particles primarily affect the initial Newtonian necking by increasing the fluid viscosity. In the viscoelastic regime, particles do not affect the thinning until the onset of the blistering instability, which they accelerate. We find that the transition from one regime to another, which corresponds to the coil-stretch transition of the polymer chains, strongly depends on the particle content. Considering that the presence of rigid particles increase the deformation of the liquid phase, we propose an expression for the local strain rate in the suspension, which rationalizes our experimental results. This method could enable the precise measurement of local stresses in particulate suspensions.

The Influence of Ethyl Branch on Formation of Shish-Kebab Crystals in Bimodal Polyethylene under Shear at Low Temperature

Formation of shish-kebab crystals using a bimodal polyethylene system containing high molecular weight (HMW) component with different ethyl branch contents was investigated. In situ small-angle X-ray scattering (SAXS) and wide-angle X-ray diffraction (WAXD) techniques were used to monitor the formation and evolution of shish-kebab structure sheared at low temperature in simple shear mode and low rate. Only the bimodal PE with no branch formed shish-kebab crystals at the shear temperature of 129 °C, and the shish length increased with the crystallization time, while bimodal PE with branch has no observable shish under the same conditions. The degree of crystallization for bimodal PE with no branch increased with time up to above 7%, while those with ethyl branch increased continually up to above 23%. Furthermore, bimodal PE’s Hermans orientation factor with no branch increased to 0.60, while those with ethyl branch only increased to a value below 0.15. This study indicated that the shish-kebab crystal formed at the low temperature of 129 °C is due to the stretch of entangled chains under shear for the bimodal PE with no branch. Only partly oriented lamellar crystals were formed for the bimodal PE with ethyl branch. All the results at the shear temperatures higher, closed to, and lower than the melting point, the modulation of shish crystals formation owing to different mechanisms of the coil-stretch transition and the stretched network by changing shear temperature was achieved in the bimodal PE samples.

Entropic characterization of the coil-stretch transition of polymers in random flows.

Polymer molecules in a flow undergo a coil-stretch phase transition on an increase of the velocity gradients. Model-independent identification and characterization of the transition in a random flow has been lacking so far. Here we suggest to use the entropy of the extension statistics as a proper measure due to strong fluctuations around the transition. We measure experimentally the entropy as a function of the local Weisenberg number and show that it has a maximum, which identifies and quantifies the transition. We compare the new approach with the traditional one based on the theory using either linear Oldroyd-B or nonlinear finite extensible nonlinear elastic polymer models.

Coil-stretch-like transition of elastic sheets in extensional flows.

The conformation of a long linear polymer dissolved in fluid and exposed to an extensional flow is well-known to exhibit a "coil-stretch" transition, which for sufficiently long chains can lead to bistability. The present work reports computations indicating that an analogous "compact-stretched" transition arises in the dynamics of a thin elastic sheet. Sheets of nominally circular, square or rectangular shape are simulated in planar and biaxial flows using a finite element method for the sheet conformations and a regularized Stokeslet method for the fluid flow. If a neo-Hookean constitutive model is used for the sheet elasticity, the sheets will stretch without bound once a critical extension rate, as characterized nondimensionally by a capillary number, is exceeded. Nonlinear elasticity, represented with the Yeoh model, arrests the stretching, leading to a highly-stretched steady state once the critical capillary number is exceeded. For all shapes and in both planar and biaxial extension, a parameter regime exists in which both weakly stretched (compact) and strongly stretched states can be found, depending on initial conditions. I.e. this parameter regime displays bistability. As in the long-chain polymer case, the bistable behavior arises from the hydrodynamic interaction between distant elements of the sheet, and vanishes if these interactions are artificially screened by use of a Brinkman model for the fluid motion. While the sheets can transiently display wrinkled shapes, all final shapes in planar and biaxial extension are planar.

Effect of internal friction on the coil-stretch transition in turbulent flows.

A polymer in a turbulent flow undergoes the coil-stretch transition when the Weissenberg number, i.e. the product of the Lyapunov exponent of the flow and the relaxation time of the polymer, surpasses a critical value. The effect of internal friction on the transition is studied by means of Brownian dynamics simulations of the elastic dumbbell model in a homogeneous and isotropic, incompressible, turbulent flow and analytical calculations for a stochastic velocity gradient. The results are explained by adapting the large deviations theory of Balkovsky et al. [Phys. Rev. Lett., 2000, 84, 4765] to an elastic dumbbell with internal viscosity. In turbulent flows, a distinctive feature of the probability distribution of polymer extensions is its power-law behaviour for extensions greater than the equilibrium length and smaller than the contour length. It is shown that although internal friction does not modify the critical Weissenberg number for the coil-stretch transition, it makes the slope of the probability distribution of the extension steeper, thus rendering the transition sharper. Internal friction therefore provides a possible explanation for the steepness of the distribution of polymer extensions observed in experiments at large Weissenberg numbers.

Molecular and thermodynamics descriptions of flow-induced crystallization in semi-crystalline polymers

The flow-induced crystallization (FIC) is commonly encountered in the polymer industry as more than 70% of commercial polymers are crystalline, which needs to be processed before the final application. The complicated external flow field, i.e., shear or extensional flow, results in a great challenge in understanding the FIC phenomenon from both general thermodynamics and detailed molecular level aspects. The current tutorial first describes the general phenomenon of FIC from the aspect of different morphologies and enhanced kinetics induced by the flow. Second, characterization methods for monitoring FIC are introduced. Here, the in situ synchrotron x-ray scattering and non-equilibrium molecular dynamics simulation are selected as typical examples. Then, the theoretical descriptions of FIC are summarized from the aspects of molecular origin and thermodynamics. The coil–stretch transition theory and later developed stretched network theory are highlighted, where the former mainly accounts for polymer dilute solution and the latter for highly entangled cases. Also, multi-step features for the formation of various intermediate states during flow-induced nucleation are depicted. Despite non-equilibrium nature, the FIC can still be treated by thermodynamics, especially under weak flow conditions. The classic entropy reduction model is introduced together with later modifications. In all, understanding the fundamental mechanism of FIC is crucial for optimizing external processing parameters and internal molecular characteristics, and useful to guide current or further applied techniques.

Deformation Response of Catenated DNA Networks in a Planar Elongational Field.

A kinetoplast is a complex catenated DNA network that bears resemblance to a two-dimensional polymeric system. In this work, we use single-molecule experiments to study the transient and steady-state deformation of kinetoplasts in a planar elongational field. We demonstrate that kinetoplasts deform in a stagewise manner and undergo transient deformation at large strains, due to conformational rearrangements from an intermediate metastable state. Kinetoplasts in an elongational field achieve a steady-state deformation that depends on strain rate, akin to the deformation of linear polymers. We do not observe an abrupt transition between the nondeformed and deformed states of a kinetoplast, in contrast to the coil-stretch transition for a linear polymer.

Flexibility, Extensibility, and Ratio of Kuhn Length to Packing Length Govern the Pinching Dynamics, Coil-Stretch Transition, and Rheology of Polymer Solutions

We elucidate the influence of chemical structure on macromolecular hydrodynamics, rheological response, and pinching dynamics underlying drop formation/liquid transfer using polyethylene oxide (PEO...