Differences between stress and strain control in the non-linear behavior of complex fluids

Transcription

1 Rheol Acta (2) 49:99 93 DOI.7/s ORIGINAL CONTRIBUTION Differences between stress and strain control in the non-linear behavior of complex fluids Jörg Läuger Heiko Stettin Received: 9 November 29 / Revised: 9 February 2 / Accepted: 26 February 2 / Published online: 4 April 2 Springer-Verlag 2 Abstract Various techniques have been proposed to characterize the behavior in the non-linear regime. A new theoretical framework, as proposed recently by Ewoldt et al. (J Rheol 52(6): , 28), provides a quantitative analysis of Lissajous figures during large-amplitude oscillatory shear (LAOS). Intra- and intercycle non-linearities, strain stiffening and softening, and shear thinning and thickening are described and can be distinguished. The new LAOS framework from Ewoldt et al. has been extended to a sinusoidal stress input. Measurements on two different samples reveal significant different results for sinusoidal strain or sinusoidal stress input. For both sinusoidal inputs, the results have been verified by cyclic stress and strain loading tests. The sinusoidal input tests are analyzed as an oscillatory test by the rheometer software and firmware, whereas the cyclic loading tests are purely rotational tests. Since both types of testing give the same results, any instrumental artifacts can be excluded. This implies that complex fluids can behave differently whether periodic stress or strain input functions outside the linear visco-elastic range are applied. All tests in controlled strain and stress in rotational and oscillatory modes have been performed with the same rheometer based on an air bearing-supported electrically commutated synchronous motor. J. Läuger (B) H. Stettin Anton ar Germany, Helmuth-Hirth-Strasse 6, 7376 Ostfildern, Germany joerg.laeuger@anton-paar.com URL: Keywords Large-amplitude oscillatory shear Strain control Stress control Oscillatory flow Rheometer SRM249 Introduction Complex fluids and soft solids in particular show a rich variety in their non-linear rheological behavior. A large-amplitude oscillatory shear (LAOS) is a convenient way to capture the non-linear response of such samples (Dealy and Wissbrun 99). When the amplitude of a sinusoidal oscillation is large, the corresponding response is no longer sinusoidal but contains higher harmonic contributions (Giacomin and Dealy 998). For a pure sinusoidal strain input, the resulting stress signal contains the higher harmonic contributions, whereas for the sinusoidal stress input, the non-linearity is exhibited in the strain response. The most widely employed technique to analyze oscillatory testing in the non-linear range is Fourier transform rheology (Reimers and Dealy 996; Wilhelm et al. 998; Wilhelm 22; Heyman et al. 22; Debbaut and Buhrin 22). In Fourier transform rheology, the response signal, which in general, is not a sinusoidal function, is described by a Fourier series. In the linear range, the response function is sinusoidal and only the first coefficient corresponding to the base wave exists. At larger amplitudes, higher harmonics might occur, whereby only the odd harmonics are caused by the rheological response of the sample. The existence of even harmonics is attributed to experimental shortcomings like wall slip (Graham 995; Reimers and Dealy 998) or secondary flows (Atalik and Keunings 24). As discussed previously by Yosick et al. (998), fluid inertia

2 9 Rheol Acta (2) 49:99 93 leading to odd harmonics can be a reason for experimental errors as well. Since measurements were performed at small frequencies and small gap sizes, fluid inertia effects are neglected throughout this work. The occurrence of higher harmonic contributions can be used as a sensitive tool to characterize crossover from linear to non-linear behavior in an amplitude sweep. On the other hand, a close inspection of the moduli of the base wave, i.e., G = G and G = G, will give the same result. As soon as higher harmonics are occurring, the values of the moduli will change as well. The Fourier analysis of LOAS experiments is a nice and relatively simple approach; however, as pointed out recently by Ewoldt et al. (28), it lacks a clear physical interpretation. In particular, no clear physical meaning has been attributed to the coefficients of the higher harmonics G n and G n yet. The main reason is the fact that the Fourier transform framework is in the time domain and not in the strain or strain rate domain, in which the rheological relevant processes are occurring. As discussed in detail by Ewoldt et al., various other methods have been used to quantify the non-linear behavior in LAOS, which all have similar shortcomings like the Fourier transform rheology. Ewoldt et al. proposed a new framework to describe the complex nonlinear response in LAOS experiments. Their approach is based on the work of Cho et al. (25) inwhich a method is proposed to decompose the non-linear response in LAOS conditions into elastic and viscous parts. In the elaborate work by Ewoldt et al., they are able to distinguish different mechanisms of intra-cycle behavior like strain hardening, strain softening, shear thickening, and shear thinning. The beauty of their approach is that a mathematical treatment based on Chebychev functions of the first order and a geometrical discussion of Lissajous plots of the stress and strain waveforms lead to the same conclusions, and the newly defined moduli have a clear physical meaning. The approach of Ewoldt et al., as most other published work on LAOS, is based on the assumption of a sinusoidal strain input γ(t) = γ sin(ωt). Experimental investigations following this assumption obviously require a rheometer, which is able to generate such an input function. Historically, rheometers are used in which the strain input is applied to one side of the sample and the resulting stress is measured on the other side of the sample. It has been shown that a rheometer equipped with an electrically commutated (EC) motor is able to conduct a strain-controlled oscillation testing without the need of a separate torque measuring device (Läuger et al. 22). With such an instrument, the moment of inertia of the motor and the geometry needs to be taken into account during the control mechanism to get reliable rheological data. Although this works well for the rheological data points, i.e., for the full oscillation cycle information, it was not possible to access the real sample stress waveform but only the waveform of the total stress based on the electrical torque the rheometer applies. The capability of the real position control mode has now been extended to provide the real sample stress waveform, which offers the possibility to do real intra-cycle LOAS investigations in strain control. Such a rheometer can be operated in the accustomed control stress mode as well. However, since there are many rheometers existing that are just capable of applying a sinusoidal torque input, the question arises whether a sinusoidal stress input differs from a sinusoidal strain input. The same question was raised in the context of active microrheology by Squires and Brady (25). In their paper, they discuss a difference between constant velocity- and constant force-driven microbead non-linear rheology, whereas they did not expect a difference in stress- or strain-controlled non-linear macrorheology. In order to investigate differences between a strain and a stress sinusoidal input, the framework of Ewoldt et al. have been extended to the case of a sinusoidal stress input and measurements in both modes have been performed. Two different types of samples that are known to show different LAOS behaviors have been investigated. To further validate the findings, triangle functions in both stress and strain were conducted on the same samples and the results were compared with the results from measurements with the sinusoidal input functions. The paper is organized as follows: First, the theory is shortly recapped, and the extension to a sinusoidal stress input is outlined. Afterward, the basics of stressand strain-controlled rheometers and the principle of the used rheometer motor are described. Finally, the results on both samples are presented and discussed. Theory The framework of Ewoldt et al. delivers a physical interpretation of the non-linear contributions. The mathematical description leads to new properties that are directly related to the Lissajous figures. In the linear range of small-amplitude oscillatory shear (SAOS), a Lissajous figure of shear stress versus strain will reveal an elliptical shape in the general case of a viscoelastic material. For a pure viscous sample the shape of the Lissajous figure is circular and for a pure elastic material it is a straight line. In a plot of the shear

3 Rheol Acta (2) 49: stress versus the strain rate, a pure viscous material will show a straight line, whereas the response of a purely elastic material resembles a circle. In general, the area within a Lissajous figure of shear stress versus strain represents the lost energy, whereas the area in the shear stress versus the strain rate is related to the stored energy. In the linear regime, it does not matter if a sinusoidal stress or a sinusoidal strain input is applied. In the SAOS regime, the elliptically shaped Lissajous diagrams have a strong symmetry, which is lost outside the linear visco-elastic range. The Lissajous diagrams for the LAOS regime still show point symmetry around the origin, but the reflective or mirror symmetry is lost. As will be shown, this implies that in the LAOS regime, it is not possible to calculate the properties for a sinusoidal stress input from the results of a sinusoidal strain input and vice versa. Therefore, an extension of the existing framework covering both the sinusoidal strain and stress inputs is needed in order to describe both test types quantitatively. An extension to sinusoidal stress input conditions has been suggested independently by Ewoldt (29). Sinusoidal strain input In the strain mode, the strain input is sinusoidal: ˆγ(t) = γ e iωt or γ(t) = Im{ˆγ(t)} =γ sin(ωt). () Outside the linear regime, the stress response has higher harmonic contributions: ˆ(t) = N n e i(nωt+δ n) n= or ˆ(t) = Im{ˆ(t)} = n sin(nωt + δ n ) = γ G n sin(nωt) cos(δn ) + cos(nωt) sin(δ n ) ] (2) with n = Ĝn γ G n = Ĝn cos(δn ) G n = Ĝn sin(δn ) ; it follows (t) = γ G n sin(nωt) + G n cos(nωt)] = (t) + (t) (3) The stress response can be written in an alternative form using the viscosities η n and η n : (t) = γ η n sin(nωt) + η n cos(nωt)] (4) The dissipation of the energy density during a cycle period of the base wave outside the linear visco-elastic range is given as E d = T=2π/ω t= 2π/ω = γ 2 ω t= (t) γ(t)dt N G n sin(nωt) + G n cos(nωt)] cos(ωt)dt n= = γ 2 πg (5) This implies that in case of a full oscillation cycle, solely the loss modulus of the base wave G determines the energy dissipation (Dealy and Wissbrun 99). The G n does not contribute to the dissipated energy if a full oscillation cycle is considered. On the other side, the elastic behavior is determined by all moduli G n and G n except G. This lack of physical meaning of the individual coefficients for a full oscillation cycle is a drawback of the so-called Fourier transform rheology. According to Eq. 2, the resulting shear stress is composed of elastic and viscous parts. The elastic part is represented by the sine term (in phase), whereas the viscous part is described by the cosine term (out of phase), respectively. Based on this and further following Ewoldt et al., new variables are defined: From Eq. 3, theminimum-strain elastic shear modulus (shear modulus at zero strain) G M, which is the slope of the tangent at ; γ ] = (); ] in the Lissajous plot (γ),iscalculated. G M = d dγ = d γ = dt dt dγ = G M (ω, γ ) γ(t)= ] G M = (6) G M = ng n G M = n n cos(δ n ) G M γ (γ ) = G (7)

4 92 Rheol Acta (2) 49:99 93 In addition to the geometrical interpretation, the new quantity G M can be calculated from the Fourier coefficients G n or the amplitudes n and the phases δ n of the higher harmonic stress contributions. A second property is the large-strain elastic shear modulus (shear modulus at maximum strain) G L,which is the ratio of shear stress at maximum strain and the maximum strain. In the Lissajous plot (γ),this represents the slope of a straight line from the origin ; γ ] = ; ] to ; γ ] = (γ ); γ ]. G L = γ = G L (ω, γ ) G L ]= (8) γ =±γ G L = γ G n ( ) (n )/2] ±γ = G n ( ) (n )/2] G L = ( ) (n )/2 n cos (δ n ) G L γ (γ ) = G (9) For small amplitudes, G M = G L = G represents the normal storage shear modulus. The strain stif fening ratio is defined as S (ω,γ ) = G L G M G L S ] = () For S >, the material is strain stiffening, and for S <, it shows strain softening. In the linear range, S = holds. Similarly, the minimum-rate dynamic viscosity (viscosity at zero strain rate), which represents the slope of the tangent at the point ; γ ] = () ; ] in the Lissajous plot ( γ ), η M = d d γ = d γ = dt = η M (ω, γ ) η M = ωγ = ω dt d γ γ = ng n ( )(n±)/2] ω 2 γ ng n ( ) (n )/2] η M = ωγ η M ] = s () ( ) (n )/2 n n sin (δ n ) η M (γ ) = η (2) and the large-rate dynamic viscosity (viscosity at maximum strain rate), as the slope of a straight line from the origin to ; γ ] = ( γ ) ; γ ], are defined: η L = γ ] η L = s (3) γ =± γ γ G n (±)] η L = = ±ωγ ω η L = ωγ G n n sin (δ n ) η L (γ ) = η (4) η M and η L are the intra-cycle viscosities at the smallest and largest strain rates. Similar to that of the elastic part, a shear thickening ratio is calculated: T (ω,γ ) = η L η M η L T] = (5) For T >, the sample depicts shear thickening, and for T <, it shows a shear-thinning behavior during an oscillation cycle, respectively. T = in the linear range. In the SAOS case, the Lissajous diagrams have an elliptical form. The mirror symmetry implies (γ =) γ ( =) = γ (γ =γ ) = γ ( = ) γ γ = γ γ= = = γ γ = γ γ=γ = = γ = (6) = (7) The distance of the points, ] and, γ ]fromthe origin can be calculated with the left part of Eq. 6. The right part of Eq. 6 gives the slopes of the tangents for the same points. The slopes are reciprocal to each other. Similarly, the distance of the points,γ ]and,γ] from the origin can be calculated with the left part of Eq. 7. TherightpartofEq.7 gives the slopes of the tangents for the same points. The slopes are again reciprocal to each other. In the LAOS regime, the mirror symmetry is lost and these conversions between a sinusoidal strain and a sinusoidal stress input are no longer possible.

5 Rheol Acta (2) 49: Sinusoidal stress input The framework for the sinusoidal strain input is applied to a sinusoidal stress input: J L = γ J L = =± = J L (ω; ) J n J L ] = (24) ˆ (t) = e iωt or (t) = Im { ˆ (t) } = sin (ωt) (8) Now, the strain response will have higher harmonic contributions outside the linear range: γ(t) = γ n sin(nωt δ n ) <δ < π 2 π <δ n < +π = G n sin(nωt) cos(δ n) cos(nωt) sin(δ n )] γ n = Jn = J n sin(nωt) J n cos(nωt)] J n = J n cos(δ n) J n = J n sin(δ n) = γ (t) + γ (t) (9) γ (t) = ω nj n cos (nωt) + nj n sin (nωt)] = γ (t) + γ (t) (2) Strain γ(t) and strain rate γ (t) can be separated in elastic and viscous parts. The strain rate response can be written in an alternative form using the fluidities n and n : γ (t) = n n cos (nωt) + n n sin (nωt)] (2) With the storage J n and the loss compliances J n as the coefficients in the series expansion, a small-strain compliance J M and a large-strain compliance J L are defined: J M = d γ d = J M (ω; ) = = d γ dt J M = nj n dt ] d J M = = (22) J M = n γ n cos (δ n ) J M ( ) = J (23) J L = γ n cos (δ n ) J L ( ) = J (25) For a better visualization, it is helpful to plot the harmonic input function γ() on the x-axis in a Lissajous diagram. Now, J M is the slope of the tangent at γ ; ] = γ (); ], and J L is the slope of a straight line from the origin γ ; ]=;]toγ ; ] =γ ( ); ]. The stress-softening ratio R is defined as R (ω, ) = J L J M J L R] = (26) R < shows intra-cycle stress stiffening (similar to S > ); and R >, stress softening (similar to S < ). As S =, R = represents a linear behavior. The fluidities n and n canbeusedtodefinea small-rate f luidity M and a large-rate f luidity L : ψ M = d γ d = d γ dt ψ M = ω = ψ M (ω; ) = dt ] d ψ M = = s n 2 J n (27) ψ M = ω n 2 γ n sin (δ n ) ψ M ( ) = ψ (28) ψ L = γ = ψ L (ω; ) =± ψ L = ω nj n ψ L ] = s (29) ψ L = ω n γ n sin (δ n ) ψ L ( ) = ψ (3) A Lissajous plot γ () is now the most appropriate way to visualize these newly defined fluidities. ψ M is the slope of the tangent at γ ; ] = γ () ; ], ψ L is the slope of a straight line from the origin γ ; ] = ; ] to γ ; ] = ] γ ( ) ;.

6 94 Rheol Acta (2) 49:99 93 The shear-thinning ratio Q follows as Q (ω, ) = ψ L ψ M ψ L Q ] = (3) Q > occurs for ψ L >ψ M (similar to T < ). The fluidity at a large strain rate exceeds the fluidity at a small strain rate. This represents a shear-thinning behavior since the viscosity at a large strain rate is smaller compared to the viscosity at a small strain rate. Shear thickening occurs for Q < (similar to T > ). Q = represents the linear behavior. Relations between properties for strain and stress inputs In the limiting case of small amplitudes, the newly defined properties equal the well-known moduli (G), compliances (J), viscosities (η), and fluidities (ψ). For the complex shear modulus Ĝ and the complex compliance Ĵ, the following well-known relations hold: Ĝ = Ĵ G + ig = J ij Ĝ = Ĵ (32) The sign of the imaginary part of the compliance is defined as being negative in order to have a positive absolute value for the compliance. Similar relations between the complex viscosity ˆη and the complex fluidity ˆ exist: With Eqs. 3 and 32, the relations for sinusoidal strain and stress inputs can be converted. From the real and imaginary parts, the corresponding quantities in the linear range can be calculated: J L J M G L = Ĵ L 2 = G M = Ĵ M 2 (37) ψ L ψ M η L = ω 2 Ĵ L 2 = η M = ω 2 Ĵ M 2 (38) For a strain input, the four new quantities defined in Eqs. 6, 8,, and3 can be combined to two new complex quantities according to Eq. 34. Similar for a stress input, the four new quantities defined in Eqs. 22, 24, 27, and 29 are combined in two new complex properties as described in Eq. 35. In the linear range, every complex strain quantity equals an inverse stress quantity. As mentioned, the defined quantities can be visualized in Lissajous plots. For each data point with a sinusoidal strain input, two lines in a (γ ) diagram and two lines in a ( γ ) diagram are representative. In Fig., an example for Lissajous plots for one measuring cycle under LAOS conditions with a sinusoidal strain input for a solution of an entangled polymer is shown. In the ˆη = ˆψ η iη = ˆη = ψ + iψ ˆψ (33) The sign of the imaginary part of the viscosity is now defined as being negative in order to have a positive absolute value for the fluidity. Based on the relations Ĝ = ω ˆη and ˆψ = ω Ĵ,more general complex quantities can be defined for a sinusoidal strain input, Ĝ L = G L + iωη L Ĝ M = G M + iωη M (34) and for a sinusoidal stress input, Ĵ L = J L i ψ L ω Ĵ M = J M i ψ M ω. (35) In the linear range, the following equations hold: Ĝ L = = Ĝ Ĵ L M = and Ĵ M Ĝ L = = Ĝ M = Ĵ L Ĵ M (36) Fig. Lissajous plots of stress versus strain (left) and stress versus strain rate (right) for the SRM249 sample. Four lines representing the defined properties for LAOS measurements for a sinusoidal strain input are depicted. Upper plots show a measurement in the linear, and the lower plots show a measurement in the non-linear regime

7 Rheol Acta (2) 49: (γ ) diagram, a straight line goes from the origin to the point P =(γ ); γ ] and has the slope G L. The slope of the tangent at P 2 = (γ = ) ; ] is G M.Inthe ( γ ) diagram, a line from the origin to P 3 = ] ( γ ) ; γ has the slope η L. The tangent at P 4 = ( γ = ) ; ] has a slope of η M. The corresponding linear equations have the form: P : (γ ) = G L γ (39) P 2 : (γ ) = G M γ + (γ = ) (4) point P 7 = γ ( ) ; ] has a slope of ψ L. The slope of the tangent at point P 8 = γ ( = ) ; ] is ψ M.The corresponding linear equations are P 5 : γ () = J L (43) P 6 : γ () = J M + γ ( = ) (44) P 7 : γ () = ψ L (45) P 8 : γ () = ψ M + γ ( = ) (46) P 3 : ( γ ) = η L γ (4) P 4 : ( γ ) = η M γ + ( γ = ) (42) For each data point in an experiment with a sinusoidal stress input, the defined quantities are represented by two lines in the γ () diagram and two lines in the γ () diagram. In Fig. 2, an example for Lissajous plots for one measuring cycle under LAOS conditions with a sinusoidal stress input for a solution of entangled polymers is shown. In the γ () diagram, the line between the origin and point P 5 = γ ( ); ] has the slope J L. The slope of the tangent at P 6 = γ ( = ) ; ] is J M. In the γ () diagram, the line from the origin to the Fig. 2 Lissajous plots of strain versus stress (left) and strain rate versus stress (right) for the SRM249 sample. Four lines representing the defined properties for LAOS measurements for a sinusoidal stress input are depicted. Upper plots show a measurement in the linear, and the lower plots show a measurement in the non-linear regime Rheometer motor control The described methodology can now be applied for a sinusoidal strain or a sinusoidal stress input. However, with a rheometer, one has to be careful that these conditions are met. Otherwise, the non-linearities are in both the stress and the strain and the mathematical treatment outlined in the last section is not applicable. It is therefore crucial to understand the working principle of a modern rheometer. The main components of a rotational rheometer are the motor with its supporting bearing system and the force measurement. Historically, there are two principles used for research-grade rotational drag flow rheometers (Macosko 994). In one, a displacement or speed (strain or strain rate) is applied to the sample by the motor, and the resulting torque (stress) is measured separately by an additional force sensor. In this type of instrument, which is commonly referred to as a controlled strain or controlled strain rate (CR) rheometer or separate motor transducer system, the electrical current used to generate the displacement or speed of the motor is not used as a measure of the electrical torque. With the other type of instruments, often called controlled stress (CS) rheometer or combined motor transducer system, a certain electrical current is applied onto the motor assembly. The current builds up a magnetic field that produces an electrical torque resulting in a rotation of the drive shaft. In such a design there is no separate torque sensor needed, since the torque signal is directly calculated from the motor current. The movement of the motor shaft is measured by an angular displacement sensor, which in most types of CS rheometers, is an optical encoder. Most CS rheometers are based on the so-called drag cup motor, which was already used in the first CS instrument, the Deer rheometer built in 968 (Davis et al. 968; Davis 969). While most CS instruments still employ the same motor principle like the original Deer rheometer, in 995, a rheometer based on a different

8 96 Rheol Acta (2) 49:99 93 motor system was introduced (Läuger and Huck 2). Newer generations of rheometers employing the new motor technology were made available in 999 and 24. Before this new motor principle is described, the more traditional systems are discussed for a better understanding. Traditional controlled strain rheometers In a typical strain-controlled rheometer, a direct current (DC) motor with some sort of closed-loop servo position control is used. This motor, which is a synchronous drive, applies the programmed speed or deformation to one of the measuring tools. With such a DC motor, the commutation of the motor is done by a mechanical contact. In a newer embodiment of a traditional controlled strain rheometer, a DC motor without mechanical contacts has been used. In such a motor design, the commutation of the motor is done electrically instead of mechanically. It is therefore called electrically commutated or brushless DC. Although a CR rheometer is commonly thought of being independent of the sample, it still requires a control mode for setting the rotational speeds and angles in a fast and accurate way. Therefore, a measurement of the angular speed or the position of the drive shaft is needed. Only an accurate measuring sensor on the drive shaft in combination with a fast control loop allows a precise control of angle (strain) or angular velocity (strain rate). In older CR rheometers, the speed in steady rotation can be measured by a tachometer, while in dynamic oscillation or in step strain tests, the angular position changes can be detected by a rotary variable capacitance transformer (RVCT), which consists of two fixed capacitive plates attached to the static motor housing and a moveable plate mounted onto the rotor shaft (Macosko 994). In a newer embodiment of a CR rheometer, an optical encoder is used for detecting the angular movement. Although the transient response of the DC servo motor in such rheometers is quite fast, one limiting factor in conducting transient measurements in CR control is the precision of strain or speed control, which has to be achieved by a feedback loop. Macosko (994) reported that in a step strain experiment, a DC motor with RVCT feedback showed a significant strain overshoot at a strain of γ =. or % before reaching the desired value. By observation of the plate movement with a highspeed camera, it has been found for a CR rheometer that the actual strain in a step strain experiment can overshoot the commanded strain (γ = 2. or 2%) by almost 2%. The total time to reach the final strain was found to be more than 3 ms (Gevgilili and Kaylon 2). This shows that the drive of a CR rheometer is obviously not infinitely fast but has a time lag to reach a final commanded strain or strain rate. In addition, if larger strains are to be set, significant strain overshoots might occur, leading to imperfect strain histories. This fact is sometimes neglected, and the command values are taken as the actual angles or speeds in CR rheometers. Since the motor just sets the angular movement, an additional torque measuring device is needed to measure the resulting force on the opposing measuring tool. The most popular one is the force rebalance transducer (FRT; Macosko 994 and references therein). The FRT is basically an air bearing-supported DC drag cup motor that tries to prevent the transducer shaft from any deflection. The current used by the motor is a measure of torque. In addition to the motor control time, the time response of the torque transducer can give a significant contribution to the total time resolution in which the instrument can perform fast transient measurements. The advantage of such a design is that the measured torque at the transducer is not influenced by the torque needed to accelerate the driving motor and the moving geometry. Therefore, the moment of inertia of the motor is not influencing the torque measurement. However, at small strains, the actual movement of the transducer motor cannot be neglected. Similarly, at high frequencies, the moment of inertia of the transducer motors becomes relevant as well. If a controlled strain rheometer is used to perform a test requiring preset stresses (CS test), the performance is normally very limited, since there is a feedback loop that has a quite large time constant. In addition, the torque transducer has a finite response time of the order of 3 ms. Therefore, only some limited stress capabilities in rotational testing are available, and in particular, no sinusoidal stress input can be generated. Traditional controlled stress rheometers In a controlled stress rheometer, the current of the motor is used as a measure of the torque applied to the sample. The movement of the shaft is measured by a rotary encoder, which in most cases, is an optical encoder on the same side of the motor. The most straightforward and traditional use of the CS rheometer is to apply an electrical current to the motor that, in turn, produces a torque and to measure the movement of the shaft by means of a rotary encoder. However, it is important to note that such a

9 Rheol Acta (2) 49: motor, in general, applies the motor (or electrical) torque to the total system consisting of a rotor, a rotor shaft, a bearing, a measuring geometry, and a sample. Under transient conditions, only part of the motor torque is reaching the sample (sample torque). This implies that a controlled stress rheometer without special control routines is not controlling the sample s stress; it just controls the electrical torque of the motor. Depending on the characteristics of the motor, the measuring geometry, and the type of rheological test, the electrical torque and the sample s torque can be quite different. A resonance phenomenon as described by Baravian et al. (27) further limits the useable frequency range toward higher frequencies especially in the case of a soft solid. One strategy to extend this limit is to reduce the moment of inertia of the rheometer motor and the geometry. However, even a reduction of the moment of inertia by a factor of will lead to a shift of the resonance frequency by a factor of only 3.3. On the other hand, a doubling of the geometry diameter shifts the resonance frequency by a factor of 8 in the case of cone-and-plate geometries and by factor of 6 for parallel-plates geometries. Due to faster electronics, it is possible to use the position information from the rotary encoder as a feedback signal for the control loop to adjust the motor speed or the angular position to the desired value. If such a feedback is fast enough, controlled strain or strain rate experiments are possible to some extent with a CS instrument. Most controlled stress rheometers nowadays are based on a drag cup motor, which goes back to Davis et al. (968), Davis (969) and Plazek (968). A detailed description of an early drag cup motor-based rheometer can be found in Franck (985). A comprehensive review on the history of one line of drag cup-based rheometers was published recently (Barnes and Bell 23). In a drag cup motor, an alternating current (AC) produces a fast, rotating (e.g., 4 Hz) magnetic field on the stator, which itself generates a force in a metal (e.g. aluminum) cup by inducing eddy currents in the metal cup. These eddy currents, in turn, produce their own magnetic field, which is forced to follow the rotating field. Hence, the cup is moved by a drag force with the rotating stator field, thereby generating a torque. The rotational speed of the magnetic field on the stator is given by the frequency and number of poles. The torque can be varied by changing the strength of the stator field by varying the current through each set of poles. The torque produced by a drag cup motor depends on the magnitude and frequency of the current in the stator and the electric resistance of the drag cup. The magnetic field on the rotor has to be induced by the stator field. Typical buildup times for the magnetic field on the rotor are of the order of several milliseconds, thus limiting the behavior at very fast torque changes, which are required for oscillatory tests at higher frequencies and for the control of fast changes in rotational speeds or angles. The drag cup motor is also called an asynchronous motor, since the rotor is not moving synchronously with the magnetic field of the stator. The stator field rotates with a high frequency. The rotor speed is always lower than the speed of the stator field. This difference in the two speeds is needed to induce the magnetic field in the rotor. It is not possible to use this rotor field to control the motor. Therefore, the current in the stator can not be divided into a magnetizing and a torqueproducing part. An assumption of a quadratic relation between the stator current, which is a relatively highfrequency AC, and the torque is made, i.e., M I 2 s, with M being the torque and I s being the stator current. This assumption is more or less valid in stationary situations. However, it is not possible to get useful information about the distribution of the components of the stator current directly after a change in the current and therefore in the torque. This fact leads to limited abilities of a drag cup motor in transient responses for torque adjustment, which is especially crucial for CR tests like steps in strain or shear rate. In these tests, the torque needs to be adjusted rather fast in order to have the rotor following the step in the angle or speed. Due to better electronics and faster feedback possibilities, the movement can be controlled faster in more modern implementations of drag cup motors. This holds especially if the type of sample is known, i.e., if it is, for example, a Newtonian liquid. In these cases, the controller can be optimized for the actual specific condition. However, the disadvantages remain on principle. In particular, it is not possible to produce a sinusoidal strain input in the case of LAOS testing on complex fluids. Controlled stress and strain rheometers based on an air bearing-supported EC synchronous motor An EC motor is a DC motor. In the EC motor, the rotor rotates synchronously with the rotating field on the stator; thus, the name synchronous motor is often used to designate servo motors of this design. The ECmotorisalsooftenreferredtoasabrushlessDC motor since it resembles a DC shunt motor turned inside out. In the EC motor, the current is commutated electronically and there are no brushes or other mechanical contacts to excite the motor. Here, the motor is

10 98 Rheol Acta (2) 49:99 93 excited by special permanent magnets with a high flux density located on the rotor. The permanent magnet poles on the rotor are attracted to the rotating poles of the stator by their opposite magnetic polarity. The magnetic poles of the stator are produced by an electric current flowing through a coil system located on the stator. The flux of the current carrying windings of the coil system rotates with respect to the stator, but like the DC motor, the current-carrying flux remains in position with respect to the field flux rotating with the rotor. The major difference is that the synchronous EC motor maintains position by an electrical commutation rather than a mechanical commutation. Like for the DC shunt motor, torque is proportional to the strength of the permanent magnetic field and to the field created by the current-carrying coils. The magnetic field in the stator rotates at a speed proportional to the frequency of the applied voltage. This is called a synchronous motor since the rotor rotates at the same speed, i.e., synchronously, with the stator field. The rotor field is produced by highly energetic permanent magnets, and each one of it is mounted at a fix position on the rotor disc. Since the positions, shapes, and strengths of these permanent magnets are known, also the rotor field is well known. The EC control makes use of this knowledge of the rotor field. Therefore, it is possible to adjust the electro-magnetical torque in such a way that it is linear to the total amount of the stator current, i.e., M I s. In this case, a change of the stator current will be followed by a change of the torque almost instantaneously. The strain or shear rate can be adjusted in a very fast way and without any overshoots. In combination with a high-resolution optical encoder, real strain and strain rate control are possible. In a traditional strain rheometer, a similar motor concept is used. In distinction to a traditional controlled strain rheometer, no separate torque transducer is needed, but the electrical current of the motor, I s,is used as a measure of the torque. Like for CS rheometers, both presetting and measuring of the corresponding properties are done from the same side of the rheometer. The described motor setup basically combines the advantage of both the traditional controlled strain rheometer with a fast motor control and the traditional controlled stress instrument with the ability to take the motor current as a measure of the torque. In the case of a traditional CR rheometer, where the torque is measured separately from the motor, the torque value is not influenced by the torque needed to accelerate the motor. In a setup that uses the motor current as a measure of the torque, the effect of accelerating the motor needs to be accounted for. As already described, one way is to reduce the moment of inertia of the motor. However, this can have the drawback that the motor is very sensitive on any change of the moment of inertia caused by, for example, a large and heavy measuring geometry. The moment of inertia of a given mechanical system should be constant. However, effects, like resonance, electro-magnetic couplings between stator and rotor, and thermal expansion, can change the value of the moment of inertia. Using an EC motor and its constant magnetic field on the rotor, the moment of inertia can be determined quite accurately for a given configuration. The actual value of the moment of inertia is considered by the electronics in the control mechanism. Therefore, the set stress in controlled stress mode or the resulting stress in controlled strain mode is the sample stress and not the total stress the instrument applies. Using the actual moment of inertia value during the control mechanism also eliminates the already discussed resonance peak that often limits the frequency range of traditional controlled stress rheometers toward high frequencies. This was always integrated in the calculation of the rheological data points, i.e., for the full oscillation cycle information. However, it was not possible to access the real sample stress waveform but only the waveform of the total stress based on the electrical torque the rheometer applies. The capability of the rheometer has now been extended to provide the real sample stress waveform, which offers the possibility to do real intra-cycle LOAS investigations in strain and stress control. Historically, CS rheometers are controlling the amplitude of the oscillation. In a controlled stress mode, an instrument applies a sine wave with torque amplitude. However, when a strain-controlled mode is used, the instrument still applies a torque or stress sine wave. Generally, a strain-controlled oscillatory test in a CS rheometer consists of the following steps: applying one full oscillation cycle with an arbitrary stress amplitude, measuring the strain amplitude, adjusting the stress in the next oscillation cycle, and repeating this routine until the desired strain amplitude is reached. This method can be referred to as amplitude control. It has been shown earlier that rheometers with EC motors are able not just to perform this traditional amplitude control but also to carry out strain oscillation also with a real position or true strain control (Läuger et al. 22). This oscillation technique was called direct strain oscillation (DSO). DSO does not require a full (or even part of an) oscillation cycle but uses a real-time position control and adjusts to the desired strain directly on the sine wave. Therefore, the actual movement of the measuring system follows directly the required change in strain during each individual oscillation cycle. Since

11 Rheol Acta (2) 49: then, the term DSO sometimes has been used also for the strain amplitude control methods. In this paper, we use the expression DSO, as it was introduced, exclusively for the strain position control method. Including DSO, there are three different ways to perform an oscillatory test: () stress control by applying a sinusoidal stress input and controlling the stress amplitude (controlled shear stress or CSS), (2) strain control by applying a sinusoidal stress input and controlling the strain amplitude (controlled shear deformation or CSD), and (3) strain control by controlling the strain position during an oscillation cycle (direct strain oscillation or DSO) and therefore applying a sinusoidal strain input. As will be shown in Results and discussion, only method 3 (DSO) creates a real sinusoidal strain input in LAOS measurements. Samples and methods An entangled polymer solution and a gel-like xanthan gum solution have been investigated. The standard reference material SRM249 was introduced in 2 by the National Institute of Standards and Technology as a non-newtonian reference fluid. It consists of polyisobutylene dissolved in 2,6,,4- tetramethylpentadecane. The details of the sample are described by Schultheisz and Leigh (22). There were some errors in the absolute values of the G and G moduli in the original measurements by Schultheisz and Leigh (Läuger et al. 25). In the actual certificate for the SRM249 samples, the original data for the oscillatory measurements are replaced by data from one of the authors of this paper (Certificate for SRM249 2). Xanthan gum, which is a high-molecular-weight polysaccharide, was dissolved at room temperature in water with a xanthan gum concentration of 4%. The two samples have been chosen since they are known to show a different behavior of G and G by increasing the amplitude from SAOS to LAOS conditions. While the polymer solution shows decreasing moduli with increasing amplitude, the xanthan solution has a strong overshoot in the loss modulus before both moduli decrease with increasing amplitude. Various MCR3 and MCR5 rheometers from Anton ar with Peltier temperature control at 25 C and cone-and-plate geometries with a 5-mm diameter have been employed for both samples. The cone angles were for the SRM249 sample and 2 for the xanthan solution. The newest software and firmware (RheoPlus V3.4), together with the waveform monitor and the DSO modules, have been used. Results and discussion Polymer solution (SRM249) In Fig. 3, a frequency sweep measured with the DSO method at a strain of % is shown. The insets display the Lissajous plots (γ )and ( γ ) at various frequencies. Since with the selected strain, the measurement is in the SAOS range, all the Lissajous figures reveal an elliptical shape. As expected, the enclosed area decreases in the (γ ) diagram, while the enclosed area increases in the ( γ ) diagram while moving from small to high frequencies, reflecting the transition from a liquid-like behavior at small frequencies to a solid-like behavior at high frequencies. Figure 4 depicts an amplitude sweep measured at a frequency of rad/s by all three oscillation techniques on the SRM249 sample. Over the whole measurement range, G is above G at the frequency chosen. Both moduli exhibit a smooth transition from the linear into the non-linear region. In the LAOS regime, a deviation between measurements made with the DSO method on one side and with CSD and CSS on the other side can be seen. CSD and CSS measurements reveal exactly the same results. It is interesting to note that for the DSO measurement, the ratio of the slopes of G and G has a value of exactly 2 in the non-linear region. The normalized stress and strain waveforms and the corresponding Lissajous plots at a strain of,% are displayed in Fig. 5 for DSO and CSD measurements. In the DSO measurement, the strain is sinusoidal, whereas the corresponding sample stress has non-sinusoidal shape, reflecting the non-linearity. In the CSD measurement, the stress is sinusoidal and the strain shows some non-linearity, although to a lesser extend, compared to the stress in the DSO result. A closer inspection of the waveforms by a fast Fourier transformation reveals that in case of DSO, the strain is indeed sinusoidal and does not contain any significant higher harmonic contributions. The amplitude ratio of the third harmonic in the strain, which is the largest of the higher harmonics, to the base strain wave is smaller than.%. However, the sample stress wave in case of the CSD (and the CSS) measurement has also some small non-linearity. The amplitude ratio of the third harmonic in the stress to the base stress wave is.4%. While the rheometer applies a sinusoidal electrical torque, the resulting strain becomes non-linear at large amplitudes; i.e., the strain waveform is not sinusoidal anymore. This means the stress acting on the samples is also not sinusoidal anymore. It is interesting to note that this effect occurs at a relatively small oscillation frequency and at a high sample viscosity. Therefore, it is crucial to look at the

12 92 Rheol Acta (2) 49:99 93 Angular Frequency Fig. 3 Frequency sweep measured on the SRM249 sample with DSO at %strain. Lissajous figures (γ) (closed symbols) and ( γ) (open symbols) at angular frequencies of.,,, and rad/s actual sample stress waveform when constructing and analyzing LAOS measurements with a sinusoidal stress input wave. A stress waveform constructed from the electrical torque might always show a perfect sinusoidal form. Although the sample stress input wave is slightly non-sinusoidal, this small deviation should not be responsible for the larger difference in the Lissajous figures. The ratio of the sum over all amplitudes of the higher harmonics to the amplitude of the base wave is about 5% (in stress response) in the case of a strain input and just 5% (in strain response) in the case of a stress input. At even higher strains of 2,5%, the values are 22% for the strain and 7% for the stress input. The described effect could explain the smaller G values in the non-linear range for the CSD test. The Lissajous plot of the stress versus the strain shows that the shape of the curve is more circular in the case of CSD compared to that of DSO, reflecting a somewhat less pronounced non-linearity. This indicates that a sinusoidal strain input leads to larger non-linear effects compared to a sinusoidal stress input for the SRM249 sample. The intra-cycle results can be analyzed quantitatively by either using the Fourier coefficients and calculating the new properties as outlined in Theory or graphically directly from the Lissajous plots by obtaining the slopes of the respective lines. The examples already shownin Theory represent the Lissajous figures that belong to the DSO and CSD measurements at strain amplitudes of % and,%. Figure shows the Lissajous diagrams for the DSO measurement at % and,% strain, whereas in Fig. 2, the respective G' G'' % 5 Strain γ Fig. 4 Amplitude sweep on the SRM249 sample at an angular frequency of rad/s for all three the oscillation techniques. DSO (triangles), CSD (squares), and CSS (circles). The dotted line marks a strain of,% (see text)

13 Rheol Acta (2) 49: Fig. 5 Normalized strain (left) and stress (right) waveforms and Lissajous diagram (γ) (bottom) ata strain of,% (indicated by the line in Fig. 4)forDSO (closed symbols) andcsd (open symbols) γ s 6 Period Time t per s 6 Period Time t per γ plots for the CSD measurements are depicted. In Table, the values calculated by the use of the Fourier coefficients are listed. In the linear regime at % strain, all ratios, S, T, R, andq, are close to zero as expected. At larger strain amplitudes, the following intra-cycle behavior occurs: strain stiffening (S > ) and shear thinning (T < ) for a sinusoidal strain input and stress stiffening (R < ) and shear thinning (Q > ) fora sinusoidal stress input. In the bottom part, the relations comparing the strain and stress inputs are shown. If both methods reveal the same information, the properties Ĝ M Ĵ M and Ĝ L Ĵ L should be equal to, which is the case at small but not at larger strain amplitudes as expected. While the absolute amount of shear thinning is similar for sinusoidal strain and stress inputs, the amount of intra-cycle stiffening is larger for the sinusoidal strain input compared to the sinusoidal stress input. Table Calculated properties according to the equations in Theory forthe SRM249 sample at % and,% strain amplitude G M η M G L DSO % s s DSO,% J M ψ M s J L η L S T ψ L s R Q CSD % CSD,% Strain G M J M G L J L G M J M G L J L % ,%

14 922 Rheol Acta (2) 49:99 93 Xanthan gum solution Figure 6 shows an amplitude sweep on a xanthan solution at an angular frequency of rad/s. At small strains, the storage modulus is larger compared to the loss modulus, indicating a solid-like behavior. At larger strains, the loss modulus shows a strong increase before it deceases at even higher strain values. Without having any overshoot, the storage modulus starts to decrease at the same strain at which G increases. Toward larger strains, both moduli decrease further, with G having a larger slope compared to G.InFig.6, measurements performed with the three different oscillation techniques as described in Rheometer motor control are plotted. The general trend is the same for all the three oscillation techniques. However, a closer inspection shows that there is a difference in G at strains from 2% up to 4% between DSO on one side and CSS and CSD on the other side. In the DSO measurement, the G values run slightly higher in this region. The curves show that the data points with the CSD method are taken at the same strains as for the DSO method, which indicates that with the CSD technique, the strain amplitude control works well. Compared to the DSO and CSD measurements, the CSS test reveals only a few data points in the non-linear region. The obvious reason is that a slightly increasing stress leads to a large jump in strain when the sample has lower moduli in the non-linear region. Although increasing the number of data points per decade of stress in the crossover region from linear to non-linear behavior is possible, this is a major drawback of stress amplitude sweeps in general. In the CSS test, the results in this transition 3 2 G' G'' % 4 Strain γ Fig. 6 Amplitude sweep at an angular frequency of ω = rad/s for three different oscillation techniques: DSO (triangles), CSD (squares), and CSS (circles). The arrows indicate the strain at which the Lissajous diagrams in Fig. 7 are plotted regime depend on the exact setting of the measuring conditions. In a separate experiment not shown here, it has been confirmed that the results of a CSS data point in the crossover region depend on the number of oscillation cycles. The intra-cycle behavior at various strains can be investigated by looking at Lissajous plots with stress versus strain and stress versus strain rate as depicted in Fig. 7, where the Lissajous diagrams are plotted for all the three oscillation techniques. The data are normalized onto the base wave, i.e., the first harmonic. At small strains, the sample behaves like a visco-elastic solid. The area within the ellipse of the stress/strain plot is small, i.e., the dissipated energy is small, whereas the area within the ellipse of the stress/strain rate plot is large, i.e., the stored energy is large. With increasing strain, the area gets larger in the stress/strain plot (increasing dissipated energy) and the area in the stress/strain rate plots decreases (decreasing stored energy). With increasing strain, the shape changes from an elliptical to a non-elliptical form, indicating the crossover from a linear to a non-linear behavior. By comparing the results of the three different oscillation techniques, again, differences between DSO on one side and CSD and CSS on the other side can be seen. CSD and CSS reveal basically the same Lissajous plots at the same strain values. This is not surprising since with CSD, a sinusoidal stress input is applied as in the case of CSS. CSD controls onto the amplitude of the strain response but applies a sinusoidal stress. In the DSO measurement, an extra peak occurs in the Lissajous presentation of stress versus strain for larger strains. As shown in more detail in Fig. 8, this peak starts to evolve just after G reached its maximum at a strain of about 2%. The peak increases up to a strain of about 4%. The size of the peak is reduced at larger strains but still visible at strains up to,% as already shown in Fig. 7. At strains for which the peak in the stress/strain plot is at or close to its maximum, a kind of bow or loop is formed at the end of the elliptical figure in the stress/strain rate presentation. Both features, the peak and the bow, are not occurring in the CSD (and CSS) measurements. In order to understand the differences between the different features occurring in the Lissajous plots, the waveforms at a strain of 4% are depicted in Fig. 9 for the DSO (sinusoidal strain input) and CSD (sinusoidal stress input) measurements. The waveforms for DSO and CSD are significantly different. In CSD, the stress shows a nice sinusoidal shape. All non-linearity is in the strain wave. The strain waveform is not sinusoidal but has a shoulder. On the other hand, DSO produces a sinusoidal strain input,