On the wall slip phenomenon of elastomers in oscillatory shear measurements using parallel-plate rotational rheometry: II. Influence of experimental conditions

Accepted Manuscript On the wall slip phenomenon of elastomers in oscillatory shear measurements using parallel-plate rotational rheometry: II. Influence of experimental conditions Bastian L. Walter, Jean-Paul Pelteret, Joachim Kaschta, Dirk W. Schubert, Paul Steinmann PII:

S0142-9418(17)30254-4

DOI:

10.1016/j.polymertesting.2017.05.036

Reference:

POTE 5042

To appear in:

Polymer Testing

Received Date: 1 March 2017 Revised Date:

20 May 2017

Accepted Date: 24 May 2017

Please cite this article as: B.L. Walter, J.-P. Pelteret, J. Kaschta, D.W. Schubert, P. Steinmann, On the wall slip phenomenon of elastomers in oscillatory shear measurements using parallel-plate rotational rheometry: II. Influence of experimental conditions, Polymer Testing (2017), doi: 10.1016/ j.polymertesting.2017.05.036. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

On the wall slip phenomenon of elastomers in oscillatory shear measurements using parallel-plate rotational rheometry: II. Influence of experimental conditions Bastian L. Waltera,∗, Jean-Paul Peltereta , Joachim Kaschtab , Dirk W. Schubertb , Paul Steinmanna a Chair

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of Applied Mechanics, Friedrich-Alexander-University Erlangen-Nuremberg, Egerlandstraße 5, 91058 Erlangen, Germany b Institute of Polymer Materials, Friedrich-Alexander-University Erlangen-Nuremberg, Martensstraße 7, 91058 Erlangen, Germany

Abstract

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The use of parallel-plate rotational rheometry to characterize ex situ pre-prepared samples of rubber-like polymers is motivated by, for example, the investigation of magneto-rheological elastomers. When exceeding a critical excitation amplitude in oscillatory shear experiments, these elastomeric samples are prone to slip at the sample-plate contact interface. This phenomenon, known as wall slip, starts to occur at the sample’s outer rim and leads to an imperfect force transfer onto the sample. This results in a systematic error of measured rheological material quantities. A thorough investigation is presented to reveal how this phenomenon is affected by selected experimental conditions, namely the static axial preload and measuring frequency. For this purpose disc-shaped samples composed of an unfilled silicone rubber are prepared by casting and examined by means of a controlled stress rotational rheometer equipped with a serrated rotor configuration. The oscillatory strain sweep experiments suggest that wall slip, exclusively present at the serrated rotor surface, is significantly influenced by the static preload. In contrast, only a slight frequency dependence is observed within the examined experimental conditions. Further insights into the wall slip mechanism were attained by two novel methodologies. It is shown that it is possible to produce a master curve for the various applied preloads. This demonstrates that the physical mechanism behind wall slip is independent of the axial force. Furthermore, we derive an empirical model for the criterion governing the onset of wall slip. This links the critical stress at which wall slip is initiated to the static friction condition and geometrical aspects of the rotor configuration. From this it is anticipated that the conditions for reliable experiments involving ex situ pre-prepared samples composed of low damping elastomers can, in the future, be estimated a priori.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1. Introduction

17 18

The wall slip phenomenon is an experimental artifact that 19 was first systematically addressed independently by Mooney 20 [1] and Reiner [2] [3, 4]. It has been thoroughly investigated 21 in many different systems such as uncured rubbers, polymer 22 melts, polymer solutions and suspensions, gels, microgels, col- 23 loidal gels/glasses, and pastes (for example, see the very recent 24 review article by Hatzikiriakos [5], and the references therein). 25 Although the physical mechanism leading to wall slip is not 26 identical for all investigated systems, the experimental impact 27 is similar. For example, there is a systematic underestimation of measured rheological material quantities (such as the viscos- 28 29 ity), or an artificial yield point may be present. This paper continues the investigation of the wall slip phe- 30 nomenon that was commenced in [6]. The experimental arti- 31 fact was observed in oscillatory shear measurements conducted 32

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Keywords: wall slip, adhesive failure, measuring artifact, large amplitude oscillatory shear, silicone rubber, parallel-plate rotational rheometry

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∗ Corresponding

Author Email address: [email protected] (Bastian L. Walter) URL: http://www.ltm.uni-erlangen.de (Paul Steinmann)

Preprint submitted to Polymer Testing

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with ex situ pre-prepared elastomeric samples using parallelplate rotational rheometry. To the best of the authors’ knowledge the literature on the wall slip phenomenon with respect to cross-linked elastomeric materials is sparse; [6] is the only publication available to date that systematically explores this topic. Therein, wall slip, a measuring artifact caused by an adhesive failure at the sample-plate interface, is detected using two rotor surfaces and two different sample preparation techniques. Both smooth and serrated rotors were tested, and curing was performed in situ (within the rheometer to fix the sample to the measuring plates) and ex situ (by casting). For the ex situ pre-prepared samples wall slip was found to arise exclusively at the serrated plate surface. Using a highspeed camera, it was observed that the sample’s lower surface remained firmly bonded to the smooth stationary base plate. The imperfect force transfer leads to a substantial decrease in the storage modulus and a pronounced increase in the loss modulus. This is associated with the complex slip and stick-slip interaction at the sample-plate interface. For the smooth rotor configuration, a qualitatively similar system response was obMay 27, 2017

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2. Experimental setting

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2.1. Materials and sample preparation 113 An unfilled room-temperature-curing 2-component silicone114 rubber (RTV-2) is used in this study. This is done to avoid dis-115 sipative effects and possible viscoelastic nonlinearities caused116 by any particles present in the elastomer. By doing this (and117 through the careful choice of experimental configuration) we118 ensure that any strain-dependent dissipative effects detected in119 the system response can be solely attributed to the presence of120 wall slip. The raw materials, namely component A (containing121 the cross-linker) and component B (including the Pt-catalyst),122 consist of vinyl-terminated polydimethylsiloxane (PDMS) and123 are mixed in the ratio 2:1. All ex situ pre-prepared disc-shaped124 samples (20 mm, t = 1 mm) are fabricated according to the125 standardized methodology presented in [6]. 126

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used, and slip is initiated only on the rotor-sample interface. The rotor surface which has a pitted size of 0.5 mm and the overall experimental methodology are shown in [6]. The onset and development of wall slip was examined by oscillatory strain sweep experiments performed in the strain range of 10−5 to 1 at 25 ◦C and the direct strain oscillation (DSO) mode of the rheometer has been utilized. Each experiment was conducted at least three times to ensure that the results were repeatable. The normal force FN and angular frequency ω were varied in the range of 1 to 40 N and 0.05 to 50 rad s−1 , respectively. It is crucial to note that the applied preload results in the partial penetration of the rotor tips into the pre-prepared sample’s upper surface. By performing quasi-static compression tests at up to FN = 45 N at 25 ◦C, this was quantified to be within the range of 25 to 300 µm (1 to 40 N).

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served. However, significant sample buckling and dislocation 86 was also observed for this plate configuration. In contrast to 87 the pre-prepared samples, the in situ cured sample firmly estab- 88 lished that the ‘no-slip’ condition exhibited a nearly linear vis- 89 coelastic response over the entire applied range of shear strains. 90 In this work, a comprehensive and systematic study is pre- 91 sented in order to reveal how the wall slip phenomenon of cross- 92 linked elastomers is affected by selected experimental condi- 93 tions. In particular, we focus on the influence of the applied 94 static preload and the measuring frequency. As the former di- 95 rectly affects the static friction conditions, it is hypothesized 96 that it will significantly influence the onset and development of 97 wall slip. In conditions of slip, it is proposed that the system 98 response may be frequency-sensitive due to the velocity depen- 99 dence of the coefficient of dynamic friction. 100 The remainder of the manuscript is organized as follows: The experimental settings, including the material, sample prepara101 tion and characterization methods, are summarized in section 2. The experimental results and corresponding discussion are di-102 vided in three subsections. The influence of the static axial 103 preload is described in section 3.1, while section 3.2 is dedi104 cated to the study of the influence of the measuring frequency. 105 Thereafter an analysis of the critical shear stress required to ini106 tiate wall slip is conducted in section 3.3. Lastly, a summary 107 of the study and the conclusions drawn from it are provided in 108 section 4.

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3. Results and discussion

3.1. Influence of the static axial preload 3.1.1. Evolution of the storage and loss modulus Figure 1 shows the evolution of the storage G0 and loss G00 modulus2 attained by oscillatory strain sweep experiments conducted with ex situ pre-prepared disc-shaped samples at FN =1, 3, 10 and 30 N, and ω = 10 rad s−1 . We also present a set of data for in situ cured samples using a smooth rotor configuration at FN = 10 N and the same angular frequency (data taken from [6]). Significant differences between the slip and no-slip condition are evident, and are discussed in detail by Walter et al. [6]. Overall, the highly reproducible evolution of both G0 and G00 are qualitatively similar for all applied preloads. Visible is a strain range wherein the system response is linear, and both moduli are independent of the applied strain amplitude. At high strains (once slip is initiated) the system response becomes nonlinear, and exhibits a pronounced peak in the loss modulus in conjunction with a significant decrease in the storage modulus. Furthermore, there is a cross-over present in G0 and G00 that is located very close beyond the peak maximum of G00 . From the comparison presented in fig. 1 it is possible to determine at which strain the extracted data is no longer reliable. The authors would like to reinforce that, for the results obtained under conditions of slip, the extracted material parameters (G0 , G00 and others) are not indicative of the true material behavior [6]. The slip present at the sample-rotor interface is a violation of the boundary conditions of the equations characterizing the rheological experiment. The results obtained in this regime, regardless of their reproducibility, are thus unreliable. They do, however, provide some qualitative representation of the influence of slip (and the conditions affecting slip) on the system response. An increase in the applied preload yields higher values of both moduli within the linear regime. Furthermore, a variation of FN from 3 N to 30 N results in a significant shift of the peak

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2.2. Oscillatory rheology 128 Oscillatory shear measurements are conducted by means129 of the controlled stress1 rotational rheometer MCR 502 (An-130 ton Paar, Austria). A parallel-plate configuration (20 mm)131 with a smooth stationary base plate and a serrated rotor132 (PP20/MRD/Ti/P2) is utilized. Pronounced sample buckling133 and dislocation was previously observed when a smooth rotor134 was employed [6]. Therefore a serrated rotor was preferentially135 136

1 For general information regarding rheometer motor control mechanisms, i.e. controlled stress (CS) and controlled strain rate (CR), see [7] and references therein.

2 Both G 0 and G 00 are computed from the fundamental wave of the total system response; higher harmonic contributions are neglected.

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10

10 1 N

10 N

3 N

30 N

6

10 N (no slip) (Walter et al. 2016)

5

10

G'

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10

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G''

3

2

10

RTV-2 (2:1), serrated rotor,

= 10 rad s

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10

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10

-3

0

10

-2

RTV-2 (2:1),

, T = 25°C

10

-1

10

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145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168

0

10

1

1

, T = 25°C, serrated rotor

10

2

10

3

10

4

10

5

(Pa)

Figure 2: Evolution of G0 and G00 , parameterized in terms of the measured shear stress amplitude σ0 . The data are obtained by oscillatory strain sweep experiments on RTV-2 (2:1) ex situ pre-prepared disc-shaped samples using a serrated rotor (smooth stator) at FN = 1, 3, 10 and 30 N and ω = 10 rad s−1 .

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in G (both its onset and maximum value) to higher strains and172 higher values of G00 . The onset of wall slip is delayed by ap-173 proximately one order of magnitude to larger excitation ampli-174 tudes (from γ0 ≈ 5 × 10−3 to γ0 ≈ 5 × 10−2 ). The transition175 from a linear to nonlinear system response for G0 is affected in176 a similar manner. Associated with an increase in the preload is a measurable change in the magnitude of both moduli in the linear regime. This is closely related to the gap setting and the partial penetration of the rotor tips into the sample’s upper surface (discussed in [6]). The application of a very low static axial preload results in only slight tip penetration leading to a small sample-plate177 contact area and only local deformation. The experimental out-178 come at high preloads is affected by the compression modulus179 of the material and the locally non-uniform stress distribution180 generated by the rotor tips penetrating into the sample. 181 In general, an increase in FN leads to an increase in the static182 friction, and the sample-plate contact area (due to penetration183 of the rotor tips). Collectively these are responsible for the pro-184 nounced shift of the onset of wall slip. The maximum normal185 force that can be applied by the MCR 502 (and which is typi-186 cal for this class of rotational rheometers) is FN = 50 N. With187 respect to this normal force limit it is hypothesized that wall188 slip, if present, cannot be eliminated (without the occurrence189 of cohesive failure) but is rather shifted to larger applied shear190 deformations or stresses. The adhesive failure that occurs when slip is initiated (at the sample’s outer rim) is characterized by a critical shear stress. To gain further insight into this critical stress, it is beneficial to re-parameterize the experimental data shown in fig. 1 in terms of the measured shear stress amplitude. This is presented in fig. 2.

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10

= 10 rad s

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(-)

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Figure 1: Influence of the applied static axial preload FN = 1, 3, 10 and 30 N on the evolution of the storage and loss moduli (G0 , G00 ) obtained by oscillatory strain sweep experiments on RTV-2 (2:1) ex situ pre-prepared disc-shaped samples using a serrated rotor (smooth stator) at ω = 10 rad s−1 and T = 25 ◦C. The error bars are within the symbol size. Additionally, there is a set of data present obtained for RTV-2 (2:1) in ‘no-slip’ condition (cured in situ, smooth rotor, same batch) taken from [6]. Raw waveform data for select data points is 169 shown in the supplementary document.

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G', G'' (Pa)

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G', G'' (Pa)

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It appears that the data in fig. 1 and fig. 2 are of comparable shape and, thus, could be shifted in the horizontal and vertical directions to create a master curve3 . This would imply that the physical mechanism governing the wall slip phenomenon is identical for all applied preloads. A corollary is that if slip is initiated then the evolution of the measured system response (due to wall slip) is solely influenced by the applied normal force. The horizontal and vertical factors used to shift both moduli, respectively denoted aF and bF and defined as aF =

σmax 0 (F N ) ref σmax 0 (F N )

,

bF =

G00(σ0 =20 Pa) (FN ) G00(σ0 =20 Pa) (FNref )

(1)

are determined from the shear stress amplitude at the peak max00 imum in G00 (σmax 0 ) and G at σ0 = 20 Pa (within the linear region). Figure 3 showcases the resulting master curve obtained for a reference axial preload of FNref = 30 N and ω = 10 rad s−1 . It is evident that only one aF and bF are necessary to shift both the storage and loss modulus to create an accurate master curve that is valid in the slip and no-slip regimes. However, within the wall slip regime (above aF σ0 = 5 × 103 Pa) there is a marginal (but negligible) deviation visible in the evolution of both moduli. It is hypothesized that this might be related to the area of local slip. Similar to the time-temperature superposition, it is possible to predict the system response under conditions that have not been directly measured. Figure 4 shows the horizontal shift factor for the master curve presented in fig. 3, as well as for a lower frequency (ω = 0.1 rad s−1 ). As it is linked to the friction conditions, aF is 3 Compare with the time-temperature superposition valid for thermorheological simple polymer melts.

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F

10

10

6

5

1 N

10 N

3 N

30 N

not correlate linearly with the non-dimensional gap value. At high preloads bF is also affected by the compression modulus of the material and a locally non-uniform stress distribution at the sample plate interface. It has been determined that the power law " #n d bF = b0F − m (3) dref

b G' F

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b G''

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200 201 RTV-2 (2:1),

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, T = 25°C, F

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10

-1

10

0

10

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10

2

10

3

10

ref N

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10

provides a reasonable fit to the data. As a side note, it is possible to confirm the initial sample thickness from this correlation. The material thickness under no-load conditions was estimated to be t = 1.039 mm, which is in the range expected from the sample preparation technique.

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b G', b G'' (Pa)

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a

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0

(Pa)

RTV-2 (2:1)

G00

Figure 3: Master curve of and presented in terms of the measured ref = 30 N at shear stress amplitude σ0 for a reference axial preload of FN ω = 10 rad s−1 after horizontal (aF ) and vertical (bF ) shifting. The data are obtained by oscillatory strain sweep experiments on RTV-2 (2:1) ex situ preprepared disc-shaped samples using a serrated rotor (smooth stator) at FN = 1, 3, 10 and 30 N. The original data (without shifting) is shown in fig. 2.

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Linear Regression

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0.01

(F /F N

ref N

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Figure 4: Horizontal shift factor aF obtained according to eq. (1) as a funcref = 30 N220 tion of a non-dimensional stress to create the master curve for FN ref = 30 N at221 at ω = 10 rad s−1 . The horizontal shift factor obtained for FN 222 ω = 0.1 rad s−1 is also shown. 197

(d/d

1.15 ref

1.20

1.25

1.30

) (-)

Figure 5: Vertical shift factor bF obtained according to eq. (1) to create the ref = 30 N at ω = 10 rad s−1 in fig. 3. These data points (and master curve for FN therefore fit function) are also valid for ω = 0.1 rad s−1 .

3.2. Influence of the measuring frequency

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1.05

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204

serrated rotor, T = 25°C

0.01

1.00

From fig. 1, fig. 3 and fig. 4 it can be concluded that the static axial preload is one key parameter that governs the overall system response due to wall slip. This is a reasonable finding, as the preload directly affects the static friction condition and indirectly influences the sample-plate contact area.

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F

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n = 6.86

0.95

where d represented the measured gap between the rotor tips and the stationary base plate. This factor can be motivated by the expression governing the geometry of the sample-plate contact area. Using a regression analysis, the horizontal shift factor was found to be proportional to this reduced stress (linear correlation with slope 0.96) and insensitive to the angular frequency.202

1

= 1.15

0.4

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192

F

serrated rotor, T = 25°C

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b

= 0.990

m = 0.14

0.5

FNref

= 30 N

Fit (equ. 3)

0.7

0.6

represented as a function of the non-dimensional stress

N

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G0

ref

F

1.0

223

Figure 5 showcases the vertical shift factor associated with224 the master curve shown in fig. 3. It is observed that it does225 4

3.2.1. Evolution of the storage and loss modulus Figure 6 shows a set of data comparable to fig. 1 but for oscillatory strain sweep experiments conducted at ω = 0.1, 1 and 10 rad s−1 and a fixed static axial preload of FN = 10 N. Data from [6], indicating the ‘no-slip’ condition, is also presented. As before, the reproducibility of the storage modulus is very high. However, there is a data scatter visible in the loss modulus at low strains that increases with decreasing measuring frequency. This is due to resolution issues pertaining to the rheometer at low shear strains and stresses. Similar to fig. 1 a distinct linear system response evolves into a pronounced nonlinear regime with a comparable trend in the evolution of the complex moduli. Within the linear regime, an increase in ω leads to measurably higher values in the loss modulus, whereas the storage modulus is only slightly affected by the applied frequency. When compared to the case in which the preload was varied, the point of transition from a linear to

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G', G'' (Pa)

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10 rad s

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RTV-2 (2:1)

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N

-2

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nonlinear response was hardly affected by a two-order of magnitude change in the angular frequency. From visual inspection, it appears that an increase in ω shifts the onset of wall slip only slightly towards smaller excitation amplitudes. Furthermore, higher measuring frequencies yield a broader peak in G00 , a small shift of both its peak maximum and the cross-over (of G0 and G00 ) towards larger shear deformations, and a measurable change of the slope of the storage modulus within the nonlinear regime. This frequency dependent response (crossover excluded), in particular the evolution of the loss modulus, can be likely misinterpreted since similar trends have been observed in particle reinforced elastomers (frequency dependence of the Payne effect) [8]. For instance, an increasing measuring frequency leads to a broader maximum in G00 . The vertical offset of the loss modulus in the linear regime is caused by the viscoelastic nature of the low damping silicone rubber, which still possesses a measurable frequency dependency within the range of ω = 0.1 to 10 rad s−1 . The reasons for the differences in the evolution of G0 and G00 and, therefore, in the frequency dependent development of the wall slip phenomenon remain unclear. It is hypothesized that this deviation might be related to the coefficient of kinetic friction and its velocity dependence (for example, see [9, 10])4 and/or to strain position control issues directly linked to the use of a controlled stress rotational rheometer utilizing its DSO mode. As discussed by Walter et al. [6] for experiments conducted at ω = 10 rad s−1 , wall slip disturbs the real-time strain position

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where Rc is the minimal radius at which slip commences. The relationship between the nominal pressure P¯ and the resulting normal force is " #−1 3 P¯ = FN R2c − RRc . (5) 2

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3.3. Analysis of the critical shear stress required to initiate wall slip It is desirable that a deeper analysis of the wall slip condition be conducted, and the expected influence of experimental and material parameters on the experimental outcome be determined. To motivate how this is performed, we utilize the standard theory of clutch design [11, 12] to compute the total torque applied to the sample surface. Considering an infinitesimal annulus at radius r on the frictional interface, the applied normal force for a smooth plate and rotor configuration is dFN = P (r) dA = P (r) 2πrdr. The force associated with friction is therefore dF = µ (r) dFN , while the resulting torque is then dT = rdF = µ (r) P (r) 2πr2 dr. What remains is to define a pressure distribution profile P (r) and the friction coefficient µ (r). Due to the parallel-plate configuration, we assume that the slipping annulus at r > Rc remain parallel, thereby leading to a uniform wear rate at a constant velocity. The pressure distribution applied to a sample of overall radius R is then [11, 12]   ¯  if 0 ≤ r ≤ Rc  P (4) P (r) =  R c   otherwise P¯ r

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Figure 6: Influence of the applied measuring frequency ω = 0.1, 1 and 10 rad s−1 on the evolution of G0 and G00 obtained by oscillatory strain sweep269 experiments on RTV-2 (2:1) ex situ pre-prepared disc-shaped samples using a270 serrated rotor (smooth stator) at FN = 10 N and T = 25 ◦C. The data scatter visible in G00 at low strains in conjunction with low frequencies is due to resolution issues pertaining to the rheometer. Additionally, there is a set of data present obtained for RTV-2 (2:1) in ‘no-slip’ condition (cured in situ, smooth rotor, same batch) taken from [6]. Raw waveform data for select data points is shown in the supplementary document.

control onto the desired strain sine waveform resulting in a measurably non-sinusoidal strain. This effect is assumed to be further exacerbated at higher frequencies in conjunction with large excitation amplitudes. In summary, the wall slip phenomenon seems to be only slightly frequency dependent. As opposed to the case of changing preload (section 3.1.1), it is not possible to create a master curve of the data shown in fig. 6 for differing frequencies. From that it is hypothesized that the physical mechanism that governs the total system response due to wall slip is affected by the applied angular frequency. This may be related to, amongst other factors, the velocity dependence of the coefficient of kinetic friction. Although not directly in the scope of this work, the waveform data for the experiments is shown and analyzed in the supplemental document. This provides further insights into the influence of the normal force and frequency on the system response.

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Furthermore, for simplicity it is assumed that the kinetic friction coefficient µd , applicable in the region of slip, is related to the static friction coefficient µs by a constant factor    if 0 ≤ r ≤ Rc µs µ (r) =  (6)  µd = κµs otherwise. By considering the non-slip and slip regions separately, the total torque is " 3 #" #−1 R R2 − R2c 3 2 slip T smooth = 2πµs FN c + κRc Rc − RRc . (7) 3 2 2

4 The sample-plate interaction within the wall slip regime is very complex for several reasons. It is therefore not possible to extract a reliable coefficient of kinetic friction from the conducted experiments.

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Observe that while the torque and normal force are measured, there remain three unknown quantities. At the onset of slip, when Rc = R, eq. (7) reduces to

10

4

F

= 10 N

N

Linear Regression (slope = 0)

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-2

10

serrated rotor, T = 25°C

-1

10

0

10

(Pa) crit

(rad s

(b) 10

4

Fit to 0.1 & 10 rad s

10

R

)

1

= 0.19

2

= 0.95

2

linear space, value

RTV-2 (2:1)

10

2

(equ. 12) S

10

10

1

10 rad s

3

1

1

1

0.1 rad s

serrated rotor, T = 25°C

1

10

0

10

F

N

1

10

2

(N)

Figure 7: Influence of (a) the measuring frequency ω (0.05 to 50 rad s−1 ) at FN = 10 N and (b) the static axial preload FN (1 to 40 N) at ω = 0.1 and 10 rad s−1 on the critical shear stress amplitude σcrit 0 computed by the evolution of the viscous shear stress amplitude γ0 G00 (see Appendix A).

serrated rotor a I II d

h

t

smooth stator Figure 8: Sketch of the partial penetration of the rotor tips into the samples upper surface. I qualitatively depicts the expected real sample-plate contact and, II presents the simplified sample-plate contact geometry used to approximate the total contact area Acont as a function of the applied preload. The sample thickness and the measured gap are respectively denoted by t and d, and the penetration depth is given by h.

The total sample-plate contact area, as approximated using a geometric simplification (fig. 8, case II), is " # Aplate h2 cot α 4 , (9) Acont (h) = Atip sin α 316

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linear space, value

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which has a single unknown, namely the static friction coefficient. A similar relationship could be derived for the serrated rotor configuration, for which the contact area Acont = Acont (r) has a more complex description, in order to deduce the static coefficient of friction between the sample and rotor. Raw torque data measured during the experiment gives little insight into the slip initiation point (i.e. strain at which the dissipation artificially increases). Therefore an alternative approach that uses the stress-strain data, for which the critical shear stress σcrit 0 can be reliably determined, is utilized. Details on how this examination is conducted are provided in Appendix A. Visible in fig. 7 are the values of the critical shear stress, as determined from the experimental data in fig. 1 and fig. 6. The influence of the measuring frequency (0.05 to 50 rad s−1 ) at FN = 10 N is presented in fig. 7(a). Figure 7(b) shows the corresponding influence of the static axial preload (1 to 40 N) at both ω = 0.1 and 10 rad s−1 . Overall, the computational error involved in determining the critical stress is small, demonstrating the method’s robustness. The results are influenced by numerous experimental uncertainties such as the slight variation in the sample thickness (that also affects the contact area), and the relatively small number (3 - 10) of examined samples. With respect to fig. 7(a) it appears, that there is no clear relationship between the measuring frequency and the computed critical shear stress required to initiate slip. In contrast, fig. 7(b) demonstrates that there is a clear correlation between the static axial preload and the shear stress required to initiate slip. This is in accordance with the data previously shown in fig. 1, as well as fig. 3 and fig. 4. Increasing the preload leads to higher values of the critical shear stress. This is due to an increase in both static friction and sample-plate contact area. This observation raises the question as to whether some further information, such as the coefficient of friction, can be extracted from fig. 7(b). The contact area Acont is governed by the penetration depth of the rotor tips h and geometrical aspects of the serrated rotor surface, such as the slant angle 0 < α < 90◦ of the square pyramid shaped tips. Figure 8 illustrates the partial tip penetration 0 ≤ h ≤ H (where H is rotor tip height) into the sample’s upper surface. Assuming a constant sample and rotor geometry, the primary factors that affect the penetration depth are the material’s compression modulus and the applied static axial preload.

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4π Rµs FN 3

0

crit T smooth =

where Aplate and Atip are, respectively, the projected area of317 the measuring plate πR2 (plate radius R) and a single tip (base318 6

length 2H cot α). The ratio between these values provides an estimate of the number of tips that penetrate the sample’s upper surface. It is crucial to note that this approximation results in a

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σcrit 0 (F N ) 320 321 322 323 324 325 326 327 328

341

µ s FN H 2 = . 2πR2 h2

(10)342 343

Therein we have approximated the uniform normal force along344 the slanted rotor tips to be FN cos α, with α = 45◦ , and the345 contact area is given by eq. (9). The tip penetration, dependent on the applied preload, can346 be expressed as h(FN ) = t − d(FN ). As is shown in fig. 9, this relationship between d and FN is established by fitting the ex-347 perimental data of a quasi static compression test. This test was348 conducted with the serrated rotor configuration and preloads up349 to 45 N (T = 25 ◦C). 350 351 1.1

352

serrated rotor, T = 25°C

RTV-2 (2:1)

353 1.0

354

0.9

t =

355

1.036 mm

356

D = 0.377 mm

0.8

357

b = 59.5 N

358

c = 0.516

359

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360 361

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362 363

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Figure 9: Quasi-static compression test conducted with the serrated rotor con-368 figuration (FN = 0.1 to 45 N, T = 25 ◦C). The experimental data is fitted using eq. (11) to detect the sample thickness t and partial penetration of the rotor tips369 370 h(FN ) = t − d(FN ).

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A nonlinear regression function that fits the experimental372 data is 373 t−D d(FN ) = (11)374 h F ic + D , 375 1 + bN 329

376

where the sample thickness t, and b, c and D are fit parameters. 377 Note that for infinitesimally small (as well as very large) values 378 of the axial force, the gradient is near zero. 379 Combining eq. (10) with h(FN ) obtained from fig. 9 using 380 eq. (11), the expression for the critical shear stress is  −2  2  µ F H t − D  s N  . t − + D σcrit h i c  0 (F N ) = 2R2 π  1 + FN b

381 382

(12)383 384 385

332 333 334 335 336 337

4. Conclusions

The wall slip phenomenon of cross-linked elastomeric materials, as characterized by means of parallel-plate rotational rheometry, has been investigated with respect to the influence of the applied static axial preload (FN = 1 to 40 N) and measuring frequency (ω = 0.05 to 50 rad s−1 ). Oscillatory strain sweep experiments were conducted using ex situ pre-prepared disc-shaped samples in conjunction with a serrated rotor configuration, for which wall slip occurs exclusively at the serrated rotor surface [6]. A robust class of computational methods have been introduced to determine the critical shear strain and corresponding shear stress at which slip is initiated. Lastly, a simple empirical model has been derived to fit the critical shear stress as a function of the applied preload. It takes geometrical aspects of the measuring plate into account, and from this it was possible to approximately determine the coefficient of static friction. The total system response was found to be qualitatively similar for all experimental conditions. There was a distinct linear regime for the complex moduli that turned into a pronounced nonlinear regime as a result of wall slip. The loss modulus exhibited a pronounced peak and a significant decrease in the storage modulus was also observed. However, results were affected by artifacts related to the use of a parallel-plate configuration, the partial penetration of the serrated rotor surface and wall slip. The point of initiation of wall slip was considered to be that at which the loss modulus deviates 10 % from the value obtained in the linear viscoelastic regime. Due to its dependence on the static friction, wall slip was found to be very sensitive to the applied preload. For instance, a variation from 3 to 30 N resulted in a delay of the onset of wall slip by approximately one order of magnitude to larger excitation amplitudes. Furthermore, the preload dependent experimental data (the storage and loss moduli), presented in terms of the measured shear stress, could be shifted in horizontal and vertical direction to create a master curve for a fixed measuring frequency. This suggested that the physical mechanism that governs the onset and development of slip is identical for all applied preloads. The corresponding horizontal shift factor was found to be a function of both the applied preload and the penetration depth of the rotor tips. Motivated by these observations, it was possible to extract an approximate coefficient of static friction for the used sample and rotor materials. For this computation is was necessary to take the geometrical aspects of the measuring plate into account. The estimated friction coefficient was found to be of the correct order of magnitude.

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Fit (equ. 11 )

rotor). This therefore validates the methodology used to first determine the critical shear stress related to the onset of slip. A corollary of this is that in order to estimate σcrit 0 (F N ) for the serrated rotor geometry, it is only necessary to perform a quasistatic compression test (to fit the parameters in eq. (11)) and provide a reasonable, but conservative estimate of µs . However, this approach applies only to low damping elastomers and has not been proven to be valid for high damping elastomers.

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slight overestimation of Acont . 338 Assuming no deformation other than the penetration of the339 serrated rotor surface, the critical shear stress amplitude is 340

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Fitting the data shown in fig. 7(b) using eq. (12) results in an386 approximated coefficient of static friction of µs = 0.19 (±6 %).387 Similar values were obtained for silicone rubber on steel (µs =388 0.188 and 0.16) by Yang et al. [13]. This value appears to389 be within a realistic order of magnitude for the coefficient of390 static friction expected for silicone rubber on titanium (serrated391 7

398 399 400 401 402 403 404 405

406

407 408 409 410

20

Exp. data (F

N

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= 10 rad s

1

, T = 25°C)

Fitted model

Linear region

15

Acknowledgement The support of this work by the European Research Council (ERC) through the Advanced Grand 289049 MOCOPOLY is gratefully acknowledged. The authors would like to thank Dr. Z. Starý for his comments and discussions related to this work.

0.05

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In contrast to the preload, the onset and development of wall slip was found to be only slightly frequency dependent within the range of applied measuring frequencies. Furthermore, we found no clear relationship between the applied frequency and the computed critical shear stress. It was also not possible to shift the complex moduli to create a master curve similar to the preload. As a final note, it can be concluded that wall slip, if present in experiments conducted by parallel-plate rotational rheometry, cannot be eliminated without the occurrence of any cohesive failure. It can, however, be delayed to larger shear deformations. Further work is being conducted to study the influence of material properties (such as the influence of cross-linking density) on the onset and development of wall slip.

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Exp. data (F

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Fitted model Linear region

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It has been shown for LAOS experiments conducted with unfilled elastomeric systems that the increase in the loss modulus G00 is a reliable indicator of the onset of wall slip [6]. It is therefore necessary to develop tools to detect the critical shear strain amplitude γ0crit at which slip is initiated. The method should be accurate, robust and applicable to a wide range of experimental conditions. Illustrated in fig. A.10 is an example of the result of our approach that computes the strain at which the viscous shear stress σv = γ0G00 (or, correspondingly, the loss modulus) is no longer linearly dependent on the applied strain amplitude. There exist a number of possible approaches to achieve this goal. We will first outline the current approach and subsequently briefly discuss some alternative methods that were developed and tested, but later discarded. Instead of determining the critical shear strain directly from the experimental data, we choose to fit to the data a model (for G00 in particular) that quantitatively reproduces the overall nonlinear system response. This model is then used to compute the viscous shear stress σv and, by means of some indicators that will be described later, used to estimate γ0crit . Some necessary characteristics of the model, as governed by the measured system response, are that it should have a linear region (corresponding to the real material behavior) and a nonlinear region (which captures the response due to slip). Such a model does not necessarily have to be phenomenologically correct as we are modeling an experimental artifact and not the true material behavior. Furthermore, since we are only interested in determining the point of onset of slip, it is not strictly necessary that the entire slip response is modeled exactly. However, using a model that can match this recorded response will be of benefit as it can easily be assured that the transition between the linear and nonlinear regions is accurately represented.

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Appendix A. Computation of critical point

10

10

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0

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(b) Fit in logarithmic space Figure A.10: Viscous shear stress σv obtained by oscillatory strain sweep experiments on RTV-2 (2:1) pre-prepared samples at FN = 10 N and ω = 10 rad s−1 using a serrated rotor (smooth stator); experimental data taken from [6]. In (a) the model is fitted in linear space, (b) showcases the fit in logarithmic space.

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461 462 463 464 465 466 467 468 469

σv (γ (γ)) = Gv (γ) + γ

,

(A.5)

γ dσv (γ) γ dGv (γ) dσv (γ (γ)) = ≡1+ dγ σv (γ) dγ Gv (γ) dγ

Appendix A.1. Model in linear and logarithmic space Comparing the captured experimental data to numerous models presented in the literature, it was determined that the Kraus model [14] for the loss modulus exhibited response characteristics that are qualitatively similar to that seen in the data (also see [15] for further details). This model has been lightly modified with respect to the exponent n (instead of 2m) to improve the quality of the fit. For the sake of clarity, the loss modulus G00 and the shear strain amplitude γ0 are further written as Gv and γ, respectively. The loss modulus Gv , defined as a function of the shear strain amplitude γ in linear space is given by h im h i 2 γγc max (A.1)473 Gv (γ) = Ginf − Ginf h in v + Gv v 474 1 + γγc

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Since rheological data is often interpreted in logarithmic space, it is pertinent to consider interpreting the system response and the determination of γ0crit from this viewpoint. Using the fundamental logarithmic identities and the logarithmic d log ( f (x)) x d f (x) derivative d log10 (x) = f (x) dx , it is possible to transform the 10 previously described model for the loss modulus and viscous shear stress from linear to logarithmic space. Denoting logarithmic quantities as (•) = log10 (•), the viscous shear stress and its gradient are given by

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The use of a fitted model, as opposed to directly using the recorded data, to determine the onset of slip is necessary as it is expected that the critical point lies between the recorded data points. Two alternative approaches that we evaluated were based on (i) fitting a polynomial function (piecewise-defined spline) to the entire dataset, and (ii) performing a linear regression on the linear region of the data. However, both of these methods were highly susceptible to the influence of experimental errors at low strains due to resolution issues pertaining to the rheometer. This noise induced oscillations in the fitted polynomials (even if damping was applied) and led to an ill-fit in the regression approach. Reliably post-processing the fitted curves as determined by these methods for the critical point, using the later detailed indicators, proved to be challenging. The primary benefit of using the current approach is that it is significantly more resistant to the effect of experimental noise that is prevalent at low strains.

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where Ginf v is the loss modulus at infinitely low or infinitely high475 strain, Gmax is the maximum value of the loss modulus occurv ring at the critical strain amplitude γc , and m, n are exponents that govern the shape of the local maximum in the loss modulus. The gradient of this function with respect to the shear strain is h in h i " γ #m γγc [n − m] − m dGv (γ) , = −2 Gmax − Ginf h h im i2 v v dγ γc γ 1 + γγc (A.2)

, (A.6)

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with the loss modulus Gv and its gradient in linear space given by eq. (A.1) and eq. (A.2). The linear response for the loss modulus and viscous shear stress, as defined in the logarithmic space, are dGv (γ) e , (A.7) Gv (γ) = mGv γ + cGv , mGv = dγ γ=γ min
cGv = Gv γmin − mv γmin

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(γ) dσ v ev (γ) = mσ γ + cσ , mσ = σ dγ γ=γ min
cσ = σv γmin − mσ γmin

,

(A.8)

where γmin denotes the minimal logarithmic strain at which experimental recordings were taken. Appendix A.2. Model fitting and computing γ0crit To fit the modified Kraus model to the experimental data, a binary generic algorithm [16, 17] is used to compute the coefficients of eq. (A.1) that minimize the difference in the loss modulus between the experimental data and the fitted curve. In order to remove the majority of the noisy data from later consideration, a conservative approximation to the critical point b γ0crit is made by finding the first experimental point that does not satisfy ev (γ) | < 1 % . |σv (γ) − σ

(A.9)

and the viscous shear stress and its gradient are given by σv (γ) = Gv (γ) γ

and

dσv (γ) dGv (γ) = Gv (γ) + γ dγ dγ

Thereafter minimization of an error function using the bisection γ0crit . In linear method [18] renders the critical strain γ0crit > b space, two possible definitions of relative error functions are given by

. (A.3)

The linear region of the loss modulus and viscous shear stress, f are simply with quantities denoted by (•), ev (γ) = Ginf G v 471 472

ev (γ) γ ev (γ) = G and σ

.

    ev (γ) 2 − tolval × σ ev (γ) 2 = 0 , eval (γ) = σv (γ) − σ (A.10) " #2 " #2 dσv (γ) de σv (γ) de σv (γ) egrad (γ) = − − tolgrad × =0 dγ dγ dγ (A.11)

(A.4)

Note that in linear space the viscous shear stress curve passes through the origin and has a slope of gradient Ginf v . 9

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[15] J. L. Leblanc, Filled polymers: science and industrial applications, CRC Press, 2009, iSBN 9781439800423. [16] D. E. Goldberg, Genetic algorithms in search, optimization and machine learning, 1st Edition, Addison-Wesley Longman Publishing Co., Inc., Massachusetts, USA, 1989, iSSN 978-0201157673. [17] R. L. Haupt, S. E. Haupt, Practical Genetic Algorithms, 2nd Edition, John Wiley & Sons Inc., New Jersey, USA, 2004, iSBN 0-471-45565-2. [18] W. H. Press, Numerical recipes 3rd edition: The art of scientific computing, Cambridge university press, 2007. [19] M. D. Graham, Wall slip and the nonlinear dynamics of large amplitude oscillatory shear flows, Journal of Rheology 39 (4) (1995) 697–712. [20] M. J. Reimers, J. M. Dealy, Sliding plate rheometer studies of concentrated polystyrene solutions: Nonlinear viscoelasticity and wall slip of two high molecular weight polymers in tricresyl phosphate, Journal of Rheology 42 (3) (1998) 527–548. [21] C. Klein, H. W. Spiess, A. Calin, C. Balan, M. Wilhelm, Separation of the nonlinear oscillatory response into a superposition of linear, strain hardening, strain softening, and wall slip response, Macromolecules 40 (12) (2007) 4250–4259. [22] K. Hyun, M. Wilhelm, C. O. Klein, K. S. Cho, J. G. Nam, K. H. Ahn, S. J. Lee, R. H. Ewoldt, G. H. McKinley, A review of nonlinear oscillatory shear tests: Analysis and application of large amplitude oscillatory shear (LAOS), Progress in Polymer Science 36 (12) (2011) 1697–1753.

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[1] M. Mooney, Explicit formulas for slip and fluidity, Journal of Rheology 2 (2) (1931) 210–222. [2] M. Reiner, Slippage in a non-newtonian liquid, Journal of Rheology 2 (4) (1931) 337–350. [3] S. G. Hatzikiriakos, J. M. Dealy, Wall slip of molten high density polyethylene. i. sliding plate rheometer studies, Journal of Rheology 35 (4) (1991) 497–523. [4] W. B. Russel, M. C. Grant, Distinguishing between dynamic yielding and wall slip in a weakly flocculated colloidal dispersion, Colloids and Surfaces A: Physicochemical and Engineering Aspects 161 (2) (2000) 271– 282. [5] S. G. Hatzikiriakos, Slip mechanisms in complex fluid flows, Soft Matter 11 (40) (2015) 7851–7856. [6] B. Walter, J.-P. Pelteret, J. Kaschta, D. W. Schubert, P. Steinmann, On the wall slip phenomenon of elastomers in oscillatory shear measurements using parallel-plate rotational rheometry: I. detecting wall slip, Polymer Testing submitted (-) (2017) —. [7] J. Läuger, H. Stettin, Differences between stress and strain control in the non-linear behavior of complex fluids, Rheologica Acta 49 (9) (2010) 909–930. [8] A. Lion, C. Kardelky, P. Haupt, On the frequency and amplitude dependence of the payne effect: Theory and experiments, Rubber Chemistry and Technology 76 (2) (2003) 533. [9] K. A. Grosch, The relation between the friction and visco-elastic properties of rubber, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 274, The Royal Society, 1963, pp. 21–39. [10] B. N. J. Persson, Theory of rubber friction and contact mechanics, The Journal of Chemical Physics 115 (8) (2001) 3840–3861. [11] J. E. Shigley, Mechanical Engineering Design, 5th Edition, McGrawHill, Pennsylvania Plaza, New York, New York, USA, 1989, iSBN 9780070568990. [12] R. C. Juvinall, K. M. Marchek, Fundamentals of Machine Component Design, 5th Edition, John Wiley & Sons, Inc., Hoboken, New Jersey, USA, 2012, iSBN-13 9781118012895. [13] Z. Yang, H. P. Zhang, M. Marder, Dynamics of static friction between steel and silicon, Proceedings of the National Academy of Sciences 105 (36) (2008) 13264–13268. [14] G. Kraus, Mechanical losses in carbon-black-filled rubbers, Journal of Applied Polymer Science: Applied Polymer Symposium 39 (1984) 17.

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where tolval and tolgrad and respectively define the allowable539 difference in the value and gradient of the fitted model when540 compared to the linear response. Similarly composed functions541 542 with tolerances tolval and tolgrad can be defined for the logarith-543 544 mic space. The choice of comparing data in the linear or logarithmic545 546 space, and using value- or gradient-based error function to de-547 crit fine γ0 is largely based on the way in which the data is to be548 interpreted. In concept, with suitably selected tolerances each549 method should render the same result. In practice, for the data550 551 used in this work, it was found that choosing tolval = 10 %,552 tolgrad = 20 %, tolval = 3 % and tolgrad = 15 % produced qualita-553 tively similar values for the critical strain. With the exception of554 one case the variation in the results was marginal. It was found555 556 that the logarithmic space comparison with a value-based error557 function was very sensitive to the underlying data (as can be558 determined from the comparatively low tolerance) which lead559 to an unreasonably high scatter in the predicted critical strain.560 561 Due to the equivalence of outcome for the other three (more robust) methods, all further results shown in later text will be extracted using the eq. (A.10) as defined in the linear space with tolval = 10 %.

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I.I. Influence of the static axial preload Figure I showcases (a) the normalized steady state stress waveforms, and (b) the corresponding normalized Lissajous figures, both recorded in the ‘no-slip’ regime close to the onset of wall slip. Within the margin of experimental uncertainty the normalized stress response for all applied preloads fits nearly exactly to the normalized data as obtained by experiments using the in situ cured sample preparation technique at ω = 10 rad s−1 and FN = 10 N (data taken from [6]). It can therefore be concluded that the normalized linear viscoelastic response of the tested elastomer is largely independent of the applied preload. The small area within the normalized Lissajous figure (fig. I (b)) again demonstrates the low-dissipative nature of the RTV-2 silicone rubber at the specified frequency. Figure II presents the normalized stress waveforms and corresponding normalized Lissajous figures within the ‘slip’ regime. More specifically, the data is shown at and beyond the peak maximum in the loss modulus at γ0 /γ0max = 1.0 and 3.3. It is evident that the overall non-sinusoidal stress response and dissipative nature of the wall slip phenomenon are independent of the applied preload; this is discussed further in [6]. Note that the examined data points are measured at strain amplitudes that differ by an order of magnitude. The overall result presented here aligns with the development of the master curve shown in fig. 3. This reinforces the hypothesis that the applied preload is one of the dominant factors influencing the governing physical mechanism for the total system response due to wall slip.

1 N 3 N 10 N 30 N

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It has been shown that the wall slip phenomenon, as observed in the characterization of cross-linked elastomers, leads to a significant non-sinusoidal stress response in conjunction with a pronounced additional contribution to the phase shift. We now focus on the steady state waveform data of low damping elastomers at distinct characteristic points related to the peak maximum in G00 (at γ0max ). It appears that there is a close connection the normalized stress response σ(ω, t)/σ0 and the development of slip [6]. Therefore, the experimental data displayed in fig. 1 and fig. 6 of section 3, i.e. the overall system response for a single steady state deformation cycle, are further analyzed.

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/

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0.5

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Figure I: Comparison of (a) the normalized steady state stress waveform and (b) the corresponding normalized Lissajous figure obtained by oscillatory strain sweep experiments on RTV-2 (2:1) ex situ pre-prepared disc-shaped samples. The data is obtained using a serrated rotor (smooth stator) at FN = 1, 3, 10 and 30 N and ω = 10 rad s−1 , and is within the no slip regime close to the onset of wall slip. Note that the depicted data points represent a portion of the 512 sampling points taken per period. Additionally, there is a set of data present obtained for RTV-2 (2:1) in the ‘no-slip’ condition (cured in situ, same batch) taken from [6].

1

at the maximum in G''

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1.0

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Figure II: Influence of the applied static axial preload on (left) the normalized steady state stress waveform and (right) the normalized Lissajous figure. The data is obtained by oscillatory strain sweep experiments on RTV-2 (2:1) ex situ pre-prepared disc-shaped samples using a serrated rotor (smooth stator). The applied strains correspond to (a) the peak maximum in G00 (γ0 /γ0max = 1) and (b) beyond the peak maximum in G00 (γ0 /γ0max = 3.3). Note that the depicted data points represent a portion of the 512 sampling points taken per period, and that there is no experimental data for FN = 30 N at γ0 /γ0max = 3.3.

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Figure III shows a set of normalized steady state waveform618 data, comparable to fig. I, underlying the experimental data in619 the linear system response for fig. 6 of section 3.2. Visually,620 there is no difference in the normalized stress response for each621 applied measuring frequency. Therefore, it can be concluded622 that the normalized linear viscoelastic response for the tested623 low damping elastomer is hardly affected by the applied mea-624 suring frequency (that spans two order of magnitude). In con-625 626

(a)

627 628 0.1 rad/s

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633 0.0

634 635 10 N, 10 rad/s

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637 638

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= 10 N, T = 25°C

640

0

641

*

t

642 643

(b)

644 645

D

0.1 rad/s

1.0

646

1.0 rad/s

647

10 rad/s

0

(-)

TE

0.5

EP

0.0

10 N, 10 rad/s

-0.5

648

649 650 651 652 653

( no slip, Walter et al. 2015)

654

RTV-2 (2:1), no slip regime,

-1.0

AC C

serrated rotor, F

-1.0

-0.5

0.0

/

0

N

= 10 N, T = 25°C

655 656 657

0.5

1.0

(-)

658 659 660

Figure III: Comparison of (a) the normalized steady state stress waveform and 661 (b) the corresponding normalized Lissajous figure obtained by oscillatory strain sweep experiments on RTV-2 (2:1) ex situ pre-prepared disc-shaped samples. The data was obtained using a serrated rotor (smooth stator) at ω = 0.1, 1 and 10 rad s−1 , and FN = 10 N, and is within the no slip regime close to the onset of wall slip. Note that the depicted data points represent a portion of the 512 sampling points taken per period. Additionally, there is a set of data present obtained for RTV-2 (2:1) in the ‘no-slip’ condition (cured in situ, same batch) taken from [6]. 611 612 613 614 615

the loss modulus is maximized. For ω = 0.1 rad s−1 the nonsinusoidal stress response appears to be symmetric, even at γ0 /γ0max = 3.3. With an increase in ω the stress response becomes more asymmetric; this is evident when comparing the data extracted for low frequencies to that of ω = 10 rad s−1 . It is hypothesized that this discrepancy, particularly noticeable at high measuring frequencies, is closely influenced by two factors. For this particular material, the first is the coefficient of kinetic friction. The second is possible strain position control issues that are likely directly linked to the control mechanism of the rotational rheometer (see page 5). The latter results in a non-sinusoidal applied strain that affects the experimental outcome. In order to account for possible strain control issues, the steady state stress and strain waveforms underlying the experimental data presented in fig. 6 are investigated further using a fast Fourier transformation (FFT) analysis. In accordance with [6] and the discussion therein, the focus of the examination is on the 2nd , 3rd and 5th harmonic contributions. Figure V showcases the normalized stress and strain intensities obtained by the FFT analysis for ω = 0.1, 1.0 and 10 rad s−1 at FN = 10 N. For ω = 0.1 and 1 rad s−1 the strain is sinusoidal for the range of entire applied excitation amplitudes. There are no higher harmonics, neither odd nor even, present in the FFT results. However, at ω = 10 rad s−1 the real-time strain position control of the rheometer (when using the DSO mode) is disturbed by wall slip. This results in a non-sinusoidal strain, which is characterized by odd higher harmonics (also see [6] and corresponding discussion therein). Therefore, it can be concluded that wall slip of cross-linked elastomers as measured in parallel-plate rotational rheometry (for low measuring frequencies) leads to the evolution of odd higher harmonics in the stress response. In particular the appearance of the 3rd harmonic contribution is at the onset of wall slip, whereas from inspection the appearance of the 5th harmonic contribution coincides the peak maximum in the loss modulus. As point of comparison, wall slip in complex fluid systems is considered to be associated with the evolution of the 2nd harmonic contribution (for example, see [3, 19–22] and references therein). However, this experimental observation has been debated in the scientific community. To contribute to this discussion on wall slip, it is hypothesized that the discrepancy between the results shown here and those presented in the literature is closely related to the overall physical mechanism governing slip in each system (i.e. wall slip in fluids vs. wall slip in solids).

RI PT

I.II. Influence of measuring frequency

SC

603

trast to fig. II, there is a measurable variation in the normalized stress response for different measuring frequencies at the examined characteristic points. This deviation is more pronounced at large excitation amplitudes, that is, beyond the strain at which 3

at the maximum in G''

1.0

1.0

RI PT

ACCEPTED MANUSCRIPT

at the maximum in G'' 0.1 rad/s 1.0 rad/s 10 rad/s

0.5

0.0

0.0 0.1 rad/s 1.0 rad/s

-0.5

-0.5

/

0

-1.0

F

N

max 0

= 1.0

M AN U

10 rad/s

SC

0

0

(-)

(-)

0.5

/

0

-1.0

= 10 N, T = 25°C

-1.0

0 *

t

F

N

-0.5

0.0

/

0

max 0

= 1.0

= 10 N, T = 25°C

0.5

1.0

(-)

(a) Influence of measuring frequency at the local maximum in G00 (γ0 /γ0max = 1)

0

TE

0.0 0.1 rad/s 1.0 rad/s

-0.5

10 rad/s

-1.0

F

N

0

max 0

= 3.3

= 10 N, T = 25°C

*

0.5

0.1 rad/s

0.0

1.0 rad/s 10 rad/s

-0.5

/

0

-1.0

EP

/

0

beyond the maximum in G''

(-)

(-)

D

0.5

1.0

0

beyond the maximum in G''

1.0

F

N

-1.0

-0.5

0.0

/

t

0

max 0

= 3.3

= 10 N, T = 25°C

0.5

1.0

(-)

AC C

(b) Influence of measuring frequency beyond the local maximum in G00 (γ0 /γ0max = 3.3)

Figure IV: Influence of the measuring frequency on (left) the normalized steady state stress waveform and (right) the normalized Lissajous figure. The data is obtained by oscillatory strain sweep experiments on RTV-2 (2:1) ex situ pre-prepared disc-shaped samples using a serrated rotor (smooth stator). The applied strains correspond to (a) the peak maximum in G00 (γ0 /γ0max = 1) and (b) beyond the peak maximum in G00 (γ0 /γ0max = 3.3). Note that the depicted data points represent a portion of the 512 sampling points taken per period. The green lines represent identical data to that shown in fig. II.

4

ACCEPTED MANUSCRIPT (a) 0.25 RTV-2 (2:1), F

N

= 0.1 rad/s

0.20

strain

stress n = 2

n = 2

n = 3

n = 3

n = 5

n = 5

0.10

RI PT

n/1

(-)

0.15

I

= 10 N, T = 25°C

0.05

0.00 -4

10

-3

10

-2

0

10

-1

10

0

(-)

SC

10

(b) 0.25 RTV-2 (2:1), F

N

strain

stress

n/1

(-)

0.15

M AN U

0.20

I

= 10 N, T = 25°C

= 1.0 rad/s

n = 2

n = 2

n = 3

n = 3

n = 5

n = 5

0.10

0.05

-4

10

-3

10

-2

0

10

(-)

(c) 0.25 N

strain

stress n = 2

n = 2

n = 3

n = 3

AC C

I

n/1

(-)

0.15

n = 5

n = 5

0.10

0.05

0.00

10

0

= 10 N, T = 25°C

= 10 rad/s

0.20

10

EP

RTV-2 (2:1), F

-1

TE

10

D

0.00

-4

10

-3

10

-2

0

10

-1

10

0

(-)

Figure V: Normalized stress and strain intensities of the 2nd , 3rd and 5th harmonic contribution. The data is obtained by oscillatory strain sweep experiments (direct strain oscillation mode of the rheometer) on RTV-2 (2:1) ex situ pre-prepared disc-shaped samples using a serrated rotor (smooth stator) at FN = 10 N and (a) ω = 0.1 rad s−1 , (b) ω = 1.0 rad s−1 and (c) ω = 10 rad s−1 . As is highlighted in fig. 6, there is data scatter visible at low strains in conjunction with low frequencies. Therefore the results obtained for below γ0 = 10−4 is neglected.

5