The Loss Modulus

The modulus G iw) is defined as the stress 90° out of phase with the strain divided by the strain it is a measure of the energy dissipated or lost as heat per cycle of sinusoidal deformation, when different systems are compared at the same strain amplitude. It is plotted with logarithmic scales in Fig. 2-4. 

At high frequencies, a mechanical model such as Fig. 1-9 would be expected to approach perfect elastic behavior, as the motion of the dashpots became negligible 

At very low frequencies, G for a viscoelastic liquid should be directly proportional to CO, with a slope of 1 on a logarithmic plot. This is evident in Examples I, II, and III. The proportionality constant is the Newtonian steady-flow viscosity Tjo. as shown below. For a simple Newtonian liquid, G = cot o over the entire frequency range this would be represented by a straight line with unit slope for the solvent of the dilute solution. Example I. 

For a viscoelastic solid with linear viscoelasticity corresponding to a model with springs and dashpots such as Fig. 1 -9 or 1 -10, G should also be directly proportional to CO at very low frequencies. Such behavior is not observed for the examples on the right side of Fig. 2-4 experiments have never been carried to sufficiently low frequencies to test this prediction. 

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The loss tangent rather than the loss modulus is plotted, also at 1 Hz. 

Chains of polybutadiene were trapped in the network formed by cooling a butadiene-styrene copolymer until phase separation occurred for the styrene, effectively crosslinking the copolymer. At 25°C the loss modulus shows a maximum which is associated with the free chains. This maximum occurst at the following frequencies for the indicated molecular weights of polybutadiene ... 

Relaxations of a-PVDF have been investigated by various methods including dielectric, dynamic mechanical, nmr, dilatometric, and piezoelectric and reviewed (3). Significant relaxation ranges are seen in the loss-modulus curve of the dynamic mechanical spectmm for a-PVDF at about 100°C (a ), 50°C (a ), —38° C (P), and —70° C (y). PVDF relaxation temperatures are rather complex because the behavior of PVDF varies with thermal or mechanical history and with the testing methodology (131). 

Fig. 18. Resolution of the complex modulus G into two vectors, G the storage modulus, and G the loss modulus the phase angle is 5.

G is called the loss modulus. It arises from the out-of-phase components of y and T and is associated with viscous energy dissipation, ie, damping. The ratio of G and G gives another measure of damping, the dissipation factor or loss tangent (often just called tan 5), which is the ratio of energy dissipated to energy stored (eq. 16). 

A viscoelastic material also possesses a complex dynamic viscosity, rj = rj - - iv( and it can be shown that r = G jiuj-, rj = G juj and rj = G ju), where CO is the angular frequency. The parameter Tj is useful for many viscoelastic fluids in that a plot of its absolute value Tj vs angular frequency in radians/s is often numerically similar to a plot of shear viscosity Tj vs shear rate. This correspondence is known as the Cox-Merz empirical relationship. The parameter Tj is called the dynamic viscosity and is related to G the loss modulus the parameter Tj does not deal with viscosity, but is a measure of elasticity. 

Another resonant frequency instmment is the TA Instmments dynamic mechanical analy2er (DMA). A bar-like specimen is clamped between two pivoted arms and sinusoidally oscillated at its resonant frequency with an ampHtude selected by the operator. An amount of energy equal to that dissipated by the specimen is added on each cycle to maintain a constant ampHtude. The flexural modulus, E is calculated from the resonant frequency, and the makeup energy represents a damping function, which can be related to the loss modulus, E". A newer version of this instmment, the TA Instmments 983 DMA, can also make measurements at fixed frequencies as weU as creep and stress—relaxation measurements. 

Similar information can be obtained from analysis by dynamic mechanical thermal analysis (dmta). Dmta measures the deformation of a material in response to vibrational forces. The dynamic modulus, the loss modulus, and a mechanical damping are deterrnined from such measurements. Detailed information on the theory of dmta is given (128). 

Experimentally DMTA is carried out on a small specimen of polymer held in a temperature-controlled chamber. The specimen is subjected to a sinusoidal mechanical loading (stress), which induces a corresponding extension (strain) in the material. The technique of DMTA essentially uses these measurements to evaluate a property known as the complex dynamic modulus, , which is resolved into two component parts, the storage modulus, E and the loss modulus, E . Mathematically these moduli are out of phase by an angle 5, the ratio of these moduli being defined as tan 5, Le. 

The value of this latter parameter is proportional to the energy dissipated as heat per cycle, and is known as the loss modulus. The former quantity, Gj, is proportional to the recoverable energy, and is called the storage modulus. The two are combined to form the complex modulus, G related by the equation... 

Consistent with historical results, the loss modulus at high strain correlates well with laboratory abrasion results. The best correlations occur at high strains, i.e., on the order of 50%-125%. Abrasion losses are considered to be predominantly high-strain events. Thus, the ability of a compound to dissipate energy at high strain will improve the toughness or abrasion resistance of the compound. However, it is important in the tire industry that this increase in hysteresis at high... 

The majority of the mechanical ectra revealed the typical behaviour of gels [9] The storage modulus only minimally increased when the frequency rised. However, the loss modulus was independent of the frequency in the limited range up to 0.1 Hz only. Within that range, the ratio between G and G was higher than 10 1. The mechanical spectra were similar to those of other HMP/sucrose gels [13], In order to conq>are the gels, the values at 0.01 Hz were used because of the independency of frequency for both moduli. 

The extent of the solid-like character, i.e. the strength of the samples, can be directly described by the storage modulus G. As both moduli rised during gelation, it seemed to be more efficient to take the ratio of G to G , describing the dominant elastic character of the viscoelastic sample, as a second characteristic quantity than to use the loss modulus G" itself. 

There are also some far-fetched proposals for the LST a maximum in tan S [151] or a maximum in G" [152] at LST. However, these expectations are not consistent with the observed behavior. The G" maximum seems to occur much beyond the gel point. It also has been proposed that the gel point may be reached when the storage modulus equals the loss modulus, G = G" [153,154], but this is contradicted by the observation that the G — G" crossover depends on the specific choice of frequency [154], Obviously, the gel point cannot depend on the probing frequency. Chambon and Winter [5, 6], however, showed that there is one exception for the special group of materials with a relaxation exponent value n = 0.5, the loss tangent becomes unity, tan Sc = 1, and the G — G" crossover coincides with the gel point. This shows that the crossover G = G" does not in general coincide with the LST. 

TANDEL is the loss tangent, GSP and GDP is the loss modulus. is the storage modulus... 

Tan landa, a damping term, is a measure of the ratio of energy dissipated as heat to the maximum energy stored in the material during one cycle of oscillation. For small to medium amounts of damping. G is the same as the shear modulus measured by other methods at comparable time scales. The loss modulus G" is directly proportional to the heat H dissipated per... 

Now these expressions describe the frequency dependence of the stress with respect to the strain. It is normal to represent these as two moduli which determine the component of stress in phase with the applied strain (storage modulus) and the component out of phase by 90°. The functions have some identifying features. As the frequency increases, the loss modulus at first increases from zero to G/2 and then reduces to zero giving the bell-shaped curve in Figure 4.7. The maximum in the curve and crossover point between storage and loss moduli occurs at im. 

The term in phase with the rate of strain is related to the loss modulus and the term out of phase is related to the storage modulus ... 

For a viscoelastic solid, the loss modulus which reflects the viscous processes in the material is unaffected by the presence of a spring without a dashpot. The storage modulus includes the elastic component G(0) ... 

In the limit of high frequencies the integral for the loss modulus tends to zero as the denominator in Equation 4.50 tends to infinity. The storage modulus tends to G(oo) which is just the integral under the relaxation spectrum ... 

Now the first term on the right is in phase with the applied strain, i.e. it has the form of a sine wave. This can be equated with the storage modulus. Conversely the phase difference between the second term on the right and the applied signal is the difference between sine and cosine waves which can be equated with the loss modulus ... 

At low frequencies the loss modulus is linear in frequency and the storage modulus is quadratic for both models. As the frequency exceeds the reciprocal of the relaxation time ii the Rouse model approaches a square root dependence on frequency. The Zimm model varies as the 2/3rd power in frequency. At high frequencies there is some experimental evidence that suggests the storage modulus reaches a plateau value. The loss modulus has a linear dependence on frequency with a slope controlled by the solvent viscosity. Hearst and Tschoegl32 have both illustrated how a parameter h can be introduced into a bead spring... 

However, in such a high concentration regime we can no longer represent the relaxation times (Equation (5.92)) in terms of the intrinsic viscosity. In the low frequency limit, because there is no permanent crosslinking present, the loss modulus divided by the frequency should equate with ... 

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