High Wi Research Articles

Shear-banding and superdiffusivity in entangled polymer solutions.

Using high-resolution confocal rheometry, we study the shear profiles of well-entangled DNA solutions under large-amplitude oscillatory shear in a rectilinear planar shear cell. With increasing Weissenberg number (Wi), we observe successive transitions from normal Newtonian linear shear profiles to wall-slip dominant shear profiles and, finally, to shear-banding profiles at high Wi. To investigate the microscopic origin of the observed shear banding, we study the dynamics of micron-sized tracers embedded in DNA solutions. Surprisingly, tracer particles in the shear frame exhibit transient superdiffusivity and strong dynamic heterogeneity. The probability distribution functions of particle displacements follow a power-law scaling at large displacements, indicating a Lévy-walk-type motion, reminiscent of tracer dynamics in entangled wormlike micelle solutions and sheared colloidal glasses. We further characterize the length and time scales associated with the abnormal dynamics of tracer particles. We hypothesize that the unusual particle dynamics arise from localized shear-induced chain disentanglement.

Numerical Simulation of Heat Transfer Process of Viscoelastic Fluid Flow at High Weissenberg Number by Log-Conformation Reformulation

Viscoelastic fluids are now becoming promising candidates of microheat exchangers’ working medium due to the occurrence of elastic instability and turbulence at microscale. This paper developed a sound solver for the heat transfer process of viscoelastic fluid flow at high Wi, and this solver can be used to design the multiple heat exchangers with viscoelastic fluids as working medium. The solver validation was conducted by simulating four fundamental benchmarks to assure the reliability of the established solver. After that, the solver was adopted to study the heat transfer process of viscoelastic fluid flow in a curvilinear channel, where apparent heat transfer enhancement (HTE) by viscoelastic fluid was achieved. The observed heat transfer enhancement was attributed to the occurrence of elastic turbulence which continuously mix the hot and cold fluids by the twisting and wiggling flow motions.

Improved convergence based on linear and non-linear transformations at low and high Weissenberg asymptotic analysis

We investigate the accuracy and efficiency of linear and non-linear transformations on the truncated series solutions obtained by asymptotic methods for simple shear and uniaxial elongational viscoelastic flows. The Upper Convected Maxwell, Oldroyd-B, exponential and linear Phan-Thien and Tanner, Giesekus, and FENE-P differential constitutive models are used. The major dimensionless quantity of interest is the Weissenberg number (Wi) in terms of which the unknown dependent flow variables are expanded suitably in series form. The standard regular perturbation method is performed at the Newtonian limit (zero Weissenberg number), while regular or singular perturbation techniques are required to perform the analysis at the purely elastic limit (infinite Weissenberg number). It is shown that the asymptotic approximations at low Wi converge very slowly and have limited radius of convergence, while at high Wi the asymptotic approximations are very accurate even for Wi close to unity. Further improved convergence is achieved by applying a variety of techniques on the original asymptotic solutions, such as Euler's linear transform, Shanks’ non-linear transform, and Padé-type approximants. It is found that the accuracy and the radius of convergence of the transformed solutions increase substantially compared to the original ones. By combining the transformed solutions from both limits, excellent agreement with the exact solutions is demonstrated over the entire range of the Weissenberg number, from zero to infinity, except for a very narrow region close to Wi= 1/2 under uniaxial elongational flow. The results show that, contrary to intuition, viscoelastic phenomena at Wi=O(1) can be predicted accurately using non-linear analysis.

Non-linear dynamics of semi-dilute PAAm solutions in a microfluidic 3D cross-slot flow geometry

We have characterised the flow of semi-dilute PAAm solutions (3.3c* ≤ c ≤ 16.6c*) through a three-dimensional cross-slot device (width to depth aspect ratio 2: 1) over a range of Weissenberg (0.13 ≤ Wi ≤ 88.5) and Elasticity (1.5 ≤ El ≤ 58.4) numbers by means of micro-particle imaging velocimetry (μ-PIV) and pressure drop measurements. The flow regimes are mapped into a Wi−Re diagram. At intermediate Wi (12 ≤ Wi ≤ 45) we observe the formation of lip vortices in the inlet channels. At higher Wi (Wi ≥ 45) we observe the onset of an asymmetric flow regime, which occurs at a much higher Weissenberg number than previously reported in the literature. Increasing El in the asymmetric flow regime results in flow patterns that are markedly more time-dependent and unstable. The extensional rates measured along the centreline of the outlet channels are much lower than the corresponding nominal values. By means of an improved version of the numerical method that was used in our previous works, we have linked the measured velocity field with the “local Deborah number” field to quantify the spatial distribution of the deformation rate underwent by the polymer chains at high Wi flows near the stagnation point of the cross slot device. We measured the extensional viscosity and Trouton ratios of our polymer solutions over a range of 1 ≤ Wi ≤ 30. This work is of significance for the development of extensional rheometers for measuring the properties of low-viscosity fluids such as inks, blood, silk fibroin and protein solutions.

Numerical simulation of heat transfer enhancement by elastic turbulence in a curvy channel

Although many investigations on elastic turbulence have been conducted in recent years, two major research topics still call for in-depth mechanistic investigations. Specifically, one is heat transfer performance affected by elastic turbulence; the other is so-called high Weissenberg number problem (HWNP) in numerical simulation of viscoelastic fluid flow. Taking these two topics into account simultaneously, the coupled problem becomes heat transfer characteristic of viscoelastic fluid in elastic turbulence at high Weissenberg number (Wi) and very low Reynolds number (Re). In this work, we implement numerical simulations by embedding log-conformation reformulation algorithm into the open-source software OpenFOAM. The heat transfer process of viscoelastic fluid flow in a three-dimensional (3D) curvy channel is simulated over a wide range of Wi. For the first time, significant heat transfer enhancement induced by elastic turbulence in a curvy channel at high Wi was identified numerically. When Wi is above the critical value of O(1), the heat transfer performance is found to be dramatically improved by elastic turbulence and then approaches a saturation. From the transient analysis of flow motions in the axial and cross sections, it can be seen that the flow twists and wiggles in the curvy channel and the field synergy effect of viscoelastic fluid flow becomes more intensive than that of Newtonian fluid flow. These effects give rise to the extremely irregular flow motions in the cross section and consequently lead to heat transfer enhancement.

Convoluted models and high-Weissenberg predictions for micellar thixotropic fluids in contraction–expansion flows

This study is concerned with finite element/volume modelling of contraction–expansion axisymmetric pipe flows for thixotropic and non-thixotropic viscoelastic models. To obtain solutions at high Weissenberg numbers (Wi) under a general differential form Wiτp∇=2(1−β)D−fτp, both thixotropic Bautista–Manero micellar and non-thixotropic EPTT f-functionals have been investigated. Here, three key modifications have been implemented: first, that of convoluting EPTT and micellar Bautista–Manero f-functionals, either in a multiplicative (Conv*) or additive (Conv+) form; second, by adopting f-functionals in absolute form (ABS-f-correction); and third, by imposing pure uniaxial-extension velocity-gradient components at the pure-stretch flow-centreline (VGR-correction). With this combination of strategies, highly non-linear solutions have been obtained to impressively high Wi [=O(5000+)].This capability permits analysis of industrial applications, typically displaying non-linear features such as thixotropy, yield stress and shear banding. The scope of applications covers enhanced oil-recovery, industrial processing of plastics and foods, as well as in biological and microfluidic flows. The impact of rheological properties across convoluted models (moderate-hardening, shear-thinning) has been observed through steady-state solutions and their excess pressure-drop (epd) production, stress, f-functional field structure, and vortex dynamics. Three phases of vortex-behaviour have been observed with rise in elasticity, along with upstream–downstream Moffatt vortices and plateauing epd-behaviour at high-Wi levels. Moreover, enhancement of positive-definiteness in stress has improved high-Wi solution attenuation.

Stability of fibre yield and its component traits in tossa jute (Corchorus olitorius L.)

Stability study for fibre yield and its attributing traits was carried out in six tossa jute ( Corchorus olitorius L.) genotypes including two checks under four different ecological conditions of India over two consecutive years 2005 and 2006, I under All India Network Project on Jute and Allied Fibres. Six experimental entries in AVT-I and AVT-II responded differentially to the different environments as evidenced by their non significant difference for the different traits in the different environments. On the basis of the stability parameters Wi (Wricke’s Ecovalence) and Di (Hanson’s genotypic stability) the genotype JRO-524 (standard check) was identified as stable for fibre yield and its attributing traits over the environments. It was followed by the genotype CO-56 with high fibre yield and moderate Wi (%) and Di values. These two genotypes were identified to be suitable for high input conditions of cultivation.

Startup shear of a highly entangled polystyrene solution deep into the nonlinear viscoelastic regime

We have employed optical particle tracking velocimetry to directly visualize the deformation field of a highly entangled polystyrene/diethyl phthalate (PS/DEP) solution (entanglement density Z = 61) in a parallel plate geometry. To probe the existence of shear banding, startup shear tests were carried out deep into the nonlinear viscoelastic (NLVE) regime. By roughening the surface of the rheometer plates, wall slip was effectively suppressed. In the presence of edge fracture and prior to it, steady-state shear banding was not seen as evidenced by velocity profiles that were linear for most of the shear tests. An exception was for an extremely high Weissenberg number Wi = 255, for which weak transient banding was observed. For the PS/DEP system studied, the results suggest that in the absence of wall slip, shear banding is not a steady-state phenomenon and weak shear banding is observed only in the transient state and this could possibly correlate with the edge fracture effects.

Enhanced dispersion by elastic turbulence in porous media

We experimentally investigate hydrodynamic dispersion in elastic turbulent flows of a semi-dilute aqueous polymer solution within a periodic porous structure at ultra-low Reynolds numbers by particle tracking velocimetry. Our results indicate that elastic turbulence can be characterized by an effective dispersion coefficient which exceeds that of Newtonian liquids by several orders of magnitude and grows non-linearly with the Weissenberg number Wi. Contrary to laminar flow conditions, the velocity field, and thus the shear rate, is not proportional to the flow rate and becomes asymmetric at high Wi.

Time–Weissenberg number superposition in planar contraction microchannel flows

The flow of viscoelastic fluids in a micro-fabricated 4:1 planar contraction channel was studied through flow visualization. As the Weissenberg number (Wi) increased, the flow developed from a Newtonian-like to vortex growth, and the transient start-up flow at high Wi was found to experience all the steady patterns at lower Wi flows. The flow sequence was different depending on the fluids and channel dimensions, however, in all the cases we could reach, the steady patterns at a low Wi flow could be matched 1:1 with the transient patterns at a high Wi flow. The plot of Wi and time when the two sets (transient and steady) were matched showed a clear functional relationship, from which the time–Weissenberg number superposition could be confirmed.

Viscoelastic instabilities in micro-scale flows

Many artificial and natural fluids contain macromolecules, particles or droplets that impart complex flow behavior to the fluid. This complex behavior results in a non-linear relationship between stress and deformation standing in between Newton’s law of viscosity for an ideal viscous liquid and Hooke’s law for an ideal elastic material. Such non-linear viscoelastic behavior breaks down flow reversibility under creeping flow conditions, as encountered at the micro-scale, and can lead to flow instabilities. These instabilities offer an alternative to the development of systems requiring unstable flows under conditions where chaotic advection is unfeasible. Flows of viscoelastic fluids are characterized by the Weissenberg (Wi) and Reynolds (Re) numbers, and at the micro-scale flow instabilities occur in regions in the Wi–Re space typically unreachable at the macro-scale, namely high Wi and low Re. In this paper, we review recent experimental work by the authors on the topic of elastic instabilities in flows having a strong extensional component, including: flow through a hyperbolic contraction followed by a sudden expansion; flow in a microfluidic diode and in a flow focusing device; flow around a confined cylinder; flow through porous media and simplified porous media analogs. These flows exhibit different types of flow transitions depending on geometry, Wi and Re, including: transition from a steady symmetric to a steady asymmetric flow, often followed by a second transition to unsteady flow at high Wi; direct transition between steady symmetric and unsteady flows.

Physical Activity Guideline in Mexican-Americans: Does the Built Environment Play a Role?

Given disproportionate burden of physical inactivity among US Hispanics and emerging interests in the potential role of the built environment on physical activity, we tested the hypothesis that residing in a more walkable block group is associated with increased physical activity in a cohort of Mexican-American adults. 10,183 Mexican-American adults from Houston, TX, USA were studied. Physical activity was assessed through self-report. Geographical information systems were used to create a "walkability index" (WI). We examined the relationship between WI and physical activity using regression models. Findings for the entire study population suggested a direct association between neighborhood walkability and physical activity that approached statistical significance (High WI vs. Low WI: OR = 1.16; 95% CI 0.95-1.40). Furthermore, participants who lived in a higher WI neighborhood were more likely to meet physical activity guidelines in 2 groups: (1) men whose recreational physical activity included walking (High WI vs. Low WI: OR = 5.43; 95% CI 1.30-22.73) and (2) men whose only recreational physical activity was (High WI vs. Low WI: OR = 9.54; 95% CI 1.84-49.60). Our findings suggest gender differences in the association between the built environment and physical activity in Mexican-American adults. Attempts to encourage walking among Mexican-American adults may be easier in high-walkability neighborhoods than in low-walkability neighborhoods.

On the “viscosity overshoot” during the uniaxial extension of a low density polyethylene

An experimental investigation of the viscosity overshoot phenomenon observed during uniaxial extension of a low density polyethylene is presented. For this purpose, traditional integral viscosity measurements on a Münstedt-type extensional rheometer are combined with local measurements based on the in-situ visualization of the sample under extension. For elongational experiments at constant strain rates within a wide range of Weissenberg numbers (Wi), three distinct deformation regimes are identified. Corresponding to low values of Wi (regime I), the tensile stress displays a broad maximum, but such maximum is observed with various polymeric materials deformed at low rates and it should not be confused with the “viscosity overshoot” phenomenon. Corresponding to intermediate values of Wi (regime II), a local maximum of the integral extensional viscosity is systematically observed. Moreover, within this regime, a strong discrepancy between integral measurements and the space average of the local elongational viscosity is observed which indicates large deviations from an ideal uniaxial deformation process. Images of samples within this regime reinforce this finding by showing that, corresponding to the maximum of the integral viscosity, secondary necks develop along the sample. The emergence of a maximum of the integral elongational viscosity is, thus, related to the distinct inhomogeneity of deformation states and most probably not to the rheological properties of the material. In the fast stretching limit (high Wi, regime III), the overall geometric uniformity of the sample is well preserved, no secondary necks are observed and both the integral and the space averaged transient elongational viscosity show no maximum. A detailed but yet incomplete comparison of the experimental findings with results from the literature is presented and several open questions are stated.

Flow of dilute to semi-dilute polystyrene solutions through a benchmark 8:1 planar abrupt micro-contraction

We have studied the flow of thermodynamically ideal solutions of a high molecular weight ( M w = 6.9 MDa) atactic polystyrene in the θ solvent dioctyl phthalate (aPS in DOP) through a micro-fabricated 8:1 planar abrupt contraction geometry. The channel is much deeper than most micro-scale geometries, providing an aspect ratio of 16:1 and a good approximation to 2D flow in the narrow channel. The solutions span a range of concentration 0.03 wt.% < c < 0.6 wt.%, encompassing the dilute to semi-dilute regimes and providing a range of fluid viscosities and relaxation times such that we achieve a range of Weissenberg numbers (8.7 < Wi < 1562) and Reynolds numbers (0.07 < Re < 11.2), giving elasticity numbers between 31 < El < 295. We study the flow using a combination of micro-particle image velocimetry (μ-PIV) to characterize the flow field, pressure measurements to evaluate the non-Newtonian viscosity, and birefringence measurements to assess macromolecular strain. Flow field observations reveal three broad flow regimes characterized by Newtonian-like flow, unstable flow and vortex growth in the upstream salient corners. Transitions between the flow regimes scale with Wi, independent of El, indicating the dominance of elastic over inertial effects in all the fluids. Transitions in the flow field are also reflected by transitions in the relative viscosity (determined from the pressure drop) and the macromolecular strain (determined from birefringence measurements). The strain through the 8:1 contraction saturates at a value of ∼4–5 at high Wi. The results of these experiments broaden the limited set of literature on flow through micro-fluidic planar contractions and should be of significant value for optimizing lab-on-a-chip design and for comparison with modeling studies with elasticity dominated fluids.

Dynamics of individual molecules of linear polyethylene liquids under shear: Atomistic simulation and comparison with a free-draining bead-rod chain

Nonequilibrium molecular dynamics (NEMD) simulations of a dense liquid composed of linear polyethylene chains were performed to investigate the chain dynamics under shear. Brownian dynamics (BD) simulations of a freely jointed chain with equivalent contour length were also performed in the case of a dilute solution. This allowed for a close comparison of the chain dynamics of similar molecules for two very different types of liquids. Both simulations exhibited a distribution of the end-to-end vector, |Rete|, with Gaussian behavior at low Weissenberg number (Wi). At high Wi, the NEMD distribution was bimodal, with two peaks associated with rotation and stretching of the individual molecules. BD simulations of a dilute solution did not display a bimodal character; distributions of |Rete| ranged from tightly coiled to fully stretched configurations. The simulations revealed a tumbling behavior of the chains and correlations between the components of Rete exhibited characteristic frequencies of tumbling, which scaled as Wi−0.75. Furthermore, after a critical Wi of approximately 2, another characteristic time scale appeared which scaled as Wi−0.63. Although the free-draining solution is very different than the dense liquid, the BD simulations revealed a similar behavior, with the characteristic time scales mentioned above scaling as Wi−0.68 and Wi−0.66.

Brownian dynamics simulations of polyelectrolyte adsorption in shear flow with hydrodynamic interaction

The adsorption of single polyelectrolyte molecules in shear flow is studied using Brownian dynamics simulations with hydrodynamic interaction (HI). Simulations are performed with bead-rod and bead-spring chains, and electrostatic interactions are incorporated through a screened Coulombic potential with excluded volume accounted for by the repulsive part of a Lennard-Jones potential. A correction to the Rotne-Prager-Yamakawa tensor is derived that accounts for the presence of a planar wall. The simulations show that migration away from an uncharged wall, which is due to bead-wall HI, is enhanced by increases in the strength of flow and intrachain electrostatic repulsion, consistent with kinetic theory predictions. When the wall and polyelectrolyte are oppositely charged, chain behavior depends on the strength of electrostatic screening. For strong screening, chains get depleted from a region close to the wall and the thickness of this depletion layer scales as N(1/3)Wi(2/3) at high Wi, where N is the chain length and Wi is the Weissenberg number. At intermediate screening, bead-wall electrostatic attraction competes with bead-wall HI, and it is found that there is a critical Weissenberg number for desorption which scales as N(-1/2)kappa(-3)(l(B)|sigmaq|)(3/2), where kappa is the inverse screening length, l(B) is the Bjerrum length, sigma is the surface charge density, and q is the bead charge. When the screening is weak, adsorbed chains are observed to align in the vorticity direction at low shear rates due to the effects of repulsive intramolecular interactions. At higher shear rates, the chains align in the flow direction. The simulation method and results of this work are expected to be useful for a number of applications in biophysics and materials science in which polyelectrolyte adsorption plays a key role.

Predicting wheat quality – consequences of the ascorbic acid improver effect

Genotype and environment sets of wheat cultivars and breeding lines were tested for bread making, dough mixing, dough rheology, protein composition and thiol/disulphide composition to find methods that could identify wheat with high baking quality (high-BQ) and moderate work input (moderate-WI) requirement using the mechanical dough development system. Wheat with these properties generally had a high baking response to ascorbic acid (AA) and a GluD1a (HMW-GS 2 + 12) allelic composition. Strong wheat with high WI and high-BQ generally had low baking response to AA and a GluD1d (HMW-GS 5 + 10) allelic composition. Using protein composition data to identify wheat of high-BQ (with AA) and moderate-WI, it was best to select wheat with as high as possible percent of SDS-unextractable polymeric protein (%UPP) in flour and as low as possible %UPP in total polymeric protein. Using a dough extension test for identifying wheat of high-BQ (with AA) and moderate-WI, it was best to select wheat with intermediate values for maximum resistance to extension (Rmax) and for values of extension at Rmax as high as possible within the intermediate Rmax range. Cysteine content of protein fractions and glutathione content of flour gave mostly poor to weak correlations with all baking and mixing properties.

A Two-Step Clustering for 3-D Gene Expression Data Reveals the Main Features of the Arabidopsis Stress Response.

Summary We developed an integrative approach for discovering gene modules, i.e. genes that are tightly correlated under several experimental conditi ons and applied it to a threedimensional Arabidopsis thaliana microarray dataset. The dataset consists of approximately 23000 genes responding to 9 abiotic stress conditions at 6-9 different points in time. Our approach aims at finding relatively small and dense modules lending themselves to a specific biological interpretation. In order to detect g ene modules within this dataset, we employ a two-step clustering process. In the first step, a k-means clustering on one condition is performed, which is subsequently used in the second step as a seed for the clustering of the remaining conditions. To validate the significance of the obtained modules, we performed a permutation analysis and determined a null hypothesis to compare the module scores against, providing a p-value for each module. Significant modules were mapped to the Gene Ontology (GO) in order to determine the participating biological processes. As a result, we isolated modules showing high significance wi th respect to the p-values obtained by permutation analysis and GO mapping. In these modules we identified a number of genes that are either part of a general stress response with similar characteristics under different conditions (coherent modules), or part of a more specific stress response to a single stress condition (single response modules). We also found genes clustering within several conditions, which are, however, not part of a coherent module. These genes have a distinct temporal response under each condition. We call the modules they are contained in individual response modules(IR).

Interfacial instability between sheared elastic liquids in a channel

We consider the linear stability of the interface between two sheared elastic liquids at large Weissenberg number ( Wi) with negligible inertia. The liquids are of Oldroyd-B or UCM type and have matched viscosity. In UCM liquids, Renardy [Y. Renardy, Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids, J. Non-Newton. Fluid Mech. 28 (1988) 99–115] found a purely elastic instability for short-waves in the absence of surface tension for which the perturbation flow decays exponentially away from the interface. For UCM liquids at large Wi we show that this instability persists even though the wavelength is larger than the channel width and the disturbance occupies the entire channel. Surprisingly, the growth rate is not affected by the location of the walls, even though the mode structure is altered. This analysis suggests a reappraisal of the appropriateness of the short-wave and long-wave classifications for instabilities of viscoelastic liquids in order to accommodate the additional length scale introduced by fluid velocity and relaxation. The instability persists for Oldroyd-B liquids even as the elastic contribution to viscosity approaches zero. Surprisingly too, the inclusion of surface tension does not affect the asymptotic growth rate at large wavenumber. When more modest values of Wi are considered, we find parameter values for which arbitrarily large surface tension reduces the growth rate but does not stabilize the flow; previously proposed mechanisms based on the interface displacement are therefore inadequate to explain the instability. Because the instability is locally generated, it appears in other high Wi flows with interfaces, both in channels and in pipes.

Shear-Induced Molecular Orientation and Crystallization in Isotactic Polypropylene: Effects of the Deformation Rate and Strain

Studies of dilute polymer solutions in shear flow suggest that the mean fractional extension of molecules increases gradually with the Weissenberg number (Wi = shear rate × longest relaxation time) and approaches an asymptotic value of 0.4−0.5, while in elongational flow it approaches full contour length above a certain critical strain rate. In an entangled polymer melt, this behavior is more complex due to inter- and intramolecular interactions. In situ rheo-SAXS (small-angle X-ray scattering) and -WAXD (wide-angle X-ray diffraction) experiments were performed to investigate the effects of shear rate, shear duration, and Wi on the extent of molecular orientation/extension and crystal orientation in an isotactic polypropylene (iPP) melt. Two series of experiments were designed: (1) variation of shear rate (30, 45, and 60 s-1) at a constant shear duration (5 s) and (2) variation of shear duration (1.3, 3, and 5 s) at a constant rate (60 s-1). The degree of crystal orientation (Herman's orientation function, f) observed at 165 °C and fraction of oriented crystals (Xo) observed in a fully crystallized sample at room temperature increased with both shear rate and shear duration. Interestingly, at a constant strain (rate × duration), short-duration shear at a high rate was found to be more effective (i.e., higher f and Xo) than long-duration shear at a low rate. The longest relaxation time for the iPP sample and Wi were estimated from the dynamic moduli data. Both f and Xo were found to gradually increase with Wi and approached plateau values at high values of Wi. Results indicated that, even under a very intense shear field (or high Wi values), molecules do not extend to full contour length, and there is a limiting value for mean orientation/extension and subsequent crystal orientation in a polymer matrix. Characteristic dimensions of the shish-kebab entity formed in a sheared iPP melt at 165 °C were determined from the rheo-SAXS data. It was found that the average shish length was 700−750 nm and the average spacing between adjacent kebabs was 60−70 nm.