CM4655 Polymer Rheology Lab. Torsional Shear Flow: Parallel-plate and Cone-and-plate

Transcription

1 CM4655 Polymer heology Lab Torsional Shear Flow: Parallel-plate and Cone-and-plate (Steady and SAOS) Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University r (-plane H r (plane 1 Torsional Shear Flow: Parallel-plate and Cone-and-plate Why do we need more than one method of measuring viscosity? log Instabilities in torsional flows At low deformation rates, torques & pressures become low At deformation high rates, torques & pressures become high; flow instabilities set in o Torsional flows Low signal in capillary flow Capillary flow The choice is determined by experimental issues (signal, noise, instrument limitations. log 1

2 Shear Flow Experimental Shear Geometries y y x (z-plane (z-plane r H r r z entrance region well-developed flow exit region x (-plane (plane (z-plane (-plane (z-plane (-plane r 3 Torsional Shear Flow: Parallel-plate and Cone-and-plate We will look at two flows measurable in torsional shear: steady and smallamplitude oscillatory shear (SAOS). t 1,t t 1 a. Steady o o o t t t... Material functions: 1 (), () f. SAOS t o cost 1,t o sint 1 t o sin( t ) t t t Material functions: G (), G () or (), () (Linear viscoelastic regime) 4

3 Cox-Merz ule * ( ) ( ) * ( ) G G,, An empirical way to infer steady shear data from SAOS data. Figure 6.3, p. 193 Venkataraman et al.; LDPE 5 Steady Shear Flow Material Functions Imposed Kinematics:, constant Material Stress esponse:, Material Functions: Viscosity First normal-stress coefficient Second normalstress coefficient Ψ Ψ 6 3

4 Torsional Shear Flow: Parallel-plate Torsional Parallel-Plate Flow - Viscosity Measureables: Torque T to turn plate ate of angular rotation W cross-sectional view: z Note: shear rate experienced by fluid elements depends on their r position. r H r H r By carrying out a abinowitsch-like calculation, we can obtain the stress at the rim (r=). z r z ( r ) Correction required 3 3 d ln( T / ) T / 3 d ln 7 Torsional Shear Flow: Parallel-plate Parallel-Plate Shear ate Correction T 3 1 slope is a function of 1 3 d log T / slope d log T / ( ) d ln( T / ) 3 d ln H 8 4

5 Torsional Shear Flow: Cone-and-plate Torsional Cone-and-Plate Flow Viscosity Measureables: Torque T to turn cone ate of angular rotation polymer melt (-plane r Note: the introduction of the cone means that shear rate is independent of r. Ω Θ Since shear rate is constant everywhere, so is stress, and we can calculate stress from torque. constant 3 3Θ Ω No corrections needed in cone-and-plate 9 Torsional Shear Flow: Cone-and-plate 1 st Normal Stress (C&P) Measureables: Normal thrust F (-plane The total upward thrust of the cone can be related directly to the first normal stress coefficient. F 1 ( ) polymer melt F rdr p atm r 1 (see text pp44-5; also DPL pp5-53) 5

6 Torsional Shear Flow: Cone-and-plate nd Normal Stress (C&P) Need normal force as a Cone and Plate: function of r / r p ( N1 N) ln N (see Bird et al., DPL) MEMS used to manufacture sensors at different radial positions S. G. Baek and J. J. Magda, J. heology, 47(5), (3) J. Magda et al. Proc. XIV International Congress on heology, Seoul, 4. heosense Incorporated ( 11 Torsional Shear Flow: Cone-and-plate heosense Incorporated Comparison with other instruments S. G. Baek and J. J. Magda, J. heology, 47(5), (3) 1 6

7 Torsional Shear Flow: Cone-and-plate Start up of Steady Shear flow: obtain Steady State Choose Take frequent data points Steady state must be experimentally observed for each chosen 13 Torsional Shear Flow: Cone-and-plate Start up of Steady Shear flow: obtain Steady State Obtained in the same run as More experimental noise Ψ Ψ Steady state must be experimentally observed for each chosen 14 7

8 Torsional Shear Flow: Parallel-plate and Cone-and-plate eport Flow Curves, Ψ 1 Obtain widest range of possible within the ability of the instrument eport on reproducibility 1 1 Ψ Torsional Shear Flow: Cone-and-plate Limits on Measurements: Flow instabilities in rheology cone and plate flow High (steady shear) or (SAOS) cause these instabilities to be observed. Figures 6.7 and 6.8, p. 175 Hutton; PDMS 16 8

9 Small-Amplitude Oscillatory Shear Material Functions Imposed Kinematics: cos, sin cos Material Stress esponse: phase difference between stress and strain waves sin (linear viscoelastic regime) Material Functions: SAOS stress,, sin sin cos Storage modulus cos Loss modulus sin 17 What is the strain in SAOS flow? (, t) 1 t ( t) dt t 1 cost dt sint The strain amplitude is: The strain imposed is sinusoidal. 18 9

10 Generating Shear Steady shear b ( t ) Vt h t o h x constant x 1 Small-amplitude oscillatory shear b ( t ) h o sin sin t h o sin t h x periodic x 1 19 In SAOS the strain amplitude is small, and a sinusoidal imposed strain induces a sinusoidal measured stress (in the linear regime). 1( t) sin( t ) 1( t) sin( t ) sint cos cost sin cos sint sin cost portion in-phase with strain portion in-phase with strain-rate 1

11 3 is the phase difference between the stress wave and the strain wave (, t) 1( t -1 ) - ( t 1 ) -3 1 SAOS Material Functions 1( t) cos sin sint cost portion in-phase with strain G portion in-phase with strain-rate For Newtonian fluids, stress is proportional to strain rate: 1 1 G G is thus known as the viscous loss modulus. It characterizes the viscous contribution to the stress response. 11

12 What types of materials generate stress in proportion to the strain imposed? Answer: elastic solids Hooke s Law for elastic solids G 1 1 v 1 x x u 1 initial state, no flow, no forces deformed state, u1 1 G x Hooke's law for elastic solids x 1 x 1 f spring restoring force initial state, no force deformed state, f k 1 x 1 Hooke's law for linear springs Similar to the linear spring law 3 SAOS Material Functions 1( t) cos sin sint cost portion in-phase with strain G portion in-phase with strain-rate G For Hookean solids, stress is proportional to strain : 1 G 1 G is thus known as the elastic storage modulus. It characterizes the elastic contribution to the stress response. (note: SAOS material functions may also be expressed in complex notation. See pp of Morrison, 1) 4 1

13 Linear-Viscoelastic egime Impose: sin Measure: sin eport: sin cos Choose to obtain good, strong signal, but within the linear regime cos sin cos sin The response must be independent of the strain amplitude 5 Linear-Viscoelastic egime: Strain Sweep LVE Limit The linear regime must be experimentally determined for your material by doing strain sweeps. 6 13

14 Linear-Viscoelastic egime: Strain Sweep LVE Limit will be a function of frequency (check at low and high ). 7 Linear-Viscoelastic egime: Frequency Sweeps Should be independent of strain amplitude Frequency sweeps give the and curves Collect as a function of temperature to expand the dynamic range (tt shifting) G" 1 1 G'

15 Linear-Viscoelastic egime: Time-Temperature Superposition Data taken at different temperatures may be combined to make master curves of and versus (see earlier lecture) 1.E+7 1.E+6 G' (Pa) G'' (Pa) 1.E+5 1.E+4 1.E+3 1.E+ k Figure k (s) 8.8, p. g84 k (kpa) data from Vinogradov, 1.3E-3 16 polystyrene 3.E E-5 9 melt 4 3.E E-7 4 and a Generalized Maxwell model may be fit (see later lecture) 1.E+1 1.E+ 1.E+1 1.E+ 1.E+3 1.E+4 1.E+5 1.E+6 1.E+7 a T, rad/s 9 SAOS Material Functions tan 1 1/ 1tan 1/ 1 tan 3 15

16 Assignment: For the PDMS polymer in the lab Measure and report on the true steady shear viscosity at room temperature and other assigned temperatures, as a function of shear rate, as measured with the torsional coneplate rheometer eport and at room temperature and other assigned temperatures. You must determine the appropriate strains to stay in the linear-viscoelastic regime Check to see if the Cox-Merz rule holds for PDMS. Cox-Merz ule * ( ) ( ) See memo for additional objectives (time-temperature superposition) 31 16