Praktikum III. Experiment P4. Simulation of Polymer Behavior in Step Shear Rate Flow using the Reptation Model

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1 Praktikum III Fall Term 211 Experiment P4 Simulation of Polymer Behavior in Step Shear Rate Flow using the Reptation Model A lab report written by: Baumli Philipp baumlip@student.ethz.ch Date of the experiment: Date of writing the report: Assistant: Aparna Sreekumari Students: Baumli Philipp Marion Frey Matthias Dzung Stefan Wenk Departement of Materials ETH Zürich

2 1. Abstract In this experiment the Behavior of a polymer melt in Step Shear Rate Flow using the Reptation Model was simulated by using the commercial software MATLAB. Since polymer melts are non-newtonian fluids it was expected that the viscosity decreases with increasing shear rates, which could be verified in this experiment. Furthermore, it was found out, that with increasing reptation times the memory, i.e. the delay of the stress response of a polymer melt to an applied deformation, and the viscosity of polymer melts decrease, whereas the first normal stress coefficient and the time needed by the polymer melt to achieve zero-valued memory increase. Not only the viscosity, also the first normal stress coefficient decreases with increasing shear rates. 2. Introduction The dynamical properties of polymer melts of high molecular weight can be described by the so-called reptation theory, introduced approximately thirty years ago. It takes into account the strong influence of entanglements in the polymer melt. The word reptation comes from the imagined snake-like (snakes are reptiles) motion of the long (and therefore heavy) polymer molecules in a sort of tube which is formed by neighboring polymer molecules and therefore, since a polymer molecule cannot go through other polymer molecules without breaking bonds, confines the motion of a polymer molecule in it due to the hindrance caused by the neighboring molecules and their entanglements with the concerning polymer molecule. The neighboring polymer molecules are the obstacles to the motion of a considered polymer molecule. Polymer molecules move snake-like along each other. The neighboring polymer molecules move too. A real-life imagination could be a pit full of snakes in an adventure movie or a plate full of Italian Spaghettis. An illustration of this concept can be found in the figure below. Figure1: Basic concept of the reptation model [1] The snake-like motion of the polymer chains along the tube is termed reptation. In the absence of external forces this snake-like motion of the polymer molecules along these imagined tubes is purely diffusive (Brownian motion). A polymer molecule can snake out of the imagined tube much faster than the neighboring

3 molecules can give way to the considered polymer chain. This large scale motion out of the tube is also diffusive. It changes the entanglement network and occurs in a specific time range, the so-called reptation time τ rept. Within this time range, the tube gradually disappears. This concept is visualized in figure 2 below. Figure2: The reptation of a polymer molecule out of its tube within the time scale of the reptation time [1] Reptation also allows polymer chains to relax stresses caused by external forces and deformations. The reptation time plays a key role in the polymers` response to an applied stress, which consists of the escaping of the tube. This process is slow and the origin for the polymer melts` viscoelastic behavior. The delay of the stress response to the applied deformation is described by the memory function (or simply the memory) µ. This quantity is dimensionless. In recent times computer developments enabled scientists to model this computationally via commercial software such as MATLAB ([1]). 3. Materials and Methods For all parts of the experiment the two matlab files memoryde.m and reptde.m were used. In the first part of the experiment the memory µ was computed for different reptation times, represented by trept in the matlab file memoryde.m. The shear rate. γ, denoted as gamdt in the matlab files was kept constant at.1s -1. The goal was to find relationships between the memory and the reptation time and between the time range the polymer needed to achieve zero-valued memory. In order to achieve this, the matlab file memoryde.m was run for 11 different reptation times. These computations were summarized into two graphs, in one there was the memory plotted against the reptation time chosen, and in the second one the chosen reptation time was plotted against the time t necessary for the polymer to achieve zero-valued memory. In order to examine the low-time region, a logarithmically-scaled time- axis was used. In the second part of the experiment relationships between the first normal stress coefficient Ψ 1 (denoted as Psi1 in the matlab file) and the reptation time τ rept and between the viscosity η (denoted as eta in the matlab file) and the reptation time τ rept were to be found. To derive them, the matlab file reptde.m was run for five different reptation time values. Again, the shear rate. γ was kept constant at.1s -1. The results were summarized into two graphs. As a preparatory calculation the zero-shear

This concept is visualized in figure 2 below.

4 viscosity η was calculated from data available in appendix 2 of the experiment manual using the formula below: η = 9 i= 1 τ i g i So far the shear rate. γ was kept constant. In the third part of the experiment, this was no longer the case. Now, the behavior of the first normal stress coefficient Ψ 1 and the viscosity η was studied for five different values of the shear rate. γ. This was achieved by transferring data provided in separate DAT- files into Microsoft Excel and by summarizing the data in two graphs (Ψ 1 vs.. γ and η vs.. γ ) and modifying them, i.e. deleting data and scaling axes in order to stress the differences between the peak and the plateau values in each curve. It didn`t matter whether one worked with the peak values or the plateau values in order to detect a trend in the data, both techniques resulted in the same trend. 4. Results Out of the graph below one can see, that the memory µ decreases for increasing reptation times. The decrease is strongest in the region of small reptation times under approximately 1 seconds, and lessens strongly for larger values of the reptation time. 12 memory µvs. reptation time τ rept 1 memory µ[-] τ rept [sec] Figure 3: The memory µ in dependence of the reptation time τrept The time needed by the polymer chains to achieve zero-valued memory increases linearly with the reptation time, as one can see from figure 4 below.

Now, the behavior of the first normal stress coefficient Ψ 1 and the viscosity η was studied for five different values of the shear rate. γ.

5 τ rept [sec] τ rept [sec] vs. t[sec] t[sec] Figure 4: The reptation time τrept in dependence of the time t needed to achieve zero memory The first normal stress coefficient Ψ 1 [-] increases with increasing reptation time τ rept [s] and the viscosity η[pa s] decreases approximately linear with increasing reptation time τ rept [s] as can be seen from the two figures 5 and 6 below. For the zero-shear viscosity η, a value of η = 28` Pa s was obtained. Ψ Ψ 1 vs. τ rept [sec] τ rept [sec] Figure 5: The first normal stress coefficient Ψ 1 in the dependence of the reptation time τ rept

1 [-] increases with increasing reptation time τ rept [s] and the viscosity η[pa s] decreases approximately linear with increasing reptation time τ rept [s] as

6 η[pa s] η[pa s]vs. τ rept [sec] τ rept [sec] Figure 6: The viscosity η in dependence of the reptation time τ rept The following two graphs illustrate that the peak values of first normal stress coefficient Ψ 1 decrease with increasing shear rate. γ and that the peak values of the viscosity η also decrease with increasing with increasing shear rate. γ. However this trend is also true for all plateau values, not only for the peak values, which are used in the two graphs. Ψ 1,peak Ψ 1,peak vs. gamdt[s -1 ] gamdgamdt[s -1 ] Figure 6: The peak values of first normal stress coefficient Ψ1 in dependence of the shear rate. γ (denoted as gamdt in the graph)

stress coefficient Ψ 1 decrease with increasing shear rate. γ

7 η peak [Pa s] η peak [Pa s] vs. gamdt[s -1 ] gamdt[s -1 ] Figure 7: The peak values of the viscosity η in dependence of the shear rate γ (denoted as gamdt in the graph) Ψ Ψ 1 vs. t[s] t[s] Figure 8: The behavior of the first normal stress coefficient Ψ 1 for different values of the shear rate γ with time t η[pa s] η[pa s] vs. t[s] t[s] Figure 9: The behavior of the viscosity η for different values of the shear rate γ with time t

8 5. Discussion It could clearly be verified that the viscosity decreases with increasing shear rates as it is the case for non-newtonian fluids, such as polymer melts. This could be viewed as an indicator that the experiment was successful. It is a pity that the team decided to work on only one computer. It would have been more interesting and more informative to obtain more data by working on two computers, because the two groups could have agreed on which values to test and so more date could have been collected in order to enhance the graphs. However it wouldn`t have changed the general trends. The results generally fit the personal expectations of the author. Most interestingly is the very sharp decrease in viscosity η and the first normal stress coefficient Ψ 1 if the shear rate. γ exceeds the value of.1s -1 and the value chosen to run the simulation was 2s -1. Since viscosity stands for the resistance of a fluid to shear forces and hence to flow one could argue that bigger shear rates immediately leads to flowing/ yielding of the material, which is colloquially referred to as a giving-up of the material. The team made a mistake in the first part of the experiment: It took 11 values into the graph instead of 1 as the assistant suggested. The author thinks that no changes should be made in the experimental tasks to do in contrast to what Dr. Willeke suggested. 6. References [1] Experiment manual P4, Minirosadat Sadati and Patrick Ilg, D-MATL ETH

It would have been more interesting and more informative to obtain more data by working on two computers, because the two groups could have agreed on which values to test and so more date could have

Viscoelasticity of Polymer Fluids.

Viscoelasticity of Polymer Fluids. Main Properties of Polymer Fluids. Entangled polymer fluids are polymer melts and concentrated or semidilute (above the concentration c) solutions. In these systems polymer