RHEOMETER ALLOWING DIRECT VISUALIZATION OF CONTINUOUS SIMPLE SHEAR JN NON-NEWTONIAN FLUID

This work is supported, in part, by ACS-PRF grant (40596-AC7) and NSF grant (CTS-0115867). The United States government may have certain rights to the invention or inventions herein.

FIELD OF THE INVENTION

The present invention is directed to an improved rheometer which provides direct visualization, and to a rheometer that can be used in conjunction with shearing devices, such as rotational, torsional, sliding plate, and other type rheometers.

BACKGROUND OF THE INVENTION Rheometers are used to determine the flow characteristics or behavior of rubbers, plastics, elastomers, and other polymers and viscous materials. Rheological measurements have for decades proceeded without any visualization of the field to which materials are subjected, particularly polymeric materials. The flow of complex materials is an example of nonlinear phenomena in nature. Structured fluids such as foams, colloids, micelles, granular materials and polymers display intricate dynamic behavior that is still defying complete description. Simple fluids show Newtonian behavior such as constant viscosity so that the shear stress σ is linearly proportional to shear rate {/).

SUMMARY OF THE INVENTION The present invention is directed to an improved rheometer which provides direct visualization, and to a rheometer that can be used in conjunction with shearing devices, such as rotational, torsional, sliding plate, and other type rheometers.

In one embodiment, the present invention relates to a rheometry device comprising: an optically clear plate; a metallic cone having a longitudinal axis in a perpendicular relation to the plate and the conical surface being in an acute angular relation to a surface of the plate, wherein the cone is free to rotate about its longitudinal axis, and wherein the plate is stationary; an optically clear barrier occupying a perimeter about the plate and the cone such that the space between the plate and the cone is visible; at least one laser; wherein a beam of the laser can be

directed perpendicularly through the optically clear plate and impinging the cone surface; and at least one camera in a generally perpendicular relation to the longitudinal axis of the cone and directed to a region of space between the plate and the cone. In another embodiment, the present invention relates to a process for conducting rheometric measurements, comprising the steps of: providing an optically clear plate; providing a metallic cone having a longitudinal axis in a perpendicular relation to the plate and the conical surface being in an acute angular relation to a surface of the plate, wherein the cone freely rotates about its longitudinal axis, and wherein the plate is stationary; providing a liquid contained between the cone and plate; providing at least one particle suspended in the liquid; providing at least one laser; wherein a beam of the laser is directed perpendicularly through the optically clear plate and impinging the cone surface; providing at least one camera in a generally perpendicular relation to the longitudinal axis of the cone and directed to a region of space between the plate and the cone containing one or more of the provided particles; collecting data from the at least one camera; converting the data to velocity as a function of time; and recording the data.

BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 is a schematic of the setup of the particle tracking velocimetry, where the upper stationary plate is made of transparent glass, the lower rotating cone made of steel and θ=5°;

Figure 2 is a series of plots showing: (A) growth upon startup shear at shear rate 5 s ^"1 of both shear and normal stresses (σ and N1 respectively); (B) the velocity profiles of the well-entangled 10% PBD solution for shear rates 0.05 and 0.1 s ^"1 at any time, where the straight lines indicate linear profiles; and (C) the velocity profiles for shear rate 5 s ^' \ at initial (circles), transient (diamonds), quasi-steady state (open squares) and steady state (squares), as measured on the video monitor where the frame rate is 15 frames per second and the velocity profile for the pure oligomeric solvent, which is a Newtonian liquid, is also depicted in the plus filled squares; and

Figure 3 represents still pictures from video recording of the illuminated moving particles at (A) initial (t = 0.1), (B) transient (t = 1.0 s) and (C) steady (t = 7.8 s) states for an imposed apparent shear rate of ω/θ = 10 s ^"1 , where the distance

between the two horizontal lines, H, is again about 0.9 mm. Only the particles in focus yield the brightest spots and streaks.

DETAILED DESCRIPTION OF THE INVENTION The present invention is directed to an improved rheometer which provides direct visualization, and to a rheometer that can be used in conjunction with shearing devices, such as rotational, torsional, sliding plate, and other type rheometers.

In one embodiment, a method according to the present invention uses direct visualization of the flow field involved in rheometric measurement. Due in part to this feature, the present invention enables the probing of a velocity profile in real time during controlled-rate shear. Due to the present invention, it has been determined that when the velocity gradient is non-uniform, the velocity profile changes significantly over time upon a controlled-rate startup shear in the stress plateau region. This finding questions the widely-employed protocol of imposing controlled- rate shear as an effective method to depict constitutive flow behavior of entangled polymeric liquids and casts doubt on the existing knowledge of constitutive behavior of this large class of highly commercially valuable materials.

Using the particle-tracking-velocimetric technique of the present invention, one can demonstrate that, for an entangled polymer solution, the velocity profile in a simple shear (y) deviates strongly from the anticipated linearity, whereas these structured fluids appear to show a possible discontinuous relationship between σ and γ. The velocity gradient is non-uniform and the velocity profile changes significantly over time upon a controlled-rate startup shear in the stress plateau region that is commonly encountered for entangled polymers. This would appear to invalidate the method of controlled-rate shear that is widely employed to probe constitutive flow behavior of various complex fluids.

It has also be found that the velocity profile evolves in time, linear at the startup of shear, and attains a maximum deviation from the linearity in the transient before settling to the steady state where the velocity gradient varies explicitly along the sample thickness. In one embodiment, the process of the present invention can be used to evaluate how several hundred million tons of plastic materials get processed into final articles. This can help produce stronger plastic products ranging from grocery/garbage bags to milk bottles and children toys, in addition to lowering

the costs associated with making/producing a wide range of plastic/polymer products.

The method and apparatus of the present invention can be applied to any rheometer or shearing device where there is a need for flow visualization of the velocity profile in real time during both controlled-rate and controlled-stress shear. The present invention can also be used to provide a visual input or it can be used in combination with a typical rheometer to provide a measurement of the viscosity of the particular material being evaluated.

A cone-plate shear cell is a particularly convenient device used to explore the constitutive behavior of structured liquids including entangled polymers because it allows continuous simple shear to be produced. This device generates uniform simple shear flow for both Newtonian and non-Newtonian liquids, provided that their constitutive relations are invertible. Regarding polymeric liquids as simple fluids, researchers routinely shear their samples in a cone-plate flow cell by imposing an angular velocity ω on one of the two surfaces as shown in Figure 1. A constitutive relation is determined by measuring the corresponding torque required to maintain ω, assuming that a uniform shear rate γ = ω/θ would be created across the sample thickness. This mode of imposing a constant surface velocity has become a standard protocol known as controlled-rate shear and was thought to generate a linear velocity profile in the gap independent of the constitutive behavior.

Monodisperse entangled polymer solutions were found to display a discontinuous flow transition under constant applied shear stress at a critical value σc, where the resulting apparent shear rate was almost three orders of magnitude higher than its value below σc. The controlled-rate measurements only turned up a smooth continuous flow curve with a characteristic stress plateau. Thus, depending on which variable (surface velocity vs. applied force) is controlled in generating the simple shear, there exist two different flow curves for the same sample.

Figure 1 shows a schematic of the setup of the particle tracking velocimetry, where the upper stationary plate is made of transparent glass, the lower rotating cone made of steel and θ = 5°. The apparatus allows for a view of the velocity field at a specific radial position in the gap, and affords good spatial and time resolutions. The device of Figure 1 can be used as an accessory to a commercial rheometer, allowing in-situ observations along with rheological information.

The entangled polymer solution evaluated is a 10 weight percent 1 ,4- polybutadiene (PBD) solution, made of a high weight PBD of M _w = 1.24x10 ⁶ g/mol and Mw/M _n = 1.2 from (Polymer Source Inc. P1445-Bd) dissolved in a phenyl- terminated oligomeric butadiene of M _n = 1.0 K (Aldrich 200417). For the particle tracking purpose, the sample is seeded with silver-coated particles of 10 micron diameter (Dantec Dynamics HGS-10) at a low level of ca. 200 ppm. A controlled- rate shear rheometer (ARES) equipped with a cone-plate assembly of 25 mm diameter is employed to make velocity profile measurements, where the cone angle θ is 5°. The particle tracking velocimetry of the present invention comprises sending a laser beam along the velocity gradient direction through the gap between cone and plate and video-taping the illuminated moving particles over time with a CCD camera facing the gap as shown in Figure 1. A thin film surrounds the meniscus to allow focus of a tele-microlens onto the interior illuminated particles. The negligible perturbation of the confining film on the simple shear flow inside the cone-plate has been both estimated and experimentally verified.

We examine the flow field in the cone-plate cell at three imposed shear rates of 0.05, 0.1 and 5 s ^"1 . These rates respectively correspond to the Newtonian region, the region just before the stress plateau, and the region well inside the stress plateau. At the two low values of γ = ω/θ, i.e., two low values of angular velocity ω, the stresses grow monotonically. But at 5 s ^"1 , Figure 2A shows non-monotonic behavior in both shear (σ) and normal (N ₁ ) stresses, where σ first grew to its maximum at t = 0.5 and then dropped to its steady state value at t = 2.4 s, meanwhile the first normal stress Ni reached its maximum at t = 1.7 s and approached its steady state around t = 5.5 s. The feature of overshoots (especially of σ) is well known for entangled polymers under high shear and has been understood for decades to be the flow characteristic of a uniformly sheared sample.

As shown in Figure 2B, particle track velocimetric (PTV) measurements demonstrate that the velocity gradients are essentially uniform along the gap at the low shear rates. At the high shear rate of 5 s ^" \ i.e., when the angular velocity of the rotating cone is set to ω = 5θ = 0.44 rad/s, the velocity field no longer varied linearly across the gap as shown in Figure 20. In contrast, when a Newtonian oligomeric butadiene liquid (which was used as the solvent for the 10% PBD solution) is

subjected to the same shearing condition at 5 s ^" \ a constant value equal to the preset value prevails at all times in every layer in the gap, as shown by the plus-filled squares in Figure 2C. The actual gap distance H involved in Figure 2B and 2C was ca. 0.9 mm (Ae., about 10 mm from the cone center) and was 9 cm on the video monitor where the measurements were taken.

This first set of particle tracking velocimetric observations reveals that the entangled polymeric liquid is in a state of frustration when subjected to an average shear rate that falls in the window of flow discontinuity, from a shear rate of 0.2 to 40 s ^'1 . In other words, the entangled chains prefer to be sheared either at 0.2 s ^"1 and lower or 40 s ^"1 and higher. When the imposed angular velocity ω is such that ω/θ = 5 s ^' \ falling in the middle of this window, the system apparently satisfied this boundary condition by sampling a spectrum of shear rates. The local shear rate at the top stationary plate was smaller than 1/6 of the prescribed value of 5 s ^"1 , whereas the local shear rate at the rotating cone essentially doubled from 5 to 10 s ^"1 . Equally interesting and important, the velocity profile evolved over time.

Corresponding to the characteristic times indicated in Figure 2A, one can show that the velocity gradient is initially constant across the gap up to the point of the stress maximum, i.e., up to t = 0.5 s. The maximum deviation from this linear velocity profile occurred around the maximum of Ni, i.e., at around t = 1.7 s. The eventual steady state possessed a weaker nonlinear profile, as shown in Figure 2C, which was not reached until after Ni had stabilized at 5.5 s. In other words, the profile was still evolving at t = 2.4 s when the shear stress σ had already stabilized.

Apart from these explicit rheological and PTV measurements in Figure 2, the process of the present invention provides a direct visual impression of the variations of the velocity profile across the gap with time.

Figure 3 shows three snapshots of the particle movements for a shear rate of ω/θ = 10 s ^"1 as seen from the gap at three characteristic times corresponding to initial, transient and steady states. The exposure time of the CCD camera is δt = 1/60 s. The still pictures from video recording show the illuminated moving particles at (A) initial (t = 0.1 ), (B) transient (t = 1.0 s) and (C) steady (t = 7.8 s) states for an imposed apparent shear rate of ω/θ = 10 s ^"1 , where the distance between the two horizontal lines, H, is about 0.9 mm. Only the particles in focus yield the brightest spots and streaks. From the length δx of the streak generated by the moving light- scattering particle, one can compute the speed of the particle according to V = δx/δt.

Thus, the streak length is directly proportional to the particle speed V since δt is fixed during the video recording. The three pictures clearly show different streak length variations along the gap, indicating different velocity profiles similar to those measured in Figure 2 for 5 s ^"1 at the different times, although in real time play, it is easier to watch the particle motion at a lower shear rate of ω/θ = 2.5 s ^"1 .

The particle tracking velocimetric method of the present invention provides flow responses of entangled polymer solutions in a commonly employed shear device of cone-plate, examines the controlled-rate shear measurements that assume a uniform velocity gradient in space and examines the long-standing research tradition in the area of non-Newtonian flow of complex materials.

The following references are incorporated herein by reference in their entireties:

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Liquids that possess a linear relationship between shear stress σ and shear rate γ are known as Newtonian fluids, where the proportionality constant is viscosity η. Polymeric liquids, made of long linear chains that form entanglement since they cannot pass over one another without breaking, are an important class of non- Newtonian fluids, whose viscosity η takes a finite time to reach the steady state value and decreases with γ . This widely known shear thinning behavior was thought to occur gradually due to increased chain alignment in shear that leads to disentanglement. If the chain disentanglement event was not to occur catastrophically in shear flow as agreed upon by the current consensus, shear thinning would take place homogeneously in every layer across the sample thickness. Then, just as in the case of Newtonian fluids, imposition of V on one surface over the other stationary surface in a parallel-plate device of gap H would indeed result in a homogeneous flow with shear rate V=H prevailing in every layer.

Many structured materials, such as micellar solutions, dense suspensions, foams, liquid crystals, soft gels and glasses, granular fluids, and metals, respond to large external deformation catastrophically: The imposed shear deformation or rate of deformation can produce a spatially nonhomogeneous distribution of different states across the sample thickness. This is commonly known as shear banding. Entangled model polymer solutions have also recently been found to show a spatial variation in the shear rate measured across the sample thickness in startup shear. This finding makes it rather difficult to establish a constitutive relationship between σ and γ through rheological measurements. It also challenges the currently favored version of the theoretical description of entangled polymers in shear flow that arose from a decade of theoretical efforts based on the de Gennes' visualization of snakelike motion of a test chain within the Edwards' tube picture. Furthermore, it calls for an reexamination of previous experimental studies of similar entangled polymer solutions. Inspired by the previous results on continuous shear behavior of entangled polymer solutions, we have carried out further experiments in the present work to decipher how chains actually negotiate with one another to establish any new topological relationship during shear. Specifically, in this work, we apply large- amplitude oscillatory shear (LAOS) to illustrate some intriguing flow behavior of entangled liquids by using an effective particle-tracking velocimetric method to determine the velocity profile in LAOS. The results indicate that chain disentanglement, as the cause of shear thinning, also occurs inhomogeneously in space in LAOS.

The entangled polymer solutions under examination are a previously prepared and studied 10 wt.% 1 ,4-polybutadiene (PBD) solution, made of a high weight PBD of M _w =1;2 x 10 ⁶ g=mol and HZM _n =1:18 from Polymer Source, Inc., dissolved in a phenyl-terminated oligomeric butadiene (oBD) of M _n = 1:0 kg=mol (Aldrich 20041-7), and a newly made 10% PBD solution of a monodisperse PBD of M _w = 740 kg/mol (made at Bridgestone) in a monodisperse oBD of /Ww = 4 kg/mol (made at Goodyear). For the particle-tracking purpose, the sample was seeded with silver- coated particles of 10 μm diameter (Dantec Dynamics HGS-10) at a level of several hundred parts per million. A strain-controlled shear rheometer equipped with a cone- plate assembly of 25 mm diameter was employed in this study.

Our particle-tracking velocimetry, as illustrated in Figure 4, consists of sending a laser beam either along the velocity gradient direction or at an angle to the shearing surface and videotaping the illuminated moving particles with a black-white CCD camera (with minimum 0.3 Ix and maximum 30 frames per second) that is mounted with a DIN objective (3:2 x ) through an adaptive tube (Edmund Optics: U54-868). In scheme A, a transparent film surrounds the meniscus to allow focus of a telemicrolens onto the interior. The perturbation of the confining film on the simple shear flow inside the cone-plate cell can be evaluated and is removable by employing scheme B. In the present work, scheme A was set up for an advanced rheometrics expansion system, where a bottom cone rotates against a fixed upper plate; scheme B was set up for a Bohlin CVOR rheometer (Malvern Instruments) where an upper cone rotates. Results from both protocols will be presented below.

Let us consider oscillatory shear by sandwiching a sample between cone and plate as shown in Figure 4, where the cone would make an oscillatory torsional motion against the fixed plate, as described by angular displacement φ(t)= φo s\n{ωt) and velocity by ω = faco cos(ωt). The shear strain is usually taken to be uniform across the gap equal to y ₀ = X ₀ (ή/h(r) ∞φolθ, with X ₀ (ή = ^r ₀ , where h(r) = r tan 0« rθ is the gap distance at r. At low amplitudes γo « 1 , the storage modulus G' and loss modulus G" were obtained as a function of ω, indicating that the overall chain relaxation rate, i.e., the crossover frequency ω _c (at which G' = G"), was around 0/07 rad/s for the entangled liquid. When the oscillatory shear was applied at frequencies ω > ω _c , the entangled chains would tend to deform and orient affinely without being able to relax significantly during the shear reversal in each cycle.

We subjected the sample to LAOS for a sufficiently long time at ω = 1 rad/s > ω _c to illustrate the effect of LAOS on the molecular reorganization. In the first cycle, the shear deformation was uniform across the gap, and such a uniform deformation also prevailed for γo ≤ 1 even at long times. But, over time, something dramatic took place. One key result of our particle-tracking velocimetric measurements is shown in Figure 5. As one way to visualize the effect of LAOS, we analyzed the velocity field at the instant of the maximum ω of the rotating cone. The video recording can also be analyzed at any other moments during each cycle. The measurements typically involve playing the movie frame by frame at a fixed rate of 30 frames per second using MGI VIDEOWAVE 4 software for about one to three frames and measuring, on

a computer monitor, the displacement of the traced particles over these frames in the different layers across the sample thickness. The time resolution is therefore as good as about 0.1 s. In contrast to the linear velocity profile seen for χ ₀ = 1 and at the beginning for γ ₀ = 4, the steady state velocity profiles are highly nonlinear. The actual gap distance H was ca. 0.9 mm (i.e., about 10 mm from the cone center) and was 12 cm on the video display, where the measurements were taken and presented in Figure 5. The reader can access a movie showing the evolution of velocity profile for^o = 3, based on which the open and solid squares were obtained in Figure 5.

To remove the potential complications introduced by the confining film around the meniscus, we examine the velocity profile at LAOS of a different 10% PBD solution using a second rheometer from Malvern Instruments where an upper cone rotates against a fixed lower plate. Using scheme B as depicted in Figure 4, we obtained another set of velocity profiles as shown in Figure 6. The banding behavior is very similar to that shown in Figure 5, and the difference arises from the difference in the makeup of these two different samples. Thus, this result indicates that there is a negligible effect at the location of observation from the stationary film surrounding the meniscus in scheme A in Figure 4. To rule out the possibility that the observed banding would be due to a tiny stress gradient in the cone-plate setup, similar experiments have been conducted using parallel-disk shear cells, and the banding was also systematically observed.

Returning to Figure 5, we note the banding characteristics to depend on the amplitude of the imposed strain. The peak shear rate /O = &χo in the "fluid" layer reached 7:2 s ^"1 , whereas the "solid" top layer experienced a peak shear rate nearly tenfold smaller. In other words, the bottom layer experienced a shear strain as high as over 700%, whereas the top layer remained intact and was only undergoing deformation no greater than 100%. Upon increasing γo from 2 to 4, the fractional thickness f of the fluid layer increased from 1/6 to 1/2, which can be described by f = {γo -1)/6, implying that no part of the sample would transform to a "fluid," i.e., f = 0 when fo ≤ 1 , which is indeed the case as shown in Figure 5. To elucidate the dynamics of this striking phenomenon, we followed the evolution (i.e., the thickness growth) of the disentangled layer over time. The flow behavior of the fluid-like layer looks distinctly different from that of the solid-like layer according to the visual impression of the recorded video. We can simply watch how

the bottom layer grows in time and stop the video play to determine its thickness at the different time intervals. We first look for any amplitude effect at a fixed oscillation frequency of ω = 1 rad = s. Figure 4(a) shows that the sample reached its final state during the same time interval (ca. 100 s) regardless of the value of γo > 1. We next subjected the sample to an amplitude of γo = 3 at various oscillation frequencies that were all higher than the chain relaxation rate of ω _c = 0:07 rad/s. Independent of the oscillation frequency, the final thicknesses of the fluid layer all attained 4 cm on the video monitor, or ca. 1/3 of the total sample thickness, at approximately the same time, around 100 s as shown in Figure 7B. Since ω differed by a factor of 10, in a given duration the sample had been subjected to 10 times as many cycles at ω = 25 rad/s as it had at ω = 0.25 rad/s. Conversely, the total amount of cumulative shear strain γ\ _. was 10 times smaller at 0.25 than at 2.5 rad/s. Figure 7B shows that the growth time was hardly dependent on j\ and ω and depended only on the time during which the sample was subjected to the LAOS at ω > ω _c , indicating that flow convection played a minor role at most. Figure 7A further indicates that, as long as the strain amplitude was high enough (i.e., χo > 1 ) to produce sufficient chain orientation, the kinetics leading to the final state were essentially the same, independent of χ ₀ . This behavior is true only for the present highly polydisperse sample. Monodisperse samples display a different set of phenomena. The preceding experimental observations indicate that the entangled solution is capable of rearranging over time into different states of entanglement in the various layers in simple shear. Specifically, the LAOS produced over time a coexistence of two different states of chain entanglement, indicating that the sample could not undergo chain disentanglement and shear thinning uniformly across the sample thickness. The key controlling variable appears to be a sufficiently high level of chain orientation as determined by the amplitude of the oscillatory shear. As long as the oscillation frequency is higher than the chain relaxation rate so that the imposed LAOS is effective to produce long-lasting chain orientation, the initially well- entangled chains find themselves in a new, anisotropic environment that allows them to look for a different topological relationship. With sufficient chain orientation, the chains seem to clear out of each other's way and enter a new dynamic state of less mutual constraint in a spatially inhomogeneous fashion, where some layers absorb the imposed strain by transforming into a state of disentanglement, allowing the rest

of the sample to stay in a nearly equilibrium state of entanglement. The movie shows that it actually took roughly only one period or a few seconds for the fluid layer to become visible, whereas the time required for the transformed layer to grow to its final thickness is much longer. The kinetics governing the growth was found to be insensitive to both ω and γ _Q . Currently, there is no theoretical description available to account for the nucleation and growth of the disentangled state in shear oscillation. However, let us hope that the available theoretical framework can provide an adequate description of the reported phenomena.

In summary, the entangled polymer solution was found to respond to strain- controlled large-amplitude oscillatory shear by transforming partially into a state of lower viscosity and, thus, a lower level of chain entanglement. The response appears to show a state of frustration because the transformation (Ae., chain disentanglement) could not take place homogeneously as shown in Figures 5 and 6. The coexistence of two layers of different viscoelastic properties under a given LAOS shows that chain disentanglement nucleates unevenly in space, taking place only under the critical condition of sufficient chain orientation. Here the "nucleation" appears to occur through chain diffusion as evidenced by the observations in Figures. 7A and 7B. Ongoing work in our lab has shown other banding features that are not observed here for the present rather polydisperse solutions. Future reports will focus on the other mechanism responsible for shear banding in LAOS.

Although the invention has been described in detail with particular reference to certain embodiments detailed herein, other embodiments can achieve the same results. Variations and modifications of the present invention will be obvious to those skilled in the art and the present invention is intended to cover in the appended claims all such modifications and equivalents.