Polymers Research Group Technical Report Serieshttp://hdl.handle.net/1974/86802019-04-21T14:05:09Z2019-04-21T14:05:09ZMacromolecular Tumbling and Wobbling in Large-Amplitude Oscillatory Shear FlowJbara, Layal M.Giacomin, A. Jeffreyhttp://hdl.handle.net/1974/260882019-04-04T07:17:49Z2018-11-01T00:00:00ZMacromolecular Tumbling and Wobbling in Large-Amplitude Oscillatory Shear Flow
Jbara, Layal M.; Giacomin, A. Jeffrey
For a suspension of rigid dumbbells, in any simple shear flow, we recently solved for the diffusion equation for the orientation distribution function by a power series expansion in the shear rate magnitude. In this paper, we focus specifically on large-amplitude oscillatory shear flow (LAOS), for which we extend the orientation distribution function to the 6th power of the shear rate amplitude. We arrive at the Fourier solution for each harmonic contribution to the total orientation distribution function, separating each harmonic into its coefficients in and out-of-phase with cosnωt , ψ ′n and ψ ′′ n , respectively. We plot, for the first time, the evolving normalized alternant macromolecular orientation. Moreover, to deepen our understanding of the macromolecular motions, we distinguish and study the two types of possible rotations, tumbling and wobbling.
2018-11-01T00:00:00ZOrder in Oscillatory Shear Flow Kanso, Mona A.Giacomin, A. JeffreySaengow, ChaimongkolGilbert, Peter H.http://hdl.handle.net/1974/260872019-04-04T07:17:44Z2018-12-01T00:00:00ZOrder in Oscillatory Shear Flow
Kanso, Mona A.; Giacomin, A. Jeffrey; Saengow, Chaimongkol; Gilbert, Peter H.
We examine the second order orientation tensor for the simplest molecular model relevant to a polymeric liquid in large-amplitude oscillatory shear flow (LAOS), the rigid dumbbell suspension. For this we use an approximate solution to the diffusion equation for rigid dumbbells, an expansion for the orientation distribution function truncated after the fourth power of the shear rate amplitude. We then calculate the second order orientation tensor, and then use this to calculate the order parameter tensor. We next examine the invariants of both the second order orientation tensor and the order parameter tensor. From the second invariant of the order parameter tensor, we calculate the scalar, the nematic order, and examine its evolution for a polymeric liquid in LAOS. We find this nematic order, our main result, to be even. We use Lissajous figures to illustrate the roles of the Weissenberg and Deborah numbers on the evolving order in LAOS. We use the low frequency limit of our main result to arrive at an expression for the nematic order in steady shear flow. Our work gives a first glimpse into macromolecular order in LAOS. Our work also provides analytical benchmarks for numerical solutions to the diffusion equation for both oscillatory and steady shear flows.
2018-12-01T00:00:00ZExact Analytical Interconversion between Durometer Hardness ScalesGilbert, Peter H.Giacomin, A. Jeffreyhttp://hdl.handle.net/1974/260862019-04-04T07:17:40Z2017-02-01T00:00:00ZExact Analytical Interconversion between Durometer Hardness Scales
Gilbert, Peter H.; Giacomin, A. Jeffrey
Previous work has related Young’s modulus to durometer hardness for any standardized scale. In this paper, we build on this work to solve explicitly and exactly for the hardness in any one standardized durometer hardness scale as a function of the hardness in any other target scale. We find that when the target scale is for a flat indenter, the conversion is algebraic and straightforward. However, when the target scale is for an indenter that is not flat (conical or hemispherical), the exact explicit analytical solution requires a power series inversion, said series involving beta functions and solutions to a set of integer equations. We complete our analysis with two worked examples illustrating the use of our interconversion charts.
2017-02-01T00:00:00ZMolecular Continua for Polymeric Liquids in Large-Amplitude Oscillatory Shear FlowGiacomin, A. JeffreySaengow, Chaimongkolhttp://hdl.handle.net/1974/260852019-04-04T07:17:37Z2017-05-01T00:00:00ZMolecular Continua for Polymeric Liquids in Large-Amplitude Oscillatory Shear Flow
Giacomin, A. Jeffrey; Saengow, Chaimongkol
In this paper, we connect a molecular description of the rheology of a polymeric liquid to a continuum description, and then test this connection for large-amplitude oscillatory shear flow (LAOS). Specifically, for the continuum description we use the 6-constant Oldroyd framework, and for the molecular, we use the simplest relevant molecular model, the suspension of rigid dumbbells. By relevant, we mean predicting at least higher harmonics in the shear stress response in LAOS. We call this connection a molecular continuum, and we examine two ways of arriving at this connection. The first goes through the retarded motion expansion, and the second, expands each of a set of specific material functions (complex, steady shear, and steady uniaxial extensional viscosities). Both ways involve comparing the coefficients of expansions to then solve for the six constants of the continuum framework in terms of the two constants of the rigid dumbbell suspension. The purpose of a molecular continuum is that many well-known results for rigid dumbbell suspensions in other flow fields can then also be easily obtained, without having to first find the orientation distribution function. In this paper, we focus on the recent result for the rigid dumbbell suspension in LAOS. We compare the accuracies of the retarded motion molecular continuum (RMMC) with the material function molecular continuum (MFMC). We find the RMMC to be the most accurate for LAOS.
2017-05-01T00:00:00Z