The News and Information Publication of The Society of Rheology Volume 76 Number 2 July 2007 Rheology Bulletin Inside: John Brady is 2007 Bingham Medalist SOR2007: Salt Lake City, Utah USA Phase Angle in Oscillatory Testing ICR2008 Monterey USA Technical Program 1 How do I know if my phase angles δδ are correct ? δδ? ? Sachin S. Velankar, University of Pittsburgh and David Giles, University of Minnesota validating phase angle measurements close to 90 degrees. 1. Introduction This is important for practical purposes, not least because δ approaching 90 degrees corresponds to the terminal A large fraction of rheological testing involves small- region of most viscoelastic materials, where significant amplitude oscillatory shear, sometimes referred to as connections between the rheology and structure can be “dynamic” experiments, which yield the complex modulus made. For example, fluids such as entangled polymer G* of the sample. The results of such tests are often ′ melts and polymer solutions generally show “standard” represented in terms of the storage and loss moduli G and ′ ω2 ′′ ω ′′ liquid-like terminal behavior (G ~ and G ~ ), which G , but it is conceptually useful to instead think of G* in can be often be related quantitatively to specific δ terms of its magnitude |G*| and its phase angle . How microscale dynamic processes. In other cases, e.g. block can these quantities be validated? If a rheometer reports, copolymers, gels, or particle-filled systems, we are often δ say, |G*| = 12,755 Pa, and = 88.78 degrees, how closely interested in deviations from standard liquid-like terminal can the user trust these values? behavior to help identify, for example, an order-disorder 6 The magnitude |G*| can be validated easily with transition or a liquid-to-gel transition. A recent article in Newtonian calibration standards of known viscosity. the Rheology Bulletin has also discussed the challenge of There have also been excellent studies of the measuring terminal viscoelastic properties in the context reproducibility |G*| of molten polymers, which is of calculating the relaxation function G(t). 1-5 important for quality control applications of rheometry . When attempting to validate phase angle measurements, a Even in the absence of any rigorous validation, the significant problem is the choice of a viscoelastic standard specifications on the torque, angular displacement, and for calibration: while viscosity standards are readily frequency range of the rheometer provide guidance on available, materials with standard viscoelastic properties when the range or sensitivity of the instrument is are less easy to come by. In the absence of standards, exceeded, and hence large errors in |G*| can be expected. some laboratories validate a δ = 90 degrees using a In contrast, it is much more difficult to judge the accuracy Newtonian fluid, δ = 0 degrees using a steel test specimen, of the phase angle, δ. We are not aware of studies that and presume that phase angles between 0 and 90 degrees evaluate the accuracy of phase angle measurements. are accurate. Some rheology labs have a standard PDMS Furthermore, rheometer manufacturers do not provide putty with a broad spectrum of relaxation times whose specifications for phase angle, and hence there is no clear phase angle at a particular temperature and frequency has guidance on when the range or resolution of the rheometer been specified by the supplier. The National Institute for is exceeded. It is not difficult to understand why phase Standards and Technology (NIST) has developed standard angle specifications are generally missing: phase angle reference materials for viscoelastic measurements, most resolution and accuracy will surely depend on the quality recently the SRM 2490 and SRM 2491 polymer solution 7,8 of the torque and angular displacement signals. and melt . However most laboratories do not have these Approaching the low end of the torque or displacement fluids, and some questions have also been raised about the range, it might still be possible to measure the average validity of the viscoelastic specifications given for SRM 9 magnitudes of these signals (allowing determination of 2490 . It would be very useful for experimental |G*| within some well-defined, reasonable certainty), while rheologists to have a testing protocol, along with different the signal quality becomes too poor to accurately resolve viscoelastic calibration standards, to validate oscillatory the phase shift between them. How can a rheologist measurements on their own rheometers. establish conditions under which his or her rheometer can Here we show that a linear, monodisperse, and well- measure phase angles within some well-defined certainty? entangled polymer melt can serve as an excellent A broad validation of phase angle measurements is beyond viscoelastic calibration standard when δ is close to 90 the scope of this short article. We will concentrate on degrees. Such a material has a sharp transition to its 8 Rheology Bulletin, 76(2) July 2007 terminal region as frequency is reduced (see details ), and d G* Similarly, dG′′ dδ (7) thus is in the terminal region even when δ is still far from = + ′′ * ()δ 90 degrees, and hence still easy to measure. Phase angles G G tan close to 90 degrees can then be validated by verifying that the rheometer reproduces the expected terminal behavior In Eqs. 6 and 7 above, the left hand side is the fractional (tanδ proportional to 1/ω) at lower frequencies. error in G′ and G″, the first term on the right hand side is the fractional error in the magnitude of the complex In essence, this phase angle validation is a test of self- modulus, and the last term reflects the error in measuring consistency of the rheometer: (1) use the rheometer to phase angle. It is clear that as δ approaches 90 degrees, δ characterize the terminal behavior of the fluid when is the fractional error in G′ grows without bound even when not very close to 90 degrees, and (2) use the now- the error in phase angle dδ remains finite. This is just a characterized terminal behavior to test the rheometer quantitative way of stating that it is difficult to δ performance under adverse conditions (e.g. approaching characterize the elasticity of weakly-viscoelastic materials. 90 degrees closely, lower displacements, lower torques, The situation is reversed when δ approaches 0 degrees: etc.) then the error in G″ becomes large, whereas the error on G′ remains finite. In short, when one modulus is much 2. A brief primer on dynamic oscillatory smaller than the other, little of the total signal comes from measurements the response associated with the smaller modulus, and Theory more error is likely in its measurement. Here, we will focus on the case of δ approaching 90 degrees, where the Most dynamic oscillatory tests are performed in controlled elasticity is becoming very weak. strain mode. A sinusoidally-varying strain: The above equations suggest that rather than δ itself, it is γ = γ ()ω 0 sin t (1) better to work in terms of tanδ (or 1/tanδ) since these ω γ directly relate to the accuracy and precision of the dynamic is imposed on the sample. Here is the frequency and 0 moduli. We will reiterate this point at the end of the is the strain amplitude. The stress in the sample follows: following section. σ = σ ()ω + δ = * γ ()ω + δ (2) 0 sin t G 0 sin t Practical measurements σ * where 0 is the stress amplitude and |G | is the magnitude Assuming a parallel plate geometry with plates of diameter of the complex modulus. Most commonly, analysis is 2R and gap of h, oscillatory measurements generally performed in terms of the storage and loss moduli: depend on applying a displacement: σ = γ []G′sin()ωt + G′′cos ()ωt (3) θ = θ ()ω 0 0 sin t (8) where G′ = G* cos()δ and G′′ = G* sin()δ (4) and measuring a torque: = ()ω + δ (9) Even if the tests are performed in controlled stress mode, T T0 sin t the same equations can be used to calculate the moduli. Here θ0 and T0 are the displacement and torque amplitudes We can take the first step in error analysis as: respectively. For the parallel plate geometry, one can obtain10: dG′ = cos()δ d G* − G* sin()δ dδ (5) Rθ σ γ = 0 ; σ = 2T0 , and hence * 0 2T0h 0 0 G = = * h πR3 γ π 4θ dG′ d G 0 R 0 = − tan()δ dδ (6) ′ * G G (10) Slightly different equations can be derived for other measurement geometries10. Thus, dynamic oscillatory measurements on rotational rheometers depend on θ δ. Any linear viscoelastic fluid can be represented as a sum of Maxwell measuring three quantities: θ0, T0 and Conceptually, modes. In a linear, monodisperse, well-entangled polymer, higher order these three quantities can be measured by plotting the relaxation modes are much faster and much weaker than the longest displacement and torque signals, as illustrated in Fig. 1. mode. Therefore the terminal region, i.e. the region in which the longest δ mode dominates, ranges from from tanδ → ∞ (δ = 90 degrees) down to The factors limiting the accuracy of tan are clear from about tanδ ≈ 10 (δ ≈ 84 degrees). In contrast, in most other fluids, this diagram: As with any experimentally-determined higher order modes become important when tanδ is still large (δ is still quantities, there are some errors associated with measuring close to 90 degrees). displacement and torque. For example, if θ0 or T0 are 10 Rheology Bulletin, 76(2) July 2007 1 1 t t t 0 0 displacement displacement torque torque -1 -1 0246810 0246810 time time Fig. 1: a. The zero crossing of the sinusoidal displacement and torque signals give the phase lag, from which the phase angle δ can be calculated.