动态力学分析

236 6 Dynamic Mechanical Analysis 6 Dynamic Mechanical Analysis (DMA) 6.1 Principles of DMA 6.1.1 Introduction Dynamic mechanical analysis (DMA) yields information about the mechanical properties of a specimen placed in minor, usually sinusoidal, oscillation as a function of time and temperature by subjecting it to a small, usually sinusidal, oscillating force. The applied mechanical load, that is, stress, elicits a corresponding strain (deformation) whose amplitude and phase shift can be determined, Fig. 6.1. ISO 6721-1 [1] states that the mode of deformation governs whether the complex modulus is E*, G*, K*, or L*. The other relationships are shown below for the elastic modulus E, see also ASTM D 4092 [2]. δ ωt stress σ strain ε σ ε , σA εΑ Fig. 6.1 Sinusoidal oscillation and response of a linear-viscoelastic material; G = phase angle, E = tensile modulus, G = shear modulus, K = bulk compression modulus, L = uniaxial- strain modulus � The complex modulus E* is the ratio of the stress amplitude to the strain amplitude and represents the stiffness of the material. The magnitude of the complex modulus is: A A*E H V The complex modulus is composed of the storage modulus E´ (real part) and the loss modulus E´´ (imaginary part), Fig. 6.2. These are dynamic elastic characteristics and are material-specific; their magnitude depends critically on the frequency as well as the measuring conditions and history of the specimen. In the linear-viscoelastic range, the stress response has the same frequency (Z = 2Sf) as the deformation input excitation. The analytical parameters in dynamic tests are the amplitudes 6.1 Principles of DMA 237 of the deformation and the stress, and the time displacement G�Z between deformation and stress, and are used to determine the specimen’s characteristics [3]. A A*E H V > @ > @ 2 2 )(''E)('E*E Z�Z G˜ Z cos*E)('E G˜ Z sin*E)(''E )('E )(''Etan Z Z G Fig. 6.2 Formulae for calculating complex modulus E*, storage modulus E´, loss modulus E´´ and loss factor tan G [1, 2]G � According to ISO 6721-1 [1], the storage modulus E´ represents the stiffness of a visco- elastic material and is proportional to the energy stored during a loading cycle. It is roughly equal to the elastic modulus for a single, rapid stress at low load and reversible deformation, and is thus largely equivalent to the tabulated figures quoted in DIN 53457. In the same ISO standard [1], the loss modulus E´´ is defined as being proportional to the energy dissipated during one loading cycle. It represents, for example, energy lost as heat, and is a measure of vibrational energy that has been converted during vibration and that cannot be recovered. According to [1], modulus values are expressed in MPa, but N/mm2 are sometimes used. The real part of the modulus may be used for assessing the elastic properties, and the imaginary part for the viscous properties [3]. The phase angle G is the phase difference between the dynamic stress and the dynamic strain in a viscoelastic material subjected to a sinusoidal oscillation. The phase angle is expressed in radians (rad) [1]. The loss factor tan G is the ratio of loss modulus to storage modulus [1]. It is a measure of the energy lost, expressed in terms of the recoverable energy, and represents mechanical damping or internal friction in a viscoelastic system. The loss factor tan G is expressed as a dimensionless number. A high tan G value is indicative of a material that has a high, nonelastic strain component, while a low value indicates one that is more elastic. 238 6 Dynamic Mechanical Analysis Curves show the change in complex modulus E*, storage modulus E´, loss modulus E´´, and loss factor tan G� � In a purely elastic material (Fig. 6.3), the stress and deformation are in phase (G = 0), that is, the complex modulus E* is the ratio of the stress amplitude to the deformation amplitude and is equivalent to the storage modulus E´ (G = 0, therefore cosine 0 = 1; sine 0 = 0, therefore E* = E´). Steel is an example of an almost purely elastic material. In a purely viscous material, such as a liquid, the phase angle is 90°. In this case, E* is equal to the loss modulus E´´, the viscous part. ωt stress σ strain ε σ ε , σA εΑ Fig. 6.3 Stress–strain behavior of a purely elastic material � Viscous and elastic properties measured Typical curves of the changes undergone by amorphous thermoplastics are shown in Fig. 6.4. At low temperatures, the molecules are so immobile that they are unable to resonate with the oscillatory loads and therefore remain stiff. The macromolecular segments cannot change shape, especially through rotation about C–C bonds, and so the molecular entangle- ments act as rigid crosslinks. At elevated temperatures, the molecular segments become readily mobile and have no difficulty resonating with the load. The entanglements more or less remain firmly in place, but may occasionally slip and become disentangled. Thermosets and elastomers have additional chemical crosslinks that are retained no matter what the temperature is. Weakly crosslinked rubber has one crosslink for every 1000 atoms while cured, brittle thermosets have one for every 20 atoms. 6.1 Principles of DMA 239 Temperature/ Time C o m p le x m o d u lu s E * S to ra g e m o d u lu s E ' L o s s f a c to r ta n δ L o s s m o d u lu s E '' E*;E' E'' tan δ energy elastic entropy elastic state stateglass transition Fig. 6.4 Schematic diagram of typical DMA curves for an amorphous polymer The material is said to be in the glass state or energy elastic state at the low temperatures described above, and in the rubber or entropy elastic state at the elevated temperatures mentioned there. A change from the glass state into the rubber-elastic state is called the glass transition. When the timescale of molecular motion coincides with that of mechanical deformation, each oscillation is converted into the maximum-possible internal friction and nonelastic deformation. The loss modulus, which is a measure of this dissipated energy, also reaches a maximum. In the glass transition region, the storage modulus falls during heating to a level of one-thousandth to one ten-thousandth of its original value. Because the loss factor is the ratio of the loss modulus to the storage modulus, the drop in storage modulus suppresses the rise in the loss factor initially; the temperature at which the loss factor is a maximum is therefore higher than the temperature corresponding to maximum loss modulus. � Temperature of tan Gmax is always higher than that of E´´max. � In DMA measurements, the design of the apparatus dictates that the applied loads be small. Consequently, the materials exhibit an almost purely elastic or, at least, a linear-viscoelastic response. Because the main difference between the complex modulus and the storage modulus is the nonelastic part, the smaller the nonelastic part, the smaller the difference. E* then becomes equal to E´. Only in the glass transition, where nonelastic deformation per oscillation is a maximum, does the difference manifest itself, showing up as a decline several degrees Celsius earlier than expected. When the results of a DMA measurement are being translated to real parts, it must always be remembered that, as the magnitude and duration of loading increase, events such as the 240 6 Dynamic Mechanical Analysis glass transition occur a few degrees Celsius earlier than the DMA measurement indicates [4]. 6.1.2 Measuring Principle There are basically two types of DMA measurement. Deformation-controlled tests apply a sinusoidal deformation to the specimen and measure the stress. Force-controlled tests apply a dynamic sinusoidal stress and measure the deformation. Dynamic load may essentially be achieved in free vibration or in forced vibration. There are two designs of apparatus: í Torsion type, í Bending, tension, compression, shear type. 6.1.2.1 Free Vibration In free torsional vibration, one end of the specimen is clamped firmly while a torsion vibration disc at the other end is made to oscillate freely. The resultant frequency and amplitude of the oscillations, along with the specimen’s dimensions, are used to calculate the torsion modulus. Measurements are conducted at various temperatures to establish how the torsion modulus varies with temperature. The term torsion modulus is intended to convey the idea that the stress is not necessarily purely shear and that the observation is not necessarily a shear modulus (except in the case of cylinders). On being twisted, the flat, clamped specimen is placed in torsional stress and, to an extent depending on the way it is clamped and on its shape, its two free edges are placed in tension and its center is placed in compression. Another free-vibration method is flexural vibration. In this, the specimen is firmly clamped between two parallel oscillation arms. One arm keeps the specimen oscillating so that the system attains a resonance frequency of almost constant amplitude. The modulus is calculated from the resonance frequency, the resultant amplitude, and the dimensions of the specimen. Free-vibration apparatus (resonant) is highly sensitive and is eminently suitable for studying weak effects. The disadvantage is a drop in frequency combined with falling modulus due to elevated temperatures [5]. Frequency-dependent measurements are difficult to perform and require the use of different test-piece geometries. They thus entail considerable experimental outlay. For more information see also ASTM D 5279 [6], ISO 6721-7 [7], ISO 6721-2 [8] and ISO 6721-3 [9]. 6.1.2.2 Forced Vibration (Non-resonant) Variable frequency apparatus applies a constant amplitude (stress or deformation amplitude). The frequency may be varied during the measurement. Figure 6.5 shows the design of a torsion apparatus. 6.1 Principles of DMA 241 coupling drive air rotary bearing normal force transducer heater magnet system clamps sample vertical position control Fig. 6.5 Schematic design of a torsion vibration apparatus with variable frequency [10] � Firmly clamped at both ends, the specimen is electromagnetically excited into sinusoidal oscillation of defined amplitude and frequency. Because of the damping properties of the material or specimen, the torque lags behind the deformation by a value equal to the phase angle G� see Fig. 6.1. The observed values for torque, phase angle, and geometry constant of the specimen may be substituted into the formulae listed above to calculate the complex modulus G*, the storage modulus G´, the loss modulus G´´, and the loss factor tan G.G The specimen should be dimensionally stable and of rectangular or cylindrical cross section. Suitable specimens have modulus values ranging from very high (fiber composites) down to low (elastomers). If appropriate plane-parallel plates are attached to the drive shafts, it is also possible to measure soft, gelatinous substances and viscous liquids [11, 12]. Most types of apparatus utilize vertical loading, which allows measurements under bending, tension, compression, and shear. Usually, the same apparatus is employed, with inter- changeable clamping mechanisms applying the various types of load, Fig. 6.5. In three-point bending, the ends are freely supported and the load is applied to the midpoint, [13, 14]. To ensure direct contact with the specimen, an additional inertial member needs to be applied. This test arrangement is suitable for very stiff materials, such as metals, ceramics, and composites. It is unsuitable for amorphous polymers because they soften extensively above the Tg. It is a simple arrangement, but additional shear stress in the midpoint plane of the specimen must be taken into account. With short specimens, this gives rise to interlaminar shear stress in the neutral, usually shear-soft plane. The effect can be reduced by employing either an appropriate length/thickness ratio or a four-point arrange- ment, which is usually more complicated. Provided that bending specimens are firmly clamped, it is also possible to measure amorphous thermoplastics above the Tg. The specimen is clamped to both supports and, at its midpoint, to the push-rod (dual cantilever bending stress) [15]. Consequently, no inertial 242 6 Dynamic Mechanical Analysis member is needed. This test arrangement is used for reinforced thermosets, thermoplastics, and elastomers. Specimens that expand considerably when heated may distort in a dual cantilever arrangement, and this can falsify the readings. For such specimens, a single- cantilever or freely supported arrangement is best. While a specimen experiences alternate compression and tension mainly along its length in flexural loading, it experiences homogeneous stress down its longitudinal axis when either tension or compression is applied exclusively. Tensile mode is ideal for examining thin specimens, such as films and fibers in the low-to- medium modulus range. Clamped at top and bottom, the specimen is subjected to an underlying tensile stress to prevent it from buckling during dynamic loading [16, 17]. sample heater strain gauge 3-point bending clamped bending compression shear tension drive shaft force motor Fig. 6.6 Schematic design of a dynamic-mechanical analyzer under vertical load, and showing the various possible test arrangements � In compression mode an axial load is applied to mostly specimens held between two parallel plates [18]. Soft rubbers on gelations pastes are most suitable for this measurement. Uniaxial compression effects a one-dimensional change in the specimen’s geometry whereas bulk compression effects a three-dimensional change. Uniaxial compression can cause thin specimens to buckle. On short, thick specimens, hindered deformation at the supports makes it difficult to make accurate determination of the modulus. Like compression loading, axial shear loading is suitable for soft materials [19]. Good results are produced by a sandwich arrangement in which two specimens are subjected to cyclical shear by the displacement of a central push-rod. 6.1 Principles of DMA 243 6.1.3 Procedure and Influential Factors The stages involved in a DMA measurement are as follows: í choose a load appropriate to the problem, and a clamping device, í prepare specimen (geometry, degree of plane-parallelism), í clamp the specimen, í choose measuring parameters. Factors exerting an influence on the apparatus and specimen are: Clamps S pe ci m en g eo m et ry F requency Type of load T em pe ra tu re pr og ra m T ightening torque Influential factors The influential factors and sources of error associated with experiments are explained in detail in Section 6.2.2 with the aid of curves plots from real-life examples. 6.1.4 Evaluation Because DMA is sensitive to variations in the stiffness of a material, it may be used to deter- mine not only modulus and damping values directly, but also glass transition temperatures. It is particularly suitable for determining glass transitions because the change in modulus is much more pronounced in DMA than, for example, the cp change in a DSC measurement [20]. Owing to discrepancies between the proposals made in various standards and the information provided by apparatus manufacturers, confusion has arisen about how to determine and state glass transition temperatures in practice. Although the modulus step that occurs during the glass transition can be evaluated much in the manner of a DSC curve, it is difficult to do so in practice. This and other methods of determination are described below. Methods of determining the glass transition temperature: Evaluation of modulus step: í Step method employed for DSC curves, (start, half step height and end of glass transition), 244 6 Dynamic Mechanical Analysis í Inflection point method, í 2% offset method as set out in [21] (start of glass transition), í Tangent method as set out in [21] (start of glass transition). Evaluation of peaks from plots of loss factor and loss modulus: í Maximum loss factor, í Maximum loss modulus. 6.1.4.1 Methods of Evaluating the Modulus Step The use of the modulus step to determine the glass transition temperature is based on the standardized DSC method (ISO 11357-1 [22], Fig. 6.7) and involves ascertaining the onset, end, and midpoint temperatures. S to ra g e m o d u lu s G ' [ M P a ] Temperature [°C] S to ra g e m o d u lu s G ' [ M P a ] h 1 /2 h Tmg Teig Tefg linear plot logarithmic plot Tmg Teig Tefg Teig Extrapolated onset temperature, onset temperature: Intersection of the inflectional tangent with the tangent extrapolated from temperatures below the glass transition Tmg Midpoint temperature, glass transition temperature: Temperature of the midpoint of the inflectional tangent (half step height between Teig and Tefg), projected onto the DMA curve Tefg Extrapolated end temperature: Intersection of the inflectional tangent with the tangent extrapolated from temperatures above the glass transition Fig. 6.7 Step evaluation based on the standardized DSC evaluation Teig = Extrapolated onset temperature Tmg = Midpoint temperature Tefg = Extrapolated end temperature, linear and logarithmic plots of modulus 6.1 Principles of DMA 245 Tangents are applied to the sections of the curve above and below the glass transition step. An inflectional tangent applied to the step intersects with both these tangents at the extrapolated onset temperature Teig and the extrapolated end temperature Tefg. The midpoint temperature Tmg is determined from the half step height. The user chooses the temperatures at which to apply the tangents to be determined – this practice differs from that described in [21]. A regression curve is then calculated from values above and below these temperatures. The position of the resultant characteristic curve depends critically on how the modulus is plotted against the temperature. For step evaluation methods, logarithmic plots yield much higher Tg values than linear plots. These two different ways of plotting storage modulus have their advantages and disadvantages: Logarithmic plot Linear plot Below Tg: – Seemingly slight, often linear dependence on temperature – Tangents usually easy to apply – Clear, heavy dependence on temperature – Difficult to apply tangents, especially with nonlinear curves – Choice of contact temperatures for the tangents on the curve is subjective Logarithmic plot Linear plot Above Tg: – Mathematical stretching and thus steeper curves – Tangents difficult to apply – As for logarithmic plot below Tg Tmg: – Half step height is a linear measurement in a logarithmic plot: not appropriate – Half step height appropriate; founded more on physical structure than on technical design reasons. Tmg > Tmg Logarithmic plot of modulus: Tmg higher – subjective Linear plot of modulus: Tmg lower – appropriate � Note: Most users are interested in knowing the highest service temperature of a particular polymer. Generally the extrapolated onset temperature may be quoted for this. As the interpretation of this is partly subjective, it is preferable to use the glass transition temperature defined, in, for example, standards and procedures. Note, however, that the polymer often cannot be used at this temperature in practice. 246 6 Dynamic Mechanical Analysis In the turning point method, the glass transition is identified by mathematically identify