Storage Compliances Research Articles

Direct measurement of storage and loss behavior in AFM force–distance experiments using the modified Fourier transformation

Force–distance curve experiments are commonly performed in atomic force microscopy (AFM) to obtain the viscoelastic characteristics of materials, such as the storage and loss moduli or compliances. The classic methods used to obtain these characteristics consist of fitting a viscoelastic material model to the experimentally obtained AFM data. Here, we demonstrate a new method that utilizes the modified discrete Fourier transform to approximate the storage and loss behavior of a material directly from the data, without the need for a fit. Additionally, one may still fit a model to the resulting storage and loss behavior if a parameterized description of the material is desired. In contrast to fitting the data to a model chosen a priori, departing from a model-free description of the material's frequency behavior guides the selection of the model, such that the user may choose the one that is most appropriate for the particular material under study. To this end, we also include modified Fourier domain descriptions of commonly used viscoelastic models.

A simple method to analyze materials under quasilinear large amplitude oscillatory shear flow (QL-LAOS)

A new method to analyze materials exploring the quasilinear oscillatory regime has been described. One of its advantages is a significant reduction of the number of rheological tests needed, as compared to the originally proposed quasilinear large-amplitude oscillatory shear flow method. Furthermore, the authors improved the data analysis to a great extent, rendering it quite straightforward and physically sound. To validate the new method, we used two materials of qualitatively different rheological behavior, namely, a commercial hair gel and a solution of polyacrylamide in water/glycerol. The data obtained are presented in two distinct forms. The first one uses either the traditional Kelvin–Voigt analysis—which yields as material functions the storage and loss moduli G ′ and G ″, respectively—or the traditional Maxwell analysis, which yields as material functions the storage and loss compliances J ′ and J ″, respectively. The second form originates from a Jeffreys analysis, which gives as output two material functions, namely, the relaxation time θ 1 and the retardation time θ 2. The results of the different analyses are compared and discussed.A new method to analyze materials exploring the quasilinear oscillatory regime has been described. One of its advantages is a significant reduction of the number of rheological tests needed, as compared to the originally proposed quasilinear large-amplitude oscillatory shear flow method. Furthermore, the authors improved the data analysis to a great extent, rendering it quite straightforward and physically sound. To validate the new method, we used two materials of qualitatively different rheological behavior, namely, a commercial hair gel and a solution of polyacrylamide in water/glycerol. The data obtained are presented in two distinct forms. The first one uses either the traditional Kelvin–Voigt analysis—which yields as material functions the storage and loss moduli G ′ and G ″, respectively—or the traditional Maxwell analysis, which yields as material functions the storage and loss compliances J ′ and J ″, respectively. The second form originates from a Jeffreys analysis, which gives as ou...

Characterization of Shape Memory Polymer Estane by Means of Dynamic Mechanical Thermal Analysis Technique

Commercially available shape memory polymer (SMP) Estane (designation: ETE75DT3 NAT022) is investigated by means of dynamic mechanical thermal analysis (DMTA) technique in torsion mode using the Modular Compact Rheometer MCR-301 (Anton Paar GmbH). Amplitude sweep tests have been run below and above the glass transition temperature to establish the linear viscoelastic range (LVR) in glassy and rubbery phase of this SMP for the correct physical interpretation of DMTA data. Temperature sweep tests were performed at various frequencies to study the influence of this parameter on values of the storage and loss moduli and the storage and loss compliances as well as the viscosities. These tests have been carried out in heating mode with different rates and at different strain amplitudes. The short- and long-term behavior of SMP Estane have been studied by frequency sweep tests performed at different temperatures and data have been transformed into time-domain properties by applying time-temperature superposition principles. All these DMTA data provide the experimental basis for the study of relaxation processes, property-structure relationships, and the shape memory effect in this little-known SMP.

Applicability of Anisotropic Viscoelasticity of Paper at Small Deformations

In this paper the applicability of two-dimensional phenomenlogical linear viscoelastic modelling aimed at printing applications is presented. An iterative algorithm is used to find model parameters evaluated from dynamic mechanical testing at stationary conditions. Experiments are performed in the time domain with different load stimuli in either displacement or force control. The experiments are compared to model predictions. The results show that only small deviations were found on model parameters from the algorithm used on either the loss or the storage properties of paper. This was illustrated by converting line spectra from storage compliances to loss compliances within the frequency domain. The harmonic properties were obtained at conditions with nonlinear material behaviour. Despite this, the two-dimensional linear viscoelastic model described well the behaviour of paper at different states of stress and strain at levels adequate for web-fed printing. The usefulness of the relation between retardances and relaxances was examined by comparing model predictions with different driving stimuli resulting in either creep or relaxation. The time-dependent transverse strain of machine-made paper due to either longitudinal stress relaxation or creep was demonstrated. Finally, extensions of the present model including inelastic material behaviour were discussed.

The effect of diabetes on the viscoelastic properties of rat knee ligaments.

A series of experiments was performed to determine the effect of diabetes on the viscoelastic properties of knee joint ligaments. The experimental model was collateral ligaments from spontaneously diabetic, hyperglycemic (BBZDP/Wor) rats, and various controls including nondiabetic littermates, insulin treated diabetic rats, and alloxan treated rats. Material properties were measured using a dynamic, uniaxial loading paradigm. Ligaments were subjected to load controlled, sinusoidal tensile testing, using frequencies from 0.1 to 2.0 Hz. The resulting data were used to determine the storage and loss compliances of the ligaments. Storage compliance, which reflects tissue elastic properties, did not differ between groups. Loss compliance, which reflects the viscous component of the tissue response, was increased in the hyperglycemic animals. Thus, hyperglycemic diabetes affects tissue mechanical properties through the viscous rather than the elastic component of the response to dynamic loading. Rats treated with alloxan to induce diabetes did not show an increase in loss compliance.

A rotational rheometer for rheological studies with prescribed strain or stress history

AbstractA rheometer utilizing an eddy‐current torque transducer and an air‐bearing suspension is described. The rheometer is coupled with a computer‐based data acquisition system and permits studies of shear deformation for several strain or stress histories. A sinusoidal stress history is used to determine the shear storage and loss compliances J′(ω) and J″(ω), respectively. Step stress histories are used to determine the shear creep compliance J(t) and the recoverable complaince R(t) or more complicated linear and nonlinear rheological responses related to these. Deformation at constant strain rate is used to determine the stress growth function or the steady‐state viscosity. The rheometer may be used over the temperature range – 10–180°C, with torque from 1 to 106 dyn cm, and is adaptabel to use with a variety of sample geometrical shapes (e.g., cone and plate, parallel plate, etc.). Examples of measurements on on viscoelastic fluids and on gels below their yield strain are given.

Rheology of Blends of a Rodlike Polymer (PBO) and its Flexible Chain Analog

ABSTRACTRheological properties of isotropic solutions of rodlike poly(p-phenylene benzbisoxazole), PBO, flexible chain poly(2,5-benzoxazole), ABPBO, and their miscible blends in solution are described. Measurements include steady state properties (the viscosity and recoverable compliance as functions of shear rate), transient properties (the recoverable compliance), and dynamic mechanical properties (the loss and storage compliances as functions of frequency). The relaxation spectrum of the blends is broader than that for the rodlike chain, and tends to occur at longer times, reflecting a viscosity enhancement that occurs with the blends. The measured zero shear viscosities for rod and blend solutions are compared with predictions based on the model of Doi and Edwards.

Adhesion quality and extent of mesophase in particulates along the whole viscoelastic spectrum

AbstractThe extent of mesophase, which constitutes the boundary layer between the filler and the matrix of a composite, was defined in this paper along the whole viscoelastic spectrum of the polymeric substance constituting the matrix. The already established two‐term unfolding model1–3 was based on thermodynamic measurements of the heat capacity jumps at the glass transition regions of the filled and unfilled substances in order to define the extent of mesophase. In this paper we dispense ourselves with these delicate measurements, based on Lipatov's theory, and we use values taken from the storage and loss compliances of the composite and its constituents, along the whole viscoelastic spectrum of the composite. It was shown that the extent of the mesophase, necessary to sustain the shearing loading between phases, appearing at the glass transition zone, where the loss‐compliances presented their characteristic peaks of resonance, remained constant and almost independent of the respective values of mechanical properties at this critical region. It was shown that any set of values of the mechanical characteristic quantities, defining the composite and its constituents along the whole viscoelastic spectrum, is sufficient to evaluate the extent of the mesophase, and this boundary layer undergoes only negligible variations. Experimental evidence with typical E‐glass polystyrene and other particulates, at various volume fractions of the filler, yielded a satisfactory coincidence with the results derived by using Lipatov's theory and a constancy of the mesophase thickness along the whole temperature range.

Phase transformation, at the boundary layers between polymers, and solid surfaces

The size of the mesophase, which constitutes a boundary layer between fillers and matrix in composites, has been efficiently evaluated by the modified two-term unfolding model, which was based on delicate DSC measurements of the heat capacity jumps at the glass transitions of the composite and its constituent phases [1,2]. This model is now used to evaluate the mesophase along the whole viscoelastic spectrum of the composite, by making measurements of the storage and loss compliances or moduli of the composite and matrix and without making recourse to any other type of special measurement at the glass transition temperature of the substances.

Dynamic shear properties of a technical polystyrene melt

A dynamic viscometer of the Couette type is described which permits the determination of storage and loss shear modulus of polymer melts with high accuracy. For a technical polystyrene, results of measurements of storage and loss moduli are reported in the temperature interval between the rubbery plateau and degradation. From these data, the courses of storage and loss compliances and the course of the time-dependent shear compliance are derived and compared with the results of creep measurements.

Linear thermoviscoelasticity and characterization of noncrystalline EPDM rubber networks

AbstractAn experimental study has been made to specify how the time‐temperature superposition and the linear viscoelastic characteristics vary with the degree of crosslinking for a broad class of noncrystalline peroxide‐cured EPDM networks. A new, very sensitive method is applied to determine the horizontal and vertical shift functions in an independent way. All uncrosslinked samples are thermoelasticoviscously simple with horizontal shift functions aT of the WLF type and vertical shift functions almost independent of temperature, in agreement with recent theoretical understanding. Upon crosslinking, these materials become thermoviscoelastically complex networks, but superposition can still be accomplished by assuming different temperature dependences for the relaxational strength and the equilibrium modulus. The aT functions can be taken independent of the degree of crosslinking. The vertical shift functions bT for the relaxational strength vary with the degree of crosslinking between theoretical predictions for uncrosslinked and perfectly crosslinked EPDM networks. The equilibrium moduli of the lightly cured networks decreases with increasing temperature, which is ascribed to the presence of interchain associations between ethylene sequences in the trans state. Upon further crosslinking, these effects gradually vanish and eventually the networks can be described as viscoelastically simple with an energy elastic contribution due to the ethylene trans‐gauche transitions. The linear viscoelastic characteristics, namely the storage and loss moduli and compliances and phase‐angle master curves and the relaxation and retardation spectra are discussed as a function of the degree of crosslinking. A sol/gel analysis and equilibrium swelling measurements complete the experimental characterization of three familes of five EPDM networks each.

Determination of Dynamic Storage and Loss Compliances from Creep Data

A pair of empirical equations are proposed for the purpose of approximating the dynamic mechanical viscoelasticity functions, the storage compliance, J′(ω), and the nonviscous loss compliance, J″(ω)−1/ωη, from the value of the recoverable compliance, Jr(t), and its logarithmic slope, m(t)=d log Jr(t)/d log t. The angular frequency is ω and the shear viscosity is η. Three examples representing various kinds of polymeric, viscoelastic response are presented as tests of the effectiveness of the proposed expressions. Most of the calculated points are in error by less than 10%.

Correlation of Viscoelastic Functions for Pharmaceutical Semisolids: Comparison of Creep and Oscillatory Tests for Oil-in-Water Creams Stabilized by Mixed Emulsifiers

Liquid paraffin-in-water emulsions stabilized by mixed emulsifiers of a surfactant (cetrimide or cetomacrogol) and a long-chain alcohol were used as model systems to represent pharmaceutical semisolids. They were examined in their linear viscoelastic regions using creep and oscillatory (Weissenberg rheogoniometer) techniques. Storage and loss compliances (J′ and J″) over the frequency range 2.5 × 10−8 to 1.5 × 10−1 Hz. were calculated from creep data using numerical formulas. The transformed compliances were used to derive the other oscillatory functions, the storage and loss moduli (G′ and G″), the storage and loss viscosities (η′ and η″), and the loss tangent (tan a) at the same frequencies. The agreement between transformed functions and those obtained directly from oscillatory measurements at frequencies ranging from 7.91 × 10−4 to 1.5 × 10−1 Hz. was generally good. Discrepancies at the extremities of this range were a result of the practical difficulties involved when creep measurements are made at very short times (representing high frequencies) and evaporation of water from the parallel plates of the rheogoniometer during low frequency dynamic experiments. Neither small strain technique alone covered the entire frequency or time range necessary for an exhaustive study of semisolids. Creep data were unavoidably inaccurate at high frequencies, and the limitations of the rheogoniometer gear box prevented oscillatory measurements from being made at extremely low frequencies. The unification of data from oscillatory and creep measurements so as to provide a dynamic description over an extended frequency range (10−8–25 Hz.) is discussed. The variations of each function with frequency are considered with reference to linear viscoelastic theory; the limiting values and the shapes of the plots at low frequencies agreed well with the theoretical values for viscoelastic liquids. It was concluded that such interconversion techniques are applicable and valuable for interpreting the rheological behavior of a pharmaceutical system.

Viscoelastic properities of cross-linked solutions of poly (β-hydroxyethyl methacrylate) in diethylene glycol. V. Influence of secondary low-temperature transitions

Abstract The influence of secondary low-temperature dispersions on the storage and loss compliances of four poly(β-hydroxyethyl methacry-late) solutions in diethylene glycol was studied by comparing previously published double-transducer data with isochronal torsion pendulum data measured starting from the temperature of liquid nitrogen. It was found that the contribution of the main transition as observed in the storage compliance J'p decreased with increasing swelling degree of the samples. For a sample swollen to volume fraction of polymer v2 = 0.168, log Jp changes only about one decade, from −6 to −7; the further decrease of log J′p from −7 to −10.5 corresponds to the secondary low-temperature dispersion characteristic for polar side chains—polar diluent systems. For the latter dispersion the Arrhenius plot obtained using the double-transducer data gave δH = 24 kcal/mole. It was concluded that in some special cases the role of low-temperature secondary dispersions can be of primary importance in infl...

Investigation of Semisolid Lipophilic Preparations by Small Strain and Continuous Shear Viscometry and Their Application to Texture Profile

The effects of grade variation of white soft paraffin on the rheological properties of paraffin ointment BP, simple ointment BP, and some nonpharmacopoeial paraffinic and emulsion preparations were investigated by continuous shear and creep viscometry. The creep curves were analyzed to determine discrete and continuous spectra of retardation times and transformed mathematically to determine dynamic parameters such as storage and loss compliances. The elastic properties of the white soft paraffins influenced the properties of preparations made with these paraffins. There was no general correlation between viscosity data for the preparations, possibly due to some specific interaction such as that described for hard paraffin and liquid paraffin. Dilution of white soft paraffin with liquid paraffin shortened the retardation times but did not significantly alter the type of retardation spectrum involved (bimodal), while emulsification radically altered the retardation mechanisms. The utility of the types of viscometry and various analyses of the creep curves is discussed in terms of correlation with sensory data for texture profile evaluation. It is concluded that small strain experiments are important in texture profile evaluation.

Dynamic mechanical properties of two copolymers of styrene and n‐hexyl methacrylate

AbstractThe storage (J′) and loss (J″) shear compliances have been measured for two random copolymers of styrene and n‐hexyl methacrylate with styrene contents of 18% and 30% (by weight) in the frequency range 45–4400 Hz and the temperature range 31–107°C. The data at different temperatures were combined by the method of reduced variables, and the WLF coefficients were calculated from the temperature shift factors by the method of Pierson and Kovacs. The data were compared with earlier data for the two homopolymers. The thermal expansion coefficient of the fractional free volume, and the free volume at the glass transition temperature, varied monotonically with composition, but the fractional free volume at a reference temperature of 100°C appeared to pass through a maximum as a function of concentration. Comparison of isothermal plots of J′ at 100°C, plots of the relaxation spectrum at 100°C, the monomer friction coefficient and its temperature dependence, and isochronal plots of the storage shear moduls at 100 radians/see all show that the properties of poly(n‐hexyl methacrylate) are very slightly affected by incorporation of 18% styrene and only moderately affected by 30% styrene. By contrast, comparison of styrene–butadiene rubber with 1,4‐polybutadiene shows a very large effect of incorporation of 23.5% styrene. These differences may be associated with local packing relations of the comonomer residues and suggest that copolymer properties cannot be readily predicted from those of the component homopolymers.

Dynamic mechanical properties of poly(n‐butyl methacrylate) near its glass transition temperature

AbstractThe storage and loss shear compliances (J′, J″) of poly(n‐butyl methacrylate) (number‐average molecular weight 320,000) were measured at frequencies between 0.08 and 1.0 cycle/sec. in the temperature range of 18–33°C., encompassing the glass transition. Data were obtained both at voluminal equilibrium and as a function of time during the slow isothermal contraction which follows a quench to a temperature near the transition. The latter data were supplemented by dilatometric measurements on similar samples to determine the magnitude of the volume contraction. Correlation of the dependences of J′ and J″ on temperature and on elapsed time (at constant frequency) indicated that the relaxation mechanisms are controlled primarily by free volume. The method of reduced variables was successfully applied to the dependence of J′ on temperature and elapsed time using shift factors aT,t calculated from the modified Doolittle equation and the magnitudes of volume contraction measured dilatometrically. The dependence of J″ on both temperature and elapsed time was more complicated, indicating a change in the shape of the retardation spectrum with free volume. The behavior is qualitatively very similar to that of poly(vinyl acetate), previously studied, except that the magnitudes of J′ are considerably larger and the thermal expansion coefficient of the free volume is smaller.