Transient and Average Unsteady Dynamics of Single Polymers in Large-Amplitude Oscillatory Extension

Oscillatory rheometry has been widely used in bulk rheological measurements of complex fluids such as polymer solutions and melts. Despite recent progress on bulk oscillatory rheology, however, the vast majority of single polymer studies has focused on chain dynamics in simple on/ off step strain-rate experiments. In order to fully understand dynamic polymer microstructure and to establish connections with bulk rheology, there is a clear need to study the dynamics of single polymers in more realistic, nonidealized model flows with transient forcing functions. In this work, we study the dynamics of single polymers in large amplitude oscillatory extensional (LAOE) flow using experiments and Brownian dynamics (BD) simulations, and we characterize transient polymer stretch, orientation angle, and average unsteady stretch as functions of the flow strength (Weissenberg number, Wi) and probing frequency (Deborah number, De). Small and large amplitude sinusoidal oscillatory extensional flow are generated in a cross-slot microfluidic geometry, which is facilitated by using an automated flow device called the Stokes trap. This approach allows the conformational dynamics of single DNA molecules to be observed in oscillatory extensional flow for long times. In this way, we observe a characteristic periodic motion of polymers in LAOE including compression, rotation, and stretching between the timedependent principal axes of extension and compression. Interestingly, distinct polymer conformations are observed in LAOE that appear to be analogous to buckling instabilities for rigid or semiflexible filaments under compression. Average unsteady polymer extension is further characterized for single polymers in oscillatory extension across a wide range ofWi and De. In the limit of low Wi, average polymer stretch is interpreted using analytical results based on a Hookean dumbbell model, which can be used to define a critical Wi at the linear to nonlinear transition in oscillatory extension. These results reveal the existence of a master curve for average polymer stretch when plotted as a function of an effective Weissenberg number Wieff. Experimental results are compared to BD simulations, and we observe good agreement between simulations and experiments for transient and average unsteady dynamics. Finally, average transient dynamics in oscillatory extensional flow are further interpreted in the context of Pipkin space, defined by the two-dimensional space described by Wi and De. ■ INTRODUCTION Dynamic oscillatory rheometry has been widely used to investigate the rheological behavior of complex fluids such as polymer solutions, polymer blends, entangled polymer melts, block copolymers, and colloidal suspensions. Oscillatory shear is one of the most common bulk rheology measurements used to interrogate soft materials and complex fluids. In particular, small amplitude oscillatory shear (SAOS) is a classic method to probe the linear viscoelastic properties of materials based on established physical principles in the limit of small deformation. Using this approach, materials are interrogated under small magnitude sinusoidal strains, and the time-dependent stress response is measured. Linear viscoelastic measurements can be used to determine material functions such as the elastic modulus G′(ω) and the loss modulus G′′(ω), where both linear viscoelastic moduli are independent of the applied strain, provided that the strain amplitude is small. Although SAOS is a convenient and useful method to probe rheological behavior, it is mainly used to study only the linear response properties of materials. In most practical applications involved in processing, however, materials are exposed to highly nonlinear and nonequilibrium flow conditions, thereby inducing polymer deformation under large strains and strain rates during extrusion or related processes. Moreover, vastly different complex fluids may yield deceivingly similar linear viscoelastic signatures because small magnitude strains are insufficient to distinguish fine-scale structural differences at the nanoto microscale. Indeed, for many complex materials, the distinguishing rheological signatures may be buried in the nonlinear stress response, despite showing apparently similar linear viscoelastic responses. From this perspective, there is a strong need to characterize the nonlinear rheological properties of complex materials. In recent years, large amplitude oscillatory shear (LAOS) has been widely employed to investigate the nonlinear rheological properites of materials. LAOS can be implemented using fairly Received: July 26, 2016 Revised: September 16, 2016 Published: October 5, 2016 Article pubs.acs.org/Macromolecules © 2016 American Chemical Society 8018 DOI: 10.1021/acs.macromol.6b01606 Macromolecules 2016, 49, 8018−8030 standard experimental tools including a sliding plate rheometer (SPR) or Fourier transform (FT) rheology based on a commercial rheometer. LAOS is generally performed over a wide range of strain amplitude and frequency, thereby revealing the nonlinear rheological response of materials. In addition, LAOS is performed as a smooth and continuous process without sudden or discontinuous jumps in applied strain or strain rate, which differs from several other nonlinear transient rheological measurements such as step strain experiments. In this way, LAOS can be implemented to effectively achieve robust control over the strain input with reduced experimental noise. However, the challenge lies in interpreting the stress response obtained from LAOS experiments because the transient stress cannot generally be described using linear viscoelastic theories due to the presence of higher order harmonics or strong nonlinearities. To address this issue, Wilhelm et al. employed the Fourier transform (FT) method to analyze the shear stress response as spectra in Fourier space. Ewoldt et al. used a set of first kind orthogonal Chebyshev polynomials to characterize the nonlinear stress response, and this method was based on a nonlinear stress decomposition (SD) method. In this way, certain complex fluids can be uniquely “fingerprinted”. Rogers et al. described the nonlinear stress response as purely elastic to purely viscous sequences of processes to extract fundamental physical interpretations of dynamics. In addition to experimental work, Radhakrishnan and Underhill studied the LAOS response of a dilute polymer−nanoparticle mixture with short-ranged attractive interactions using Brownian dynamics (BD) simulations. A frequency sweep was performed with a fixed peak strain rate above the critical polymer globule−stretch transition strain rate, and transitions between different conformational states were identified. In addition to shear rheology, extensional rheology has also been used to study the nonlinear properties of complex fluids. Extensional flow is considered a strong flow that is very efficient at unraveling flexible macromolecules or aligning semiflexible biofilaments. In recent years, large amplitude oscillatory extension (LAOE) has been developed for bulk-level studies of materials. Using this approach, Rasmussen et al. and Bejenariu et al. examined the soft elasticity of polystyrene (PS) melts and cross-linked polydimethylsiloxane (PDMS) networks using a customized filament stretching rheometer (FSR). A unique periodic response for the elongational stress τxx − τrr was measured in LAOE that is analogous to the periodic response of shear stress τxy in LAOS. However, the application of bulk extensional rheometry can be quite challenging for some materials due to gravitational effects for high viscosity samples, which could limit the range of strain rates that can be explored. These problems can be mitigated by using microfluidic cross-slot devices to generate high strain rate planar extensional flow. Odell and Carrington first developed the extensional flow oscillatory rheometer (EFOR) using a cross-slot microfluidic geometry and four electronically driven piezoelectric pumps. Here, flow-induced conformational changes to macromolecules are detected using optical birefringence. Using this approach, Haward and co-workers measured the extensional viscosity of dilute polymer solutions from the transient birefringence response in flow. Recently, the geometry of the microfluidic cross-slot device for EFOR was optimized to achieve precise extensional flow profiles across the entire cross-slot region for a wide range of strain rates. Bulk rheological techniques are essential in measuring macroscopic stress in complex materials. Nevertheless, these methods can only be used to indirectly infer flow-induced deformation at the molecular scale. Single molecule techniques, on the other hand, allow for the direct observation of dynamic microstructure in nonequilibrium flow. Recent advances in single molecule fluorescence imaging have enabled the direct observation of single polymer dynamics in simple model flows such as shear flow, planar extensional flow, and linear mixed flows. However, the vast majority of single polymer studies has only focused on chain dynamics using an idealized on/off flow rate conditions or simple step forcing functions. In many processing applications, polymers experience complex time-dependent flows such as oscillatory extension for flow through porous media or in wavy-wall channels. For these reasons, there is a general need to understand the role of transient oscillatory flows on the dynamics of polymers and to establish a connection between single polymer dynamics and bulk oscillatory rheometry. In this work, we use a single polymer approach to directly probe the dynamics of DNA molecules in oscillatory extension using precisely controlled flows. Recently, Zhou and Schroeder reported the use of single polymer LAOE to construct molecular stretch−strain rate curves, which are defined as single molecule Lissajou curves. The shapes of the single molecule Lissajou curves were interpreted in the context of polymer chain conformation over a wide range of flow strength (Weissenberg number, Wi) and probing frequency (Deborah number, De). In the present article, we extend beyond this prior work by characterizing the transient stretch and transient orientation angle for single polymers in LAOE, and these quantities are analyzed using autocorrelation and cross-correlation functions to probe the underlying physics. In addition, we further study average polymer extension in LAOE, and our results reveal a critical flow strength Wi0,crit at which a linear to nonlinear transition in stretching behavior is observed. Moreover, we interpret average stretch results using an analytical model based on a Hookean dumbbell in time-dependent oscillatory extensional flow. Finally, transient polymer conformations are characterized in the context of the two-dimensional space defined byWi and De, which is generally known as Pipkin space. Taken together, these results provide new insights into the dynamics of single polymer chains in controlled time-dependent flows. ■ EXPERIMENTAL METHODS Materials. In this work, we study the dynamics of double-stranded λ-phage DNA (48.5 kbp, New England Biolabs) in oscillatory extension. λ-DNA is fluorescently labeled with an intercalating dye (YOYO-1, Thermo Fisher, Molecular Probes) with a dye-to-base pair ratio of 1:4 for >1 h in the dark at room temperature. DNA is labeled in an aqueous buffer containing 30 mM Tris/Tris-HCl (pH 8.0), 2 mM EDTA, and 5 mM NaCl. Fluorescently labeled λ-DNA is then added to an imaging buffer containing 30 mM Tris/Tris-HCl (pH 8.0), 2 mM EDTA, 5 mM NaCl, sucrose (60% w/w), glucose (5 mg/ mL), glucose oxidase (0.05 mg/mL), catalase (0.01 mg/mL), and 4% v/v β-mercaptoethanol. Sucrose is used to increase the solvent viscosity of the imaging buffer solution to 48.5 ± 1 cP at 22.5 °C, which was measured using a cone and plate viscometer (Brookfield). Glucose oxidase/catalase is used as a coupled enzymatic oxygen scavenging system to suppress photobleaching and photocleaving of fluorescently labeled DNA. The concentration of labeled DNA in the imaging buffer is ∼10−5c*, where c* is the polymer overlap concentration, thereby yielding an ultradilute solution in the absence Macromolecules Article DOI: 10.1021/acs.macromol.6b01606 Macromolecules 2016, 49, 8018−8030 8019 of polymer−polymer interactions. The contour length of fluorescently labeled λ-DNA is taken to be 21.5 μm under these conditions. Optics and Imaging. Imaging is performed using an inverted epifluorescence microscope (IX71, Olympus) coupled to an electronmultiplying charge coupled device (EMCCD) camera (iXon, Andor Technology). Labeled DNA samples are illuminated using a 100 W mercury arc lamp (USH102D, UShio) directed through a 3% neutral density filter (Olympus), a 482 ± 18 nm band-pass excitation filter (FF01-482/18-25, Semrock), and a 488 nm single-edge dichroic mirror (Di01-R488-25×36, Semrock). Fluorescence emission is collected by a 1.45 NA, 100× oil immersion objective lens (UPlanSApo, Olympus), and a 488 nm long pass filter (BLP01488R-25, Semrock) is used in the detection path. Finally, images are acquired by an Andor iXon EMCCD camera (512 × 512 pixels, 16 μm pixel size) under frame transfer mode at a frame rate of 30 Hz. Microfluidic Devices. Standard techniques in soft lithography are used to fabricate single layer microfluidic devices from polydimethylsiloxane (PDMS) with a base-to-cross-linker ratio of 5:1. PDMS devices are bonded to a glass coverslip following oxygen plasma cleaning. The PDMS/glass hybrid microfluidic device is composed of two orthogonal microfluidic channels to form a cross-slot geometry. The inlet/outlet channels are 10 mm long, 400 μm wide, and 90 μm in depth, and each channel terminates at an inlet port with a diameter of 1.6 mm. DNA imaging is performed in the region near the center of the cross-slot (imaging area approximately 80 μm × 80 μm in size). Pressure-driven flow is used to generate a planar extensional flow in the cross-slot geometry, as described below. Particle Tracking Velocimetry. The flow field is characterized prior to single polymer experiments using particle tracking velocimetry (PTV). Using this approach, fluorescent polystyrene beads (0.84 μm, Spherotech) are added to the imaging buffer and introduced into microfluidic devices. Imaging for bead tracking is performed using a standard CCD camera (Grasshopper3, Point Gray, 1920 × 1200 pixels) with an exposure time of 33 ms. Particle trajectories are determined using a particle tracker algorithm and ImageJ software. Using this analysis framework, bead positions are determined (x, y) and corresponding bead velocities (vx, vy) are calculated. Local analysis is used to determine the xand y-direction extensional strain rates (−ε̇0, ε̇0) by least-squares minimization of