Instrument for determining the complex shear modulus of soft-tissue-like materials from 10 to 300 Hz

Abstract

Accurate determination of the complex shear modulus of soft tissues and soft-tissue-like materials in the 10–300 Hz frequency range is very important to researchers in MR elastography and acoustic radiation force impulse (ARFI) imaging. A variety of instruments for making such measurements has been reported, but none of them is easily reproduced, and none have been tested to conform to causality via the Kramers–Kronig (K-K) relations. A promising linear oscillation instrument described in a previous brief report operates between 20 and 160 Hz, but results were not tested for conformity to the K-K relations. We have produced a similar instrument with our own version of the electronic components and have also accounted for instrumental effects on the data reduction, which is not addressed in the previous report. The improved instrument has been shown to conform to an accurate approximation of the K-K relations over the 10–300 Hz range. The K-K approximation is based on the Weichert mechanical circuit model. We also found that the sample thickness must be small enough to obtain agreement with a calibrated commercial rheometer. A complete description of the improved instrument is given, facilitating replication in other labs.

Introduction

Elastography refers to the generation of mapping of a mechanical property for regions or volumes of interest in a patient. For example, local strains can be derived from data acquired before and after a small quasi-static uniaxial stress is applied; ultrasound (US) radiofrequency echo data have been used (Ophir et al 1991, O'Donnell et al 1994, Zhu and Hall 2002) as well as MR data (Plewes et al 2000). The time dependence of the strain reacting to the stress is not considered to be a significant factor; thus, strain images generated are referred to as quasi-static. Dynamic elastography mappings are also made at discrete frequencies as high as 200 or 300 Hz using magnetic resonance (MR) techniques (Muthupillai et al 1995, Sinkus et al 2000) or US techniques (Rubens et al 1995). Transient elastography involves the application of localized acoustic radiation force using focused pulses of ultrasound; the power is sufficient to cause axial displacements resulting in the generation of detectable shear waves containing frequencies as high as 300 Hz (Bercoff et al 2004, Fatemi and Greenleaf 1998, Nightingale et al 2001). A more comprehensive discussion of the various forms of elastography is given by Zhang et al (2007).

Dynamic and transient elastography—whether shear waves are of single frequency or broadband—involve shear wave propagation and shear wave energy loss in the propagating medium. The frequency dependences of propagation speed and energy loss for targeted soft tissues are of interest to researchers regarding refinements of experimental techniques and testing of data reduction methods relative to generation of (complex) shear modulus maps for soft tissue organs. Complete characterization of an isotropic medium with respect to shear wave propagation (excluding nonlinear propagation) exists if the mass density is known and the complex shear moduli are known at the relevant frequencies. Thus, knowledge of the frequency-dependent complex shear modulus of soft tissues is of great interest to researchers in dynamic and transient elastography.

Instruments are available commercially for measuring complex shear moduli of viscoelastic materials. These are mostly rotational devices, where sinusoidal torque is applied to a sample disk and rotational strain is monitored. These instruments are not intended for use above frequencies of about 50 Hz for soft-tissue-like materials. Measurements can also be carried out using a dynamic mechanical analyzer, which measures the complex Young's modulus, E, from which the complex shear modulus, G, can be computed through the relation G = E/3 for viscoelastic materials (Ferry 1980, p 144); however, the accuracy of this type of uniaxial compression instrument is compromised at all frequencies for very soft materials, such as soft tissues (Hall et al 1997). Rotational devices are not intended for use above frequencies of about 50 Hz for soft-tissue-like materials because of to their very low stiffness.

Numerous custom-built instruments for measuring complex shear moduli, G, of viscoelastic materials—such as soft tissues—have been reported. More recent versions have usually been dedicated to characterizing normal brain as related brain trauma experienced by accident victims.

An elaborate apparatus and data reduction method has been described in detail by Fitzgerald and Ferry (1953) for measurement of G of viscoelastic materials, including soft-tissue-like materials. Measurements involved linear displacements and could be made over the frequency range 25–2500 Hz. However, the apparatus weighed about 100 kg and production of a reasonable facsimile today would likely be very challenging.

An instrument in which a right circular cylinder of brain tissue was subjected to sinusoidal torque at one end about its axis of symmetry has been reported (Shuck and Advani 1972). Complex shear moduli from 10 to 350 Hz were measured. The accuracy of this instrument is questionable, however, because a 36% decrease in the measured G′ (the shear storage modulus or real part of G) is reported between 100 and 200 Hz. Causality, as represented in an approximation for the Kramers–Kronig (K-K) relations (Tschoegl 1989, p 431, Eq. 8.3–36), implies a monotonic increase of G′ with increasing frequency for slowly varying G′ and G″ (see also equation (28)). (Note that the physical meaning of the K-K relations is discussed in the section ‘The exact Kramers–Kronig relations and approximations’ below.) Also, although the torque-producing apparatus is described (Shuck et al 1971), the complete instrumentation is not described adequately for others to reproduce.

An instrument for producing linear oscillations to measure G was reported by Arbogast et al (1997). The mechanical components are well described, including manufacturers. However, the custom-built electronic components are not described. Measurements of the absolute value of the shear modulus (∥G∥) are reported over the 20–160 Hz range for a silicone material manufactured by Dow Corning. These measurements are in good agreement with measurements on the same silicone material made on a commercial instrument over the frequency range 0.16–16 Hz and temperatures from −60 °C to +30 °C; the latter measurements can be used to derive ∥G∥ over the 20–160 Hz range using the Wiliams–Landel–Ferry (WLF) method (Ferry 1980, chap. 11; Williams et al 1955), facilitating the comparison. Note that the comparison involves only ∥G∥ so that agreement of the critical parameter G″/G′ = (imaginary part of G)/(real part of G) was not addressed. In a later article Arbogast and Margulies (1998) reported G′ and G″ values for porcine brainstem from 20 through 200 Hz. Using their curve fits to the data in figures 3 and 4 of that article, we found that the (approximate) K-K relations (Tschoegl 1989, p 431, Eq. 8.3–36) are not satisfied. In the course of this work, we derived the next higher order approximation of the K-K relations and were surprised to see that the results obtained by Arbogast and Margulies do conform to a higher order approximation (see appendix A). However, as shown in appendix A, there is a large difference between the result using lower order approximation and that using next higher order approximation, and the conformity may be lost with application of a still higher order approximation. Development of still higher order approximations is yet to be done.

Figure 3.

Front-view and top-view diagrams of instrument components.

Figure 4.

Overview diagram showing all components of the instrument.

Nicolle et al (2004) describe a custom-built instrument performing linear oscillations from 0.1 to 6300 Hz. The instrument is described with one simple figure and one paragraph with no information allowing replication of the instrument. A comparison of results using this instrument with results using a commercial system (Bohlin C-VOR 50; Malvern Instruments, Malvern, Worcestershire, UK) over the frequency range 0.1–10 Hz was done for porcine white matter. The frequency dependencies for G′ agree, but do not agree for G″. Assessment (by the present authors) for conformity with the K-K relations using their data in figure 8 (Nicolle et al 2004) was frustrated by the log–log plots and large uncertainties.

Figure 8.

The shear storage modulus of a 1 mm thick sample of Sylgard 527 at low frequencies measured on the TA Instruments ARES and on the instrument reported in the paper.

None of the reported methods were tested for conformity with the K-K relations. In fact, all published results that provide sufficient information appear not to conform to the K-K relations, as assessed using the lower order K-K approximation (Tschoegl 1989, p 431, Eq. 8.3–36). The test for conformity to the higher order K-K approximation was not attempted.

We designed and built the mechanical components of an instrument for measuring the complex shear modulus based on the description in Arbogast et al (1997) and a visit to view and discuss the instrument at the University of Pennsylvania. However, the electronic components, essentially all of which are available commercially, were configured and tested in our lab. Our major emphasis was on assuring accuracy of determination of both G′ and G″. A major test of the accuracy was conformity to the K-K relations. Another test was agreement with a calibrated commercial rheometer at the National Institutes of Standards and Technology (NIST) over its applicable frequency range, namely, 0.01–50 Hz. The instrument and its use are described in detail, facilitating replication in other labs.

Detailed description of the instrument components and their functions

A photograph of the mechanical components of the instrument is shown in figure 1. The ballpoint pen near the bottom of the photo is 15 cm long. Various components are labeled with letters a to z. See table 1 for model numbers and manufacturers of commercial components. a is the translatable part of a vertical micrometer linear translator, the fixed part of which is mounted on the aluminum slab b. d is an extension mounted on a providing a mount for the shear force detector e. A very low mass tongue extends downward from e and is bound to a glass microscope cover slip which in turn is bonded to the sample disk s. (s is the sample material whose complex shear modulus is to be determined.) The bottom of s is bonded to a 2 cm × 2 cm section of the glass microscope slide which is affixed to the stage f using clamp g. f is mounted on the top of a linear bearing k which allows very low friction oscillations of f, g and s. v and w are coaxial stainless steel rods screwed tightly into f. On the other (right) end of w is the (internal) moving element of the linear variable differential transformer (LVDT) h, which is fixed in the acrylic holder i. The cylindrical stainless steel jacket of the LVDT is held fixed in the acrylic block i which is mounted on the aluminum block m. x joins v to z, the driven coil assembly of the linear actuator which is suspended inside the thick cylindrical shell permanent magnet j. (j and z comprise the linear actuator.) The magnetic field of j is radial, and the winding on z is a spiral with its axis co-linear with the axes of v and w. j is press fitted into the aluminum block r which is mounted on the aluminum block c. Aluminum plate n is mounted on the translatable part of a small linear translator with its fixed part m mounted on another aluminum plate l which is mounted on the translatable part of another linear translator with the fixed part q. q is mounted on the aluminum slab c. The aluminum plate n has two projections between which a section of the rubber band o is stretched. A small bolt p is screwed into the moveable stage f and passes through a small hole near the middle of o. (The purpose of the bolt p and rubber band o is to connect the linear bearing k to the linear translation micrometer m. o, p and m allow the user to translate the sample s so that the force detector e can be centered over the sample disk; operation of both micrometer translators m and that on the right which translates the aluminum plate l allows the sample to be centered under the force detector and the rod of the LVDT to be positioned near the zero output point of the LVDT. During oscillation of the sample by the linear oscillator (peak-to-peak oscillation 20 μm or less), Hooke's law applies for the rubber band and the sinusoidal time dependence of the displacement is not compromised. The translator q provides for horizontal positioning of i so that the moving element of the LVDT can be correctly located in the fixed part of the LVDT (h). t is a 5 mm thick sponge rubber pad for aiding in mechanical isolation of the instrument. u is a 2.5 cm thick Teflon^® slab beneath t. y designates electric current carrying wires connected to the force detector e.

Figure 1.

Photograph of the main part of the instrument. The ballpoint pen is about 15 cm in length. a = translatable part of a vertical micrometer; b = aluminum slab; c = aluminum block; d = an extension mounted on a providing a mount for e; e = shear force detector; f = stage for mounting sample disk; g = clamp; h = fixed part of the LVDT; i = acrylic holder for the LVDT; j = permanent magnet of the linear actuator; k = linear bearing l = aluminum plate; m = translation micrometer; n = aluminum plate; o = rubber band; p = bolt from linear bearing to rubber band; q = linear translator; r = aluminum mount for the linear actuator; s = sample material; t = sponge rubber pad for mechanical isolation; u = 2.5 cm thick Teflon^® slab beneath t; v and w = coaxial stainless steel rods screwed into the stage for mounting the sample disk; x = fixture bolted to z and joining v to z; y = electrical signal carrying wires from the force detector to its signal conditioner/amplifier; z = driven coil assembly of the linear actuator.

Table 1.

Commercial components of the instrument not provided in the text.

ID letter or number	Description	Model number	Manufacturer
a and q	Micrometer driven translators	3″ × 4″, 1.5 cm travel	Oriel (currently Newport Corp.), Irvine, CA, USA
e	Shear force detector (miniature load cell)	BG Series ± 25 g range	Kulite Semiconductor Products, Inc., Leonia, NJ, USA
h	Linear variable differential transformer (LVDT)	0231-0000	Trans-Tek, Inc., Ellington, CT, USA
j and z	Magnet (j) and coil (z) of the linear actuator	LA15-16- 001A	BEI Motion Systems, Kimco Magnetics Division, Vista, CA, USA
k	Linear bearing (ball bearing slide)	NT53-851	Edmund Optics, Barrington, NJ, USA
m	Manual micro positioning stage	NT56-422	Edmund Optics, Barrington, NJ, USA
q	See a and q above	–	–
v and w	5.2 cm nonmagnetic core extension for h	0235-0000	Trans-Tek, Inc., Ellington, CT, USA
w	See v and w above	–	–
z	See j and z above	–	–
1	Moving part of a	–	–
2	Unregulated 75 V/15 amp dc power supply	21980	Servo Components and Systems, Ltd, UK
3	Dual dc power supply for 5	D15.200	Trans-Tek, Inc., Ellington, CT, USA
5	LVDT signal conditioner	1000-0014 (minimum delay)	Trans-Tek, Inc., Ellington, CT, USA
6	Signal conditioner/amplifier in a two-channel enclosure	E563HL in E513-2A	Ectron Corp., San Diego, CA, USA
11	Elmo motion controller	HAR-2/100	Elmo Motion Control, Inc., Westford, MA, USA

Figure 2 shows another photograph of the instrument taken from further away so that other components of the system can be seen. Letter labels are carried over from figure 1, but the numbered components were not shown in figure 1. See table 1 for model numbers and manufacturers of commercial components. 1 is the fixed part of the vertical translator for positioning the shear force detector e (figure 1) and is mounted on the aluminum slab b. 10 is a (green) ground wire connected to the ground shield 4 surrounding the wires emerging from the LVDT h. 2 is an unregulated dc power supply connected to the motion controller 11. 5 is an LVDT signal conditioner and 3 is its dual dc power supply. 6 is a signal conditioner/amplifier connected to the force detector e via wires y. 7 is a double layer of bubble wrap (used in packaging) and 8 is a 1 cm thick foam pad, both used for mechanical isolation. 9 designates wires delivering electric power to the linear actuator.

Figure 2.

Photograph of the instrument from a greater distance showing more components than in figure 1. 1 = fixed part of the vertical translator for positioning the force detector; 2 = unregulated dc power supply; 3 = dual dc power supply for the LVDT signal conditioner; 4 = ground shield for wires from LVDT; 5= LVDT signal conditioner; 6 = signal conditioner/amplifier for force detector signal; 7 = double layer of bubble wrap; 8 = 1 cm thick foam pad; 9 = wires delivering electric power to the linear actuator; 10 = ground wire connected to 4; 11 = motion controller.

In figure 3 diagrams of the instrument as viewed from the ‘front’ (upper diagram) and from the ‘top’ (lower diagram) are shown. Letters labeling components correspond to the letters in figures 1 and 2. Components m, n and o are not shown in the upper diagram to avoid possible confusing clutter; however, those components are shown in the lower diagram. Also, the sample material being tested, the microscope cover slip and the means of attachment to the tongue of the shear force detector (e), which are not shown in either diagram, are described below.

Figure 4 shows a diagram of the major components of the entire system. Data acquisition, analysis and storage are handled by the computer using software supplied by Elmo Motion Control, Inc., Westford, MA, USA, which also supplied the motion controller. The motion controller receives sinusoidal voltage bursts from the Agilent arbitrary waveform generator (Model 33250A, Agilent technologies, Inc., Santa Clara, CA, USA) and power from the unregulated dc power supply 2; the motion controller drives, via sinusoidal current bursts, the linear actuator which in turn drives the stage f and lower platen (a 2 cm × 2 cm microscope slide section in figure 3) which is bonded to the bottom of the sample material. The top of the sample is bonded to a glass microscope cover slip (catalog no 12-541A, Fisher Scientific, Pittsburgh, PA, USA), the bottom (bonding) surface of which has been roughened with diamond powder (#45, Advanced Abrasives Corporation, Pennsauken, NJ, USA) to facilitate a non-slip bond to the sample material. Figure 5 contains diagrams showing two views of the sample bonded to the two platens and the attached aluminum clamp for attaching to the tongue of the shear force detector e. The clamp is epoxied to the top of the 1.8 cm × 1.8 cm × 0.17 mm glass cover slip. The setscrew in the clamp is of stainless steel with diameter 1.45 mm, length 1.7 mm and a pitch of 31.5 threads per cm. The mass of the clamp and cover slip is as small as practicable to minimize inertial effects at higher frequencies.

Figure 5.

Side-view and top-view diagrams of the force detector, aluminum clamp, platens and sample.

Table 1 shows the model numbers and manufacturers for the commercial components of the system that are not provided in the text.

Referring again to figure 4, the measured shear force (F_ms) and displacement (y) signals are each filtered by either filter I or filter II. These eight-pole low-pass Butterworth filters (Model DP68L8B with 1.75 kHz shoulder, Frequency Devices, Inc., Ottawa, IL, USA) were manufactured at the same time to be as nearly identical as possible; however, because determination of correct phase lags is critical to the measurement accuracy, the toggle switches (figure 4) allow F_ms to be filtered by filter I and y to be filtered by filter II for one set of measurements (at some frequency) and F_ms to be filtered by filter II and y to be filtered by filter I for a second set of measurements; the results are averaged. It was found after many such measurements that the difference for the two configurations was negligible, so that we now use only one configuration. After filtering, F_ms and y are input to the LeCroy 9310A oscilloscope where display in x−y mode allows determination of the amplitudes of F_ms and y and the phase lag (ϕ) of y relative to F_ms. The information allows computation of the complex shear modulus.

Calibration of the instrument

To calibrate the force detector/amplifier system, a variety of horizontal forces were applied to the tongue of the force detector, and the shifts in dc voltage output of the signal conditioner/amplifier (item 6 in figure 2 and table 1) were determined. To accomplish the task, a 100 μm diameter nylon monofilament had a very low mass S-shaped hook on each end, and the fiber was passed over a horizontal 3 mm diameter polished stainless steel rod, one hook passing through a hole in the force detector tongue and the other hook hanging downward beyond the rod. Between the tongue and the rod, the nylon fiber was horizontal. Masses of 0.394, 0.751, 1.225, 1.619, 1.976, 2.00, and 2.370 g were placed on the hook in succession with the following results: 203, 193, 220, 185, 200, and 194 mV g⁻¹, respectively, yielding a mean of 199 mV g⁻¹ and a standard deviation of 12 mV g⁻¹. For determining F₀, the conversion factor 200 mV g⁻¹ was used.

The calibration of displacement of the lower platen was accomplished by mounting a micrometer on the shear instrument so that it could provide for a linear translation of 380 μm of the linear bearing while the dc voltage change of the LVDT signal conditioner was monitored. The resulting calibration factor was 0.567 μm mV⁻¹.

Production of samples for measurement

Almost all measurements for testing the instrument were made on a two-part silicone dielectric gel having the same manufacturer and catalog number as that used by Arbogast et al (1997), namely, Sylgard 527 (Dow Corning Corporation, Midland, MI, USA). In addition, measurements were made on a simple aqueous gelatin material, expected to have negligible loss, for comparison.

Three sample thickness ranges were produced. The microscope slides used (Catalog no 2948, Corning Glass Works, Corning, NY, USA) were found to have very consistent thickness (mean = 0.995 mm, standard deviation = 0.008 mm), and pieces were used as spacers to produce 1.00 ± 0.01 mm thick samples. The spacers were arranged so that when the cover slip was laid across the spacers a 1 mm gap existed between the 2 cm × 2 cm section and the cover slip. Samples of two smaller thicknesses were also made to test the thickness dependence of the measured values of G′ and G″.

When a sample of silicone was to be made, equal masses of parts A and B were mixed thoroughly, subjected to a 60 mmHg partial vacuum to remove any air bubbles, allowed to partially cure for about 8 h so that viscosity was great enough, and a quantity dropped onto the 2 cm × 2 cm bottom platen. Then the cover slip was dropped into place so that a circular quantity of diameter about 1.4 cm was formed. Curing of the Sylgard 527 was done at room temperature for 7 days, except for the case of one 1 mm thick sample which was cured at an elevated temperature for a few hours—in accordance with curing instructions accompanying the two-part silicone.

In the case of gelatin, preparation of the molten form (before congealing) was as follows: (1) 8.6 g of dry 200 Bloom gelatin from calfskin (Vyse Gelatin Company, Schiller Park, IL, USA) was stirred into 100 mL of doubly de-ionized water at room temperature in a 200 mL Pyrex^®; (2) the mixture was covered with a layer of Saran Wrap^® held in place by a rubber band and a small puncture put into the Saran Wrap to maintain equal pressures inside and out; (3) the mixture was heated in a ‘double boiler’ with heated water surrounding the beaker until the mixture became a solution (clarified), usually at a temperature of about 90 °C; 100 mL of the solution was poured into another Pyrex beaker and the gelatin solution cooled to 40 °C; and (4) 0.436 g of formalin was added. (Formalin contains 37% by weight formaldehyde.) The freshly prepared molten gelatin was then cooled to about 27 °C, a quantity dropped onto the 2 cm × 2 cm bottom platen and the cover slip dropped into place. After a few minutes, the gelatin was gelled, and the sample plus platens was placed in a vegetable oil medium to prevent desiccation until measurements were to be done. About 60 days are required for the formaldehyde (in the formalin) to accomplish complete cross-linking of the gelatin, but measurements can be made after as few as 3 days.

Mounting the samples in the instrument

The outward-directed glass surfaces of the platens should be clean and dry. Since the gelatin sample is stored in vegetable oil, the glass surfaces were dried with soft tissue with any remaining oil film removed with alcohol and tissue. A 0.5 to 1 mm wide ring of vegetable oil is left surrounding the gelatin disk to prevent desiccation during measurement; care should be taken that no alcohol contaminates the oil.

Before mounting the sample, the diameter of the disk was measured with a low-powered microscope equipped with a manual translator and vernier scale.

The sample is placed on the stage of the shear instrument with the microscope cover slip facing upward, and the 1 mm thick, 2 cm × 2 cm bottom platen secured to the stage with the clamp g (figure 1). Then, the aluminum clamp is centered on the microscope cover slip shown in figure 5, and the vertical micrometer translator (1 and a in figures 1 and 2) is used to lower the force detector e so that the aluminum clamp can be clamped to the tongue of the force detector using the setscrew; the clamp should contact only the tongue of the force detector. The linear micrometer translators m and q are then employed to assure that the tongue of the force detector is centered over the sample and that the moving part of the LVDT is centered in its fixed part (see top view in figure 3). Then, a small quantity of two-part ‘five-minute’ epoxy (3M Scotchweld DP-100, 3M Industrial Adhesives and Tapes, St Paul, MN, USA) is painted over the bottom of the clamp and the force detector and clamp are lowered until adequate contact is made with the upper platen of the sample (microscope cover slip). The epoxy is allowed to harden for at least 40 min after which measurements can begin.

Data acquisition and reduction

Using the Agilent arbitrary waveform generator, repeating sinusoidal pulses with at least 30 cycles are generated at the frequency of interest. The frequency is selected from N × 10 Hz for N = 1, 2, …, usually in increments of 20 Hz. The force F_ms and displacement y signals (figure 4) are displayed versus time on the oscilloscope, and the Agilent signal amplitude is adjusted so that the peak-to-peak amplitude of y is about 2% of the sample thickness. The displacement (y) signal is generally much more noisy than that of F_ms. The time base and delay of the oscilloscope are adjusted to minimize initial and final transients, and signal averaging is done over about 40 repetitions to yield well-defined sinusoidal results. (It has been verified that no change in averaged amplitude occurs for 20 through 100 signals averaged.) Then, the two average signals are displayed in x−y mode; the result is an ellipse allowing determination of the peak-to-peak amplitudes of F_ms and y and the phase lag ϕ of y relative to F_ms.

Data reduction without accounting for instrumental effects

If the thickness L of the sample is much smaller than the wavelength of shear waves in the sample medium and if the spring constant K of the force detector tongue is sufficiently large so that motion of the upper platen (microscope cover slip) is negligible, then the complex shear modulus is given by

$$G=(F_{\mathrm{ms}}∕A)∕(y∕L)$$ (1)

where A is the circular area of the sample disk. Note that the real and imaginary parts of F_ms and y vary sinusoidally and therefore can be positive or negative depending on the time. This is the data reduction method employed by Arbogast et al (1997).

Data reduction accounting for instrumental effects

As will be seen in the ‘Results and discussion’ section below, use of equation (1) for data can compromise accuracy. Over most of the range of frequencies for which measurements have been made, K is small enough relative to the shear force amplitude and L is large enough relative to the shear wavelength that G must be computed accounting for the value of K (and the mass M contacting the upper surface of the sample) and for shear wave propagation in the sample material. Shear wave propagation effects are more pronounced at higher frequencies where the wavelength is smaller. Table 2 shows K, relevant masses and the density of Sylgard 527. The sample density is needed regarding shear wave propagation.

Table 2.

Values of parameters needed for data reduction accounting for instrumental effects.

Force detector spring constant Ka	Mass of upper platen (cover slip) plus epoxied-on clamp	Mass M of upper platen plus epoxied clamp plus estimated contribution of the force detector tongue	Density of Sylgard 527
1.75 × 104 N m−1	0.26 g	0.30 g	0.97 g mL−1

^aFrom manufacturer's specifications.

Following is the derivation of the transcendental equation allowing determination of G′ and G″ from experimental results corrected for instrumental effects. Results below, referred to explicitly as ‘uncorrected’ results, correspond to use of equation (1) for data reduction; all other results have been corrected. The derivation accounts for the finite value of the force detector spring constant K, inertial effects involving the mass M attached to the top of the sample and for shear wave propagation in the sample material being measured.

Defined parameters are shown in figure 6, which depicts the involved components of the apparatus. Note that objects are not to scale in the figure. For shear waves propagating in the +x and −x directions, the displacement is

$$y(x,t)=a_0{\mathrm{e}}^{-\mathrm{i}{k}_{c}x+\mathrm{i}\omega t}+b_0{\mathrm{e}}^{-\mathrm{i}{k}_{c}x+\mathrm{i}\omega t}$$ (2)

where t is the time, ω is the angular frequency, k_c is the complex wave number and a₀ and b₀ are constants.

Figure 6.

Side-view diagram of the force detector, aluminum clamp, platens and sample (not to scale) showing F_s, F and the position and orientation of the Cartesian coordinate system relative to the derivation of equations for determining G accounting for wave propagation effects and the effect of the finite spring constant of the force detector.

We have

$$y(L,t)=a_0{\mathrm{e}}^{-\mathrm{i}{k}_{c}L+\mathrm{i}\omega t}+b_0{\mathrm{e}}^{-\mathrm{i}{k}_{c}L+\mathrm{i}\omega t}=-\frac{F_s\left(t\right)}{K}.$$ (3)

where F_s is the force exerted on the mass M by the force detector. We also define

$$k_c=k-\mathrm{i}\alpha$$ (4)

where k is the real part of the wave number and α is the shear wave attenuation coefficient. Both k and α are functions of ω.

We have

$$c_s^2=\frac{G}{{\rho }_0}\text{or}G={\rho }_0{c}_s^2={\rho }_0\frac{{\omega }^2}{k_c^2}$$ (5)

where G is the complex shear modulus, ρ₀ is the sample mass density and c_s is the sample complex propagation speed at frequency ω.

By the definition of G,

$$G=\frac{-\frac{F}{A}}{\mathrm{d}y∕\mathrm{d}x{\mid }_L}$$ (6)

where F is the force exerted on the top platen by the sample material, and A is the area of the sample making contact with the top and bottom platens. We have

$$\mathrm{d}y∕\mathrm{d}x{\mid }_L=-\mathrm{i}{k}_c(a_0{\mathrm{e}}^{-\mathrm{i}{k}_{c}L}-b_0{\mathrm{e}}^{\mathrm{i}{k}_{c}L}){\mathrm{e}}^{\mathrm{i}\omega t}.$$ (7)

From Newton's second law,

$$F\left(t\right)+F_s\left(t\right)=M\ddot{y}(L,t)\text{or}F+F_s=-M{\omega }^{2}y(L,t)$$ (8)

and

$$F=-F_s+M{\omega }^2\frac{F_s}{K}=-F_s\left(1-\frac{M{\omega }^2}{K}\right).$$ (9)

The orientation of the force detector requires that

$$F_{\mathrm{ms}}=-F_s$$ (10)

where F_ms is the detected force. Thus,

$$F=F_{\mathrm{ms}}\left(1-\frac{M{\omega }^2}{K}\right).$$ (11)

Define

$$F_{\mathrm{ms}}\equiv F_0{\mathrm{e}}^{\mathrm{i}\omega t}$$ (12)

where F₀ = ∥F_ms∥, setting the zero of the clock. (∥F_ms∥ is the magnitude of F_ms.)

Then,

$$F=\left(1-\frac{M{\omega }^2}{K}\right)F_0{\mathrm{e}}^{\mathrm{i}\omega t}.$$ (13)

Equations (5), (6), (7) and (12) give

$${\omega }^2{\rho }_0(a_0{\mathrm{e}}^{-\mathrm{i}{k}_{c}L}-b_0{\mathrm{e}}^{\mathrm{i}{k}_{c}L})=\frac{-\mathrm{i}{k}_c}{A}\left(1-\frac{M{\omega }^2}{K}\right)F_0.$$ (14)

Also,

$$y_0\equiv y(0,0)=a_0+b_0$$ (15)

(equation (2) at x = 0 and t = 0), and from equations (3), (10) and (12),

$$a_0{\mathrm{e}}^{-\mathrm{i}{k}_{c}L}+b_0{\mathrm{e}}^{\mathrm{i}{k}_{c}L}=\frac{F_0}{K}.$$ (16)

Eliminating a₀ and b₀ from equations (14), (15) and (16), the following complex equation results:

$$2\frac{\left(y_0-\frac{F_0}{K}{\mathrm{e}}^{-\mathrm{i}{k}_{c}L}\right)}{{\mathrm{e}}^{\mathrm{i}{k}_{c}L}-{\mathrm{e}}^{-\mathrm{i}{k}_{c}L}}=\frac{-\mathrm{i}{k}_c}{{\left(2\pi f\right)}^2{\rho }_0}\left(1-\frac{M{\left(2\pi f\right)}^2}{K}\right)\frac{F_0}{A}+\frac{F_0}{K}.$$ (17)

A, L, ρ₀, K and M are measured constants, f is the selected frequency in Hz, and y₀ = ∥y₀∥ e^−iϕ and F₀ are obtained experimentally with the instrument.

The transcendental equation (17) can then be solved for k_c, and G found from the relation

$$G=G^{\prime }+\mathrm{i}{G}^{″}={\rho }_0\frac{{\left(2\pi f\right)}^2}{k_c^2}.$$ (18)

Error analysis

Error analysis is done in terms of the approximate value of G, namely, G ≈ (F_ppL)/(Ax_pp) e^iφ where A is the sample area, F_pp is the peak-to-peak shear force, x_pp is the peak-to-peak displacement, L is the sample thickness and φ is the angle by which the displacement lags the shear force. We have G′ = (F_ppL)/(Ax_pp) cos φ and G″ = (F_ppL)/(Ax_pp) sin φ. Let us first estimate the fractional uncertainties in each parameter.

We have F_pp = C_FV_F where C_F is the force calibration factor and V_F is the peak-to-peak oscilloscope voltage corresponding to F_pp. The fractional uncertainty in F_pp is ΔF_pp/F_pp = [(ΔV_F/V_F)² + (ΔC_F/C_F)²]^½ where Δ means ‘uncertainty of’. From the ‘Calibration of the instrument’ section, ΔC_F/C_F = (12 mV g⁻¹)/[6^½(200 mV g⁻¹)] = 0.02 where 6 is the number of independent determinations of C_F. ΔV_F/V_F is taken to be the oscilloscope uncertainty, namely, 0.02. Thus, ΔF_pp/F_pp ≈ 0.03.

We havex_pp = C_x V_x where C_x is the calibration factor and V_x is the oscilloscope uncertainty. ΔA_Cx/C_x ≈ 0.02 and ΔV_x/V_x ≈ 0.02 again yielding Δx_pp/x_pp = 0.03.

The estimated uncertainty in L, based on measurements of thicknesses of microscope slides and cover slips with an ordinary micrometer is ΔL ≈ 0.01 mm. The values of L for the samples reported in this work were 1.00 mm, 0.43 mm, 0.23 mm and 0.22 mm; thus, the fractional uncertainties ΔL/L are 0.01, 0.02, 0.04 and 0.04, respectively.

The fractional uncertainty ΔA/A is estimated to be 0.03.

The fractional uncertainty in sinφ, i.e., Asinφ/sinφ, depends on the ratio of two oscilloscope voltage measurements using the ellipse formed in the x–y display. We have sin φ = N_F/V_F where N_F is the difference in zero crossing points when the ellipse is centered on the monitor. Since the mV per division is not changed when N_F and V_F are measured, it is assumed that the oscilloscope calibration error cancels in the ratio N_F/V_F. The fractional uncertainty in sinφ is then given by Δsinφ/sinφ = [(0.02)² + (0.01)²]^½ ≈ 0.02 where 0.02 is the fractional uncertainty in N_f due to reading error by the operator and 0.01 is the corresponding fractional uncertainty in V_F. Using the relation cos φ = (1 − sin² φ)^½ and a typical value of φ ≈ 30°, the fractional uncertainty in cos φ is Δ cos φ/cos φ ≈ 0.02.

Using the above fractional uncertainties, the fractional uncertainties in both G′ and G″ are ΔG′/G′ ≈ ΔG″/G″ ≈ [(ΔF_pp/F_pp)² + (Δx_pp/x_pp)² + (ΔA/A)² + (Δ sin φ/sinφ)² + (0.01 mm/L)²]^½ = [(0.03)² + (0.03)² + (0.03)² + (0.02)² + (0.01 mm/L)²]^½ = [0.0031 + (0.01 mm/L)²]^½. Thus, for L = 1.00 mm, 0.43 mm and 0.23 mm, ΔG′/G′ ≈ ΔG″/G″ ≈ 0.06, 0.06 and 0.07, respectively.

The exact Kramers–Kronig relations and approximations

Derivations of the K-K relations were published in the 1920s (Kronig 1926; Kramers 1927). The relations were shown to correspond to the requirement that causality be obeyed by Kronig and later, more rigorously, by Toll (1956); i.e., an output signal cannot be detected prior to occurrence of the corresponding input stimulus. The relations apply generally to physical parameters of linear phenomena which contain a real (storage) and imaginary (loss) part regarding sinusoidal oscillations. The K-K relations state that, if the real or the imaginary part of the parameter is known at all frequencies, then the other part is also determined and can be computed with the K-K relations using the known values. In the case of our experiment, the K-K relations imply that, given a shear force impulse, no responding displacement can be detected prior to the application of that impulse (causality). A derivation of the K-K relations regarding the complex shear modulus is found in the text by Tschoegl (1989; pp 426–30).

Conformity with the K-K relations is a major test of the validity of experimental results. The exact K-K relations can be expressed in various forms, one of which applies to the real and imaginary parts of the shear modulus, namely,

$$G^{\prime }\left(\omega \right)-G^{\prime }\left(0\right)=(2∕\pi )P{\int }_0^{\infty }\frac{{\omega }^2{G}^{″}\left(\beta \right)}{\beta ({\omega }^2-{\beta }^2)}\mathrm{d}\beta$$ (19)

where P ∫ indicates the principal value integral and ω is the frequency (either in radians s⁻¹ or Hz) (Tschoegl 1989, p 428, Eq. 8.3–19).

Applying the exact relation is impractical, however, because G″(ω) must be known at all frequencies to allow computation of G′(ω) − G′(0). One method for avoiding the need to know G″(ω) (or G′(ω)) for all ω is to assume a reasonable model for the viscoelastic material in terms of a mechanical circuit (electrical circuit analog) and determine a relation in which knowledge of G″(ω) over a limited range of frequencies allows computation of approximate values of G′(ω). The mechanical circuit assures that causality is obeyed.

A well-established mechanical circuit for representing viscoelastic materials is the Weichert model—or the generalized Maxwell model (Tschoegl 1989, pp 120–2). The Weichert model contains N branches, each subject to the same strain. For a viscoelastic material which has a preferred configuration—or undeformed state—all but one branch consist of a storage part and a loss part in series, the remaining branch containing a storage part only. The shear storage modulus G′(ω) and the shear loss modulus G″(ω) for the viscoelastic material have the following forms:

$$G^{\prime }\left(\omega \right)=\left\{G_e\right\}+\underset{n=1}{\overset{N-1}{\Sigma }}\frac{G_n{\omega }^2{\tau }_n^2}{1+{\omega }^2{\tau }_n^2}$$ (20)

$$G^{″}\left(\omega \right)=\underset{n=1}{\overset{N-1}{\Sigma }}\frac{G_n\omega {\tau }_n}{1+{\omega }^2{\tau }_n^2}$$ (21)

(Tschoegl 1989, p 121) where the G_n's are real positive constants and τ_n is the relaxation time associated with the nth branch. {G_e} is a positive constant corresponding to the lossless branch. For a continuous medium, it is assumed that N is very large and that the sums in equations (20) and (21) can be replaced by integrals as follows:

$$G^{\prime }\left(\omega \right)=\left\{G_e\right\}+{\int }_{-\infty }^{\infty }H\left(\tau \right)\frac{{\omega }^2{\tau }^2}{1+{\omega }^2{\tau }^2}\mathrm{d}\text{ln}\tau$$ (22)

and

$$G^{″}\left(\omega \right)={\int }_{-\infty }^{\infty }H\left(\tau \right)\frac{\omega \tau }{1+{\omega }^2{\tau }^2}\mathrm{d}\text{ln}\tau .$$ (23)

The solution of equation (22) for H(τ) to the third-order derivative with respect to ln ω is

$$H\left(\tau \right)\cong H_3^{\prime }\left(\tau \right)={\phantom{\mid }\frac{\mathrm{d}{G}^{\prime }\left(\omega \right)}{\mathrm{d}\text{ln}\omega }-\frac{1}{4}\frac{{\mathrm{d}}^3{G}^{\prime }\left(\omega \right)}{\mathrm{d}{\left(\text{ln}\omega \right)}^3}\mid }_{\omega =1∕\tau },$$ (24)

(Tschoegl 1989, p 195, Eq. 4.3–26) and the solution of equation (23) to second-order derivative is

$$H\left(\tau \right)\cong H_2^{″}\left(\tau \right)=\frac{2}{\pi }{\left[G^{″}\left(\omega \right)-\frac{{\mathrm{d}}^2{G}^{″}\left(\omega \right)}{\mathrm{d}{\left(\text{ln}\omega \right)}^2}\right]}_{\omega =1∕\tau }$$ (25)

(Tschoegl 1989, p 203, Eq. 4.3–56). Thus, from equations (24) and (25),

$$\frac{\mathrm{d}{G}^{\prime }\left(\omega \right)}{\mathrm{d}\text{ln}\omega }-\frac{1}{4}\frac{{\mathrm{d}}^3{G}^{\prime }\left(\omega \right)}{\mathrm{d}{\left(\text{ln}\omega \right)}^3}\cong \frac{2}{\pi }\left[G^{″}\left(\omega \right)-\frac{{\mathrm{d}}^2{G}^{″}\left(\omega \right)}{\mathrm{d}{\left(\text{ln}\omega \right)}^2}\right]$$ (26)

where ω can have units of radians s⁻¹ or Hz since d ln(2πf) = d[ln(2π) + ln f] = d ln f. For sufficiently slowly varying functions G′(ω) and G″(ω) (with respect to ln ω), the lowest order relation between G′(ω) and G″(ω) is applicable (Tschoegl 1989, p 431, Eq. 8.3–33), namely,

$$\frac{\mathrm{d}{G}^{\prime }\left(\omega \right)}{\mathrm{d}\text{ln}\omega }\cong \frac{2}{\pi }{G}^{″}\left(\omega \right)$$ (27)

and equation (27) can be integrated to yield

$$G^{\prime }\left(\omega \right)-G^{\prime }\left({\omega }_0\right)=\frac{2}{\pi }{\int }_{\text{ln}{\omega }_0}^{\text{ln}\omega }{G}^{″}\left({\omega }^{\prime }\right)\mathrm{d}\text{ln}{\omega }^{\prime }=\frac{2}{\pi }{\int }_{{\omega }_0}^{\omega }\frac{G^{″}\left({\omega }^{\prime }\right)\mathrm{d}{\omega }^{\prime }}{{\omega }^{\prime }}$$ (28)

where ω, ω₀ and ω′ can be in radians s⁻¹ or in Hz.

Following is the development of a higher order K-K approximation, not given in the book by Tschoegl (1989). For less slowly varying G′(ω) and G″(ω), the second-order linear differential equation in $F^{\prime }\left(\omega \right)\equiv \frac{\mathrm{d}{G}^{\prime }\left(\omega \right)}{\mathrm{d}\text{ln}\omega }$, corresponding to equation (26), can be solved for F′(ω) in terms of the experimentally determined G″(ω) and integrated to yield G′(ω). For example, experimental values of G″(ω) can be fitted with a cubic polynomial in ln ω and F′(ω) then solved for (see, e.g., Sokolnikoff and Redheffer (1966)).

Thus, if experiment yields the cubic polynomial fit

$$G^{″}\left(\omega \right)=\underset{n=0}{\overset{3}{\Sigma }}{c}_n{\left(\text{ln}\omega \right)}^n,$$ (29)

then the K-K prediction will have the form

$$\frac{\mathrm{d}{G}^{\prime }\left(\omega \right)}{\mathrm{d}\text{ln}\omega }=\underset{n=0}{\overset{3}{\Sigma }}{a}_n{\left(\text{ln}\omega \right)}^n$$ (30)

where a_n is a linear superposition of c₀, c₁, c₂ and c₃. Then, integration yields

$$G^{\prime }\left(\omega \right)-G^{\prime }\left({\omega }_0\right)=\underset{n=0}{\overset{3}{\Sigma }}{a}_n{\int }_{\text{ln}{\omega }_0}^{\text{ln}\omega }{\left(\text{ln}{\omega }^{\prime }\right)}^n\mathrm{d}\text{ln}{\omega }^{\prime }={\phantom{\mid }\underset{n=0}{\overset{3}{\Sigma }}\frac{a_n}{n+1}{x}^{n+1}\mid }_{\text{ln}{\omega }_0}^{\text{ln}\omega },$$ (31)

where ω₀ should be chosen from the range of measured frequencies and the integration constant G′(ω₀) is arbitrary.

Torque-based commercial instrument

Rheological measurements were also performed on an ARES rheometer (TA Instruments, Piscataway, NJ, USA) with parallel plate geometry (25 mm diameter). The silicone samples measured on the ARES were made from the same batch of material (cured at room temperature) as those measured with the University of Wisconsin (UW) instrument. Sylgard 527 (≈1 mm thick) was cut into a 25 mm disk and transferred onto the bottom platen; then the upper platen was gradually lowered until a full contact was established with the sample. We note that further compression leads to increased and erroneous gel modulus. Duplicate experiments showed excellent reproducibility with relative standard uncertainty of 5%.

Results and discussion

Comparisons among laboratories and instruments using 1 mm thick samples

Figure 7 shows values of ∥G∥ measured on Sylgard 527 at different times and using different instruments. The purpose is to assess the level of agreement for different instruments and methods of measurement. The reason only ∥G∥ values are compared is that those were the only values available in the article by Arbogast et al (1997). Also, no corrections for wave propagation effects or for motion of the upper platen have been done regarding any of the values in figure 7. Values measured on the instrument reported in this paper correspond to the filled squares where curing was done at room temperature (WISC unbaked sample) and the filled circles where curing was done in an oven at 70 °C for 7 h (WISC baked sample). The WISC samples were both 1 mm thick as in the case of the Arbogast et al (1997) samples. The values in filled diamonds were made at the NIST using the ARES rheometer; the silicone samples measured on the ARES were made from the same batch of material corresponding to the closed squares. The open squares were taken from figure 3 in Arbogast et al (1997) and correspond to measurements made with the instrument reported in that article. The open triangles are values derived using the WLF method (Ferry 1980, chap. 11) from measurements made over a range of temperatures (−60 °C to +30 °C) and a range of frequencies from 0.16 to 16 Hz; these values were taken from figure 4 of Arbogast et al (1997).

Figure 7.

Comparisons of ∥G∥ values for Sylgard 527 measured on different systems. All samples were 1 mm thick.

The values for the WISC unbaked sample (filled squares) agree reasonably well with those measured by Arbogast et al (1997) (open squares), the former being lower by about 8% at the lower frequency end and by about 16% at the higher frequency end. Note that it is not known how the Sylgard 527 was cured in the case of Arbogast et al (1997). On the other hand, the WISC baked sample values are all higher than those measured by Arbogast et al (1997). One possible explanation for the differences is that the curing method used by Arbogast (1997) was different from ours. Another possible explanation is that the Sylgard 527 composition was changed in 2003 with higher purity components being introduced; this could have resulted in a softer material (Lickly 2006).

The values obtained using the TA Instrument ARES (ARES unbaked sample) are the lowest values shown, but agree in slope at frequencies below about 40 Hz with values corresponding to Arbogast et al (1997) and with our values. While the ARES rheometer has an upper frequency limit of 80 Hz, the properties of this sample can only be accurate to about 50 Hz due to very low sample stiffness.

The instrument used to make the measurements between 0.16 and 16 Hz, corresponding to the open triangles, is apparently no longer available commercially, and the company recruited by Arbogast et al (1997) to make the measurements is no longer in existence.

Figures 8 and 9 compare shear storage moduli (G′) and shear loss moduli (G″), respectively, measured with the ARES system and with the University of Wisconsin instrument (WISC). Recall that the curing of the Sylgard 527 was done at room temperature and that the material used at both labs was made at the same time (same ‘batch’). Results are shown for frequencies up to 50 Hz. The shear loss moduli obtained at the two labs agree well with one another (figure 9), and the slopes of the storage moduli agree also (figure 8); however, over the frequency range the ARES values are about 0.9 kPa lower than the WISC values; recall that the WISC values were obtained using a 1 mm thick sample.

Figure 9.

The shear loss modulus of Sylgard 527 at low frequencies measured on the TA Instruments ARES and on the instrument reported in the paper. Samples were 1 mm thick.

It was surmised that the disagreement between the ARES and UW results might relate to edge effects of the UW samples. To mimic increase in the sample area, a follow-up study was done in which the thicknesses of the samples were decreased, holding the areas essentially constant. If edge effects were involved, the values of G determined would change with thickness. Results of this study are given below in the subsection ‘Dependence of measured values of G on sample thickness’.

Effect of accounting for shear wave propagation and the finite spring constant

If the spring constant of the force detector tongue were large enough, motion of the upper platen (microscope cover slip) would be negligible, i.e., would not affect the experimental results. However, as seen in the results below, the spring constant value is not sufficiently large, so its value plus the non-sample mass M attached to the force detector tongue must be accounted for. The non-sample mass equals the sum of the masses of the microscope cover slip, the clamp epoxied to the cover slip and the estimated effective mass of the force detector tongue itself. In addition, at higher frequencies inertial effects of the sample material (e.g., Sylgard 527) require correction for shear wave propagation to yield correct values for the complex shear modulus. Table 2 shows relevant masses and the density of Sylgard 527. The sample density is needed regarding shear wave propagation.

Figures 10-12 show—for unbaked (room temperature curing) 1 mm thick Sylgard 527—values for G′, G″ and tan δ ≡ G″/G′ with and without accounting for shear wave propagation and the finite value of the force detector spring constant. The filled squares, triangles and circles correspond to assuming equation (1) is valid and the open squares, triangles and circles correspond to values corrected for instrumental effects computed using the method given in the above subsection ‘Data reduction accounting for instrumental effects’. The corrected values are not well approximated using equation (1). The corrections can be significant for both G′ and G″, as seen in figures 10 and 11, where the G′ values are decreased and the G″ values are increased. Figure 12 shows the corresponding corrections to tan δ ≡ G″/G′. Thus, equation (1) should not be used for data reduction.

Figure 10.

The shear storage modulus of a 1 mm thick sample of Sylgard 527 measured between 10 and 300 Hz using equation (1) for data reduction and accounting (corrected) for wave propagation and the finite force detector spring constant. Figures 10-15 correspond to the same sample.

Figure 12.

Tan δ ≡ G″/G′ measured between 10 and 300 Hz using equation (1) for data reduction and accounting (corrected) for wave propagation and the finite force detector spring constant. Figures 10-15 correspond to the same sample.

Figure 11.

The shear loss modulus of a 1 mm thick sample of Sylgard 527 measured between 10 and 300 Hz using equation (1) for data reduction and accounting (corrected) for wave propagation and the finite force detector spring constant. Figures 10-15 correspond to the same sample.

Conformity with the Kramers–Kronig approximation

Figure 13 shows the values of G″(ω) for a 1 mm thick unbaked Sylgard 527 sample determined using the methods in the subsection ‘Data reduction accounting for instrumental effects’ (open triangles) versus ln ω. The solid line corresponds to a least-squares fit to the G″(ω) values with a cubic polynomial in ln ω.

Figure 13.

Corrected values of the shear loss modulus of the 1 mm thick unbaked Sylgard 527 versus ln f, where f is the frequency in Hz. The solid line is a cubic fit to the data. Figures 10-15 correspond to the same sample.

Figure 14 shows the values of G'(ω), again determined using the methods in the subsection ‘Data reduction accounting for instrumental effects’, versus ω The dashed line is not a fit to the data, but results from the application of equations (26), (29), (30) and (31), corresponding to the higher order K-K approximation, to predict the values of G′(ω). The agreement of the experimental values of G′(ω) with the K-K prediction is excellent, indicating that the measured G′(ω) and G″(ω) conform to the K-K relations.

Figure 14.

Corrected values for the shear storage modulus (□) of the unbaked Sylgard 527 versus frequency. The dashed line is the Kramers–Kronig prediction using the higher order approximation. Figures 10-15 correspond to the same sample.

In figure 15 prediction of G′(ω) using the lowest order K-K approximation via equation (27) is compared with the prediction using the higher order approximation via equation (26), both employing the least-squares cubic polynomial fit to G″(ω) values shown in figure 13. The two curves are nearly the same, implying that equation (26) represents a very good approximation for the levels of variation of G′(ω) and G″(ω) involved. The main difference between the two curves occurs at the lower frequency range from 10 to 60 Hz. In that range, experimental values of G′(ω) are in much better agreement with the higher order approximation than with the lower (figure 14).

Figure 15.

Comparison of the G′ prediction, based on the curve fit to G″ values in figure 13, using the lower order and higher order Kramers–Kronig approximations. Figures 10-15 correspond to the same sample.

To illustrate the importance of the instrumental corrections to obtain conformity with the K-K relations, figure 16 shows the uncorrected values of G′ and G″ from figures 10 and 11 along with the K-K prediction obtained using the uncorrected values of G″.

Figure 16.

Uncorrected values of G′ and G″ from figures 10 and 11 with the K-K prediction for G′ using the uncorrected G″ values.

Dependence of measured values of G on sample thickness

The definition of shear modulus is μ = F/(AΔθ) = FL/(AΔx) where L is the sample thickness and Δx is the sufficiently small displacement of one contact surface relative to the other. It is understood that L is small enough and A is large enough that boundary effects at the edge of A are negligible (ideally, A→∞).

Consider a square sample of area A which is driven with linear oscillations parallel to two sides. Also, consider the uniform sample to be composed of parallel rods perpendicular to the platens and bound to one another. The rods at the two edges of the square that are parallel to the oscillation direction will be minimally affected in terms of the shear forces they experience due to their immediate neighbors. However, the rods on the other two sides will experience no shear force on the open side; if there were no shear force on the other side as well, then the parameter involved would be Young's modulus, which is three times the shear modulus, thus involving a larger force to extend them than for rods removed from the edge. The presence of a meniscus at the boundary could exacerbate the problem of the force required for extension since parts of the meniscus surface would be more aligned with the force applied by the platen. The upshot is that a larger platen force F would be measured than would be the case if these boundary effects were absent, thus yielding an erroneously large values for the storage and loss components of G.

The ARES rheometer is torque-based with rotational oscillations about an axis perpendicular to—and passing through the center of—the sample disk. Regarding the above discussion of boundary effects, this circular geometry minimizes those effects. It may be a practical sacrifice that the rotational system is accurate only at sufficiently low frequencies.

An experiment was done to assess boundary effects on determinations of G using the UW instrument. (Note that since this experiment was done after the expiration date of the Sylgard 527 used in generating the results shown in figures 7-16, a new batch of Sylgard 527 was obtained for this experiment.) It is impractical with the present geometry of the UW instrument to increase the area of the sample; however, decreasing the sample thickness would seem to accomplish the same objective. Thus, we produced a sample for the ARES measurement along with three Sylgard 527 samples for UW measurements, one at a thickness of 1.00 ± 0.01 mm, one at 0.43 ± 0.01 mm (thickness of two #2 microscope cover slips) and one at 0.23 ± 0.01 mm (thickness of one #2 microscope cover slip—plus a 0.015 mm difference in thickness of the 2 cm × 2 cm base platen and the spacer slides onto which the cover slips were taped). The uncertainty in each sample thickness is estimated to be 10 μm = 0.01 mm. All three UW samples had nearly the same area.

Results are given in figures 17-20. Figure 17 shows shear storage moduli G′ for the three UW samples along with the K-K predictions for each (dashed lines). The extent of the decrease in G′ with decrease in thickness is surprisingly large. The percent drops in loss moduli were comparable. It is assumed that the values for the 0.23 mm sample are the most accurate. Figures 18 and 19 show values for the shear storage and loss moduli for the 0.23 mm sample and for the corresponding ARES values through 60 Hz. The agreement between UW and ARES values are considerably better than for the case of the 1 mm thick sample (0.23 mm values correspond to the open squares (□) in figures 18 and 19), although the frequency dependencies do not agree very well. Another 0.22 mm sample was made and the resulting values (open triangles (Δ)) are also shown in figures 18 and 19; for the latter sample, the frequency dependence of G′ agrees better with that of the ARES, but the G″ values are lower.

Figure 17.

G′ values for 1.00 mm, 0.43 mm and 0.23 mm thick Sylgard 527 samples along with the K-K predictions.

Figure 20.

G′ and G″ values for the 0.23 mm thick sample showing values through 340 Hz.

Figure 18.

Storage moduli G′ of Sylgard 527 determined with the ARES instrument (◆) and with the UW instrument for the 0.23 mm (□) and 0.22 mm (△) thick samples.

Figure 19.

Loss moduli G′ of Sylgard 527 determined with the ARES instrument (◆) and with the UW instrument for the 0.23 mm (□) and 0.22 mm (Δ) thick samples.

Figure 20 shows the (corrected) G′ and G″ values for the 0.23 mm thick sample through 340 Hz. The purpose of this figure is to show that above 300 Hz, irregularities occur in both G′ and G″, probably due to an instrumental resonance. Appendix B (figure B1) shows the corrected G′ and G″ values for a 0.47 mm thick Sylgard 527 sample where large deviations from expected values occur beyond 300 Hz.

Figure B1.

Corrected G′ (O) and G″ (■) for a 0.47 mm thick sample of Sylgard 527 including values at 320 and 340 Hz. Large deviations from expected values occur beyond 300 Hz. indicating why the upper limit of the claimed range of applicability of the instrument has been set at 300 Hz.

Gelatin cross-linked with formaldehyde

Measurements of G′ and G″ have been reported for 15% gelatin in water from 0.08 through 12.5 Hz (Laurent et al 1980, figure 1) indicating negligible loss modulus and constant storage modulus. Our gelatin is 8.1% (dry weight) gelatin and also differs in that it has been cross-linked with formaldehyde, but such cross-linking would seem to reduce loss even further; thus, we expect G″ to be very small compared to G′ and that is indeed what was found. Figure 21 shows both corrected G′ (□) and G″ (Δ) for a 0.22 mm thick sample of formaldehyde cross-linked gelatin on which we made measurements from 10 through 300 Hz. The G″ values are negligible over the entire frequency range. The predicted G′ is constant, corresponding to either the lower order K-K approximation or the higher order approximation; these results are added confirmation of the validity of our measurement system.

Figure 21.

Results for a 0.22 mm thick gelatin sample. Shown are corrected shear storage moduli (□) and loss moduli (Δ) and the Kramers–Kronig prediction based on the negligible values of G″ over the entire frequency range. Also shown are the uncorrected values for G′ (■) demonstrating the significance of corrections for instrumental effects even at lower frequencies.

In figure 21, values of G′ without correcting for instrumental effects are shown with the closed squares (■). 7% uncertainties are also shown. Note that, in this case, the uncorrected values are consistently lower than the corrected values by about 9% for frequencies below 100 Hz. The difference is due primarily to the finite spring constant K of the force detector. At the higher frequency end of the G″ values, some of the correction due to K are offset by the effect of the mass M of the upper platen and attaching clamp. (This was shown by setting M = 0 in the correction algorithm.) This example illustrates that the corrections for K and M should not be ignored, even at the lower frequencies.

Summary, further discussion and conclusions

The basic design of an instrument for measuring the shear modulus of soft-tissue-like materials was published by Arbogast et al (1997). This instrument was found to agree well with measurements of the absolute value of the complex shear modulus ∥G∥ determined by an alternative method involving the theoretical extension to higher frequencies of measurements made between 0.16 and 16 Hz. However, no comparison was made of the shear storage modulus G′ and shear loss modulus G″ separately. In a later article (Arbogast and Margulies 1998), G′ and G″ for porcine brainstem were reported, but conformity to causality via the K-K relations was not addressed. Our analysis of these results indicated that the lower order K-K relations were not satisfied, but to our surprise conformity appeared to be very good using the higher order K-K approximation developed in this paper. However, the latter conformity may be accidental since a test using still higher order K-K approximations are needed to show convergence. The derivation and application of still higher K-K approximations are yet to be done. The electronics critical to accurate determination of G′ and G″ was not described in Arbogast et al (1997) or in the related PhD thesis (Arbogast 1997).

We concentrated our efforts on the design of the electronic components which would provide sufficient accuracy of determination of the phase lag of the displacement relative to the applied shear force that conformity with an accurate K-K approximation was realized. This development involved 1 mm thick, 14 mm diameter silicone samples. The accuracy of the G determinations was questioned when disagreement was found with measurements using a calibrated commercial instrument accurate in the lower frequency range.

It was surmised that the disagreement with the commercial instrument could be due to edge effects at the boundary of the sample disk. Assuming that edge effects would be reduced as the sample thickness was reduced, measurements of G were made on thinner 14 mm diameter samples with the result that reasonably good agreement with the commercial instrument was found for samples with thickness of about 0.2 mm. Conformity with the K-K approximation was found to apply at all three thicknesses involved. Thus, there is evidence that measurements on 0.2 mm thick, 14 mm diameter disk samples yield good approximations to G values.

If this instrument is to be used for measurements on soft tissue samples, however, making such uniform 0.2 mm thick slices may be difficult, and the slicing operation itself could compromise the mechanical integrity of such thin slices. To allow thicker samples, such as 1 mm, the instrument could be modified to allow much larger areas, such as 4 cm × 2.5 cm rectangles cut from microscope slides for both platens. Corrections for M (the mass contacting the upper surface of the sample) could then become more critical.

The reported instrument allows measurements over a frequency range (10 to 300 Hz) about twice that reported in Arbogast et al (1997). Also, corrections should be made for shear wave propagation, for the finite spring constant of the force detector tongue, and for inertia of the mass between the force detector tongue and the sample material to assure optimal accuracy. Since some tissues and tissue-like materials could have G′ values much lower than those investigated in this work, the correction for shear wave propagation could be significant even for 0.2 mm thick samples. One body gel, the vitreous humor of the eye, could be a diagnostic target for ARFI or MR elastography; this material is known to become softer and softer in older patients until it becomes a liquid. An abnormal object that might be important for diagnosis via ARFI or other forms of elastography is a blood clot.

As demonstrated in figure 21 (gelatin), the corrections for the finite spring constant K of the force detector and the mass M of the upper platen and clamp could become significant; thus, it is recommended that this easily computed correction always be done to assure optimal accuracy. (We have by no means done a thorough test of the size of these corrections on a range of materials.) If thicker 1 mm samples of tissue of a much larger area are measured, all aspects of the instrumental corrections could be significant.

The instrument is described in sufficient detail that its replication in other laboratories interested in accurate determination of complex shear moduli G of soft materials, such as tissues, should be straightforward. This instrument is likely the most accurate currently available for measurement of G for soft viscoelastic materials over the 10 to 300 Hz range.

Appendix A. Kramers-Kronig approximations applied to G′ and G″ for porcine brainstem

Arbogast and Margulies (1998) made measurements of G′ and G″ for porcine brainstem. Values are shown in figures 3 and 4 of that article. Linear curve fits to the G′ and G″ values are shown from 20 through 200 Hz in those figures, and these curve fits were employed to test for conformity with the Kramers-Kronig approximations. The lower order approximation is given in equation (28) and the differential equation for the higher order approximation is given in equation (26) in this paper.

From figure 4 of Arbogast and Margulies (1998), the curve fits are G″ = (0.0094 kPa Hz⁻¹) f + 0.512 kPa = mf + b, and G′ = (0.0025 kPa Hz⁻¹) f + 1.62 kPa where f is the frequency. From equation (28) the lower order approximation is given by

$$\begin{array}{cc}\hfill G^{\prime }\left(f\right)-G^{\prime }\left(f_0\right)& =\frac{2}{\pi }{\int }_{f_0}^f\frac{G^{″}\left(f^{\prime }\right)\mathrm{d}{f}^{\prime }}{f^{\prime }}\hfill \\ \hfill & =\frac{2}{\pi }\left[\left(0.0094\mathrm{kPa}{\mathrm{Hz}}^{-1}\right){\int }_{f_0}^f\mathrm{d}{f}^{\prime }+0.512\mathrm{kPa}{\int }_{f_0}^f\frac{\mathrm{d}{f}^{\prime }}{f^{\prime }}\right].\hfill \end{array}$$

Choosing G′ (80 Hz) = 1.82 kPa, the dashed curve in figure A1 is generated, indicating considerable disagreement with the curve fit to the measured G′ values.

Figure A1.

Graphs showing the linear curve fit to G′ values (dotted line) for porcine brainstem values (Arbogast and Margulies 1998, p 804, figure 3, G′_PT values), the predicted G′ (dashed line) by applying the lower order K-K approximation to their curve fit to G″ values (Arbogast and Margulies 1998, p 804, figure 4, G_PT values) and the predicted G′ (solid line) resulting from applying the higher order K-K approximation (equation (26)), again using their linear curve fit to G″ values.

Next consider the higher order K-K approximation given by equation (26). It is easy to show that dⁿG″/(d 1n f)ⁿ = mf for all n ≥ 1. Introducing G″ and d²G″/(d ln f)² into equation (26), we have

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{G}^{\prime }}{\mathrm{d}\text{ln}f}-\frac{1}{4}\frac{d^3{G}^{\prime }}{\mathrm{d}{\left(\text{ln}f\right)}^3}& =\frac{2}{\pi }(\mathit{mf}+b-\mathit{mf})\hfill \\ \hfill & =\frac{2b}{\pi },\hfill \end{array}$$

which is a constant. The particular solution from the development in Sokolnikoff and Redheffer (1966) is dG″/d(ln f) = 2b/π = 0.326 kPa. Integration yields G′(f) – G′(f₀) = 0.326 kPa ln (f/f₀).

Setting f₀ = 80 Hz, the solid curve in figure A1 results, indicating very good agreement of the K-K prediction with the curve fit to the G values.

However, because of the large difference between the lower order and higher order K-K predictions, application of a still higher K-K approximation may not agree so well with the curve fit to experimental G′ values; i.e., convergence of the K-K approximations needs to be demonstrated. Recall that all derivatives of G″ with respect to ln f equal mf. The derivation of still higher K-K approximations has not yet been attempted.

Appendix B. More evidence for setting the upper bound of validity of measurements at 300 Hz

An example of large anomalous deviations from expected G′ and G″ values beyond 300 Hz are shown in figure B1.