# The Influence of Local Strain Distribution on the Effective Electrical Resistance of Carbon Black Filled Natural Rubber

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Research Approach

## 2. Materials and Methods

#### 2.1. Rubber Formulation

^{®}. The reinforcing filler used in all the compounds was high abrasion furnace (HAF)–N330 carbon black (CB) supplied by Cabot Corporation, Boston, MA, USA. Moreover, Naphthenic oil NYTEX

^{®}(Nynas AB, Nynashamn, Sweden) was used as a plasticizer and CBS (N–cyclohexyl–2-benzothiazolesulfenamide), was employed as the curing accelerator.

#### 2.2. Rubber Compounding and Samples Preparation

^{3}with cylindrical shoulders of 6 mm in diameter at both ends. Each cylindrical shoulder contained a brass tube (2 mm external and 1.4 mm internal diameters) in the direction of the shoulder axis for realization of future electrical contacts. Finally, in two amongst the three different samples, shape inhomogeneities, characterized by top view of an annular circle (sample c) and two semi circles (sample b) having diameters of 6 mm were implemented. The detailed geometries of the investigated samples are shown in the Figure 1.

#### 2.3. Electric Setup

_{2}(Figure 3a). Since all changes in the load were slow, a simple Ohm’s Law was used in the form

_{0}from Equation (3) in Equation (5) leads to the framing of Equation (6):

_{OA}= 30 kΩ), the customized installation is able to easily measure the resistances up to 750 MΩ and even higher. This fact guarantees that measuring method is suitable for tested samples.

## 3. Results and Discussions

#### 3.1. Local Strain Distribution and Measured EER

_{0}) and R

_{0}, the latter being the resistance of the sample before deformation. Samples containing 50 phr of CB (Figure 6b) exhibited progressive, well-distinguished increase in resistance as a function of applied strain for all the geometries of the samples. Depending on sample geometry, increase in samples resistance (ΔR/R

_{0}) attained an impressive value close to 270 times at the maximum applied strain. This fact is highly appreciated for sensitivity of sensors.

_{0}in dependence of sample geometry, exhibited a modest increase in resistance for about seven times in the best case, while the samples containing 70 phr of CB (Figure 6f) showed a very low ΔR/R

_{0}ranging between 0.02 and 0.2 times for maximum applied stress.

#### 3.2. Current Propagation Mode Switching Effect

_{e}and A

_{c}which represent the contact area between two arbitrary carbon black spheres incorporated in rubber matrix, coaxially aligned, perpendicular to the direction of strain and to direction of contraction, respectively. Taking into consideration that $\Delta {l}^{i}$ is the elongation between two particles forming couple i, and $\Delta {k}^{j}$ is the contraction between particles forming couple j (see Figure 8a),

#### 3.3. The Deformed Samples EER Estimation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A_{c} (nm^{2}) | Contact area between CB particles, aligned perpendicular to the direction of sample contraction. |

A_{e} (nm^{2}) | Contact area between CB particles, aligned perpendicular to the direction of sample strain |

A_{u} (-) | Open loop voltage amplifying coefficient |

d (mm) | Thickness of sample |

Δk (mm) | Contraction of sample |

$\Delta {k}^{j}$ (nm) | The contraction between two particles forming couple j |

Δl (mm) | Elongation of sample |

$\Delta {l}^{i}$ (nm) | The elongation between two particles forming couple i |

$\Delta {l}_{i}$ (mm) | Length of cell i for an imaginary sample partition |

ΔR/R_{0} (-) | Normalized resistance |

$\Delta {w}_{i}$ (mm) | The width of the sample in position i |

${I}_{OA}$ (A) | The input current passing through operational amplifier |

I_{s} (A) | Current going through the sample |

l (mm) | Length of sample |

ν (-) | Poisson’s ratio |

R_{s} (Ω) | Resistance of the sample |

R_{OA} (Ω) | Internal resistance of operational amplifier |

$\rho $ (Ωm) | Resistivity |

U_{s} (V) | Voltage drop across the sample |

U_{i} (V) | Inverting input voltage |

U_{o} (V) | Output voltage |

w (mm) | Width of sample |

x (nm) | The penetration depth between two arbitrary CB particles |

## Abbreviations

AC | Alternating current |

CB | Carbon black |

CBS | N-cyclohexyl-2-benzothiazolesulfenamide |

DIC | Digital Image Correlation |

DC | Direct current |

EER | Effective electric resistance |

EMT | Effective medium theory |

ERN | Equivalent resistor network |

HAF | High abrasion furnace |

NR | Natural rubber |

OA | Operational amplifier |

phr | Parts per hundred rubber |

SIC | Strain induced crystallization |

ZnO | Zinc oxide |

## References

- Wang, S.L.; Wang, P.; Ding, T.H. Piezoresistivity of silicone-rubber/carbon black composites excited by AC electrical field. J. Appl. Polym. Sci.
**2009**, 113, 337–341. [Google Scholar] [CrossRef] - Natarajan, T.S.; Eshwaran, S.B.; Stöckelhuber, K.W.; Wießner, S.; Pötschke, P.; Heinrich, G.; Das, A. Strong Strain Sensing Performance of Natural Rubber Nanocomposites. ACS Appl. Mater. Interfaces
**2017**, 9, 4860–4872. [Google Scholar] [CrossRef] [PubMed] - Liu, P.; Liu, C.X.; Huang, Y.; Wang, W.H.; Fang, D.; Zhang, Y.G.; Ge, Y.J. Transfer function and working principle of a pressure/temperature sensor based on carbon black/silicone rubber composites. J. Appl. Polym. Sci.
**2016**, 133, 42979. [Google Scholar] [CrossRef] - Ciselli, P.; Lu, L.; Busfield, J.; Peijs, T. Piezoresistive polymer composites based on EPDM and MWNTs for strain sensing applications. e-Polymers
**2010**, 10, 1–13. [Google Scholar] [CrossRef] [Green Version] - Bakošová, D. The Study of the Distribution of Carbon Black Filler in Rubber Compounds by Measuring the Electrical Conductivity. Manuf. Technol.
**2019**, 19, 366–370. [Google Scholar] [CrossRef] - Myles, T.D.; Peracchio, A.A.; Chiu, W.K.S. Extension of anisotropic effective medium theory to account for an arbitrary number of inclusion types. J. Appl. Phys.
**2017**, 117, 025101. [Google Scholar] [CrossRef] - Söderberg, M.; Grimvall, G. Conductivity of inhomogeneous materials represented by discrete resistor networks. J. Appl. Phys.
**1986**, 59, 186–190. [Google Scholar] [CrossRef] - Niklasson, G.A.; Granqvist, C.G.; Hunderi, O. Effective medium models for the optical properties of inhomogeneous materials. Appl. Opt.
**1981**, 20, 26–30. [Google Scholar] [CrossRef] - Ruehli, A.; Antonini, A.; Jiang, L. Circuit Oriented Electromagnetic Modeling Using the PEEC Techniques, 1st ed.; Wiley-IEEE Press: New York City, NY, USA, 2017. [Google Scholar]
- Robertson, C.G.; Stoček, R.; Mars, W.V. The Fatigue Threshold of Rubber and Its Characterization Using the Cutting Method. In Fatigue Crack Growth in Rubber Materials. Advances in Polymer Science; Heinrich, G., Kipscholl, R., Stoček, R., Eds.; Springer: Cham, Switzerland, 2020; pp. 19–38. [Google Scholar] [CrossRef]
- Harea, E.; Datta, S.; Stěnička, M.; Stoček, R. Electrical conductivity degradation of fatigued carbon black reinforced natural rubber composites: Effects of carbon nanotubes and strain amplitudes. Express Polym. Lett.
**2019**, 13, 1116–1124. [Google Scholar] [CrossRef] - Harea, E.; Datta, S.; Stěnička, M.; Stoček, R. Undesirable Aspects of Fatigue on Stretchable Elastomer Sensors. In Nanoscience and Nanotechnology in Security and Protection against CBRN Threats; Petkov, P., Achour, M., Popov, C., Eds.; Springer: Dordrecht, The Netherlands, 2020; pp. 95–105. [Google Scholar] [CrossRef]
- Harea, E.; Stoček, R.; Storozhuk, L.; Sementsov, Y.; Kartel, N. Study of tribological properties of natural rubber containing carbon nanotubes and carbon black as hybrid fillers. Appl. Nanosci.
**2019**, 9, 899–906. [Google Scholar] [CrossRef] - Bohm, G.G.A.; Nguyen, M.N. Flocculation of carbon black in filled rubber compounds. I. Flocculation occurring in unvucanized compounds during annealing at elevated temperatures. J. Appl. Polym. Sci.
**1995**, 55, 1041–1050. [Google Scholar] [CrossRef] - Kraus, G. Reinforcement of elastomers by carbon black. Macromol. Mater. Eng.
**1977**, 60, 215–248. [Google Scholar] [CrossRef] - Stauffer, D.; Aharony, A. Introduction to Percolation Theory, 2nd ed.; Taylor and Francis: London, UK, 2017. [Google Scholar] [CrossRef]
- Huang, M.; Tunnicliffe, L.B.; Zhuang, J.; Ren, W.; Yan, H.; Busfield, J.J.C. Strain dependent dielectric behavior of carbon black reinforced natural rubber. Macromolecules
**2016**, 49, 2339–2347. [Google Scholar] [CrossRef] - Huang, Y.; Schadler, L.S. Understanding the Strain-Dependent Dielectric Behavior of Carbon Black Reinforced Natural Rubber– An interfacial or bulk phenomenon? Compos. Sci. Technol.
**2017**, 142, 91–97. [Google Scholar] [CrossRef] [Green Version] - Gismatulin, A.A.; Kamaev, G.N.; Kruchinin, V.N.; Gritsenko, V.A.; Orlov, O.M.; Chin, A. Charge transport mechanism in the forming-free memristor based on silicon nitride. Sci. Rep.
**2021**, 11, 2417. [Google Scholar] [CrossRef] - Stoček, R.; Heinrich, G.; Gehde, M.; Rauschenbach, A. Investigations about notch length in pure-shear test specimen for exact analysis of crack propagation in elastomers. J. Plast. Technol.
**2012**, 1, 2–22. [Google Scholar] - Stoček, R.; Heinrich, G.; Gehde, M.; Kipscholl, R. Analysis of dynamic crack propagation in elastomers by simultaneous tensile- and pure-shear-mode testing. In Fracture Mechanics and Statistical Mechanics of Reinforced Elastomeric Blends. Lecture Notes in Applied and Computational Mechanics; Grellmann, W., Heinrich, G., Kaliske, M., Klüppel, M., Schneider, K., Vilgis, T., Eds.; Springer: Berlin, Germany, 2013; pp. 269–301. [Google Scholar] [CrossRef]
- Stoček, R.; Stěnička, M.; Maloch, J. Determining Parametrical Functions Defining the Deformations of a Plane Strain Tensile Rubber Sample. In Fatigue Crack Growth in Rubber Materials. Advances in Polymer Science; Heinrich, G., Kipscholl, R., Stoček, R., Eds.; Springer: Cham, Switzerland, 2020; pp. 19–38. [Google Scholar] [CrossRef]
- Trabelsi, S.; Albouy, P.A.; Rault, J. Effective Local Deformation in Stretched Filled Rubber. Macromolecules
**2003**, 36, 9093–9099. [Google Scholar] [CrossRef] - Biben, T.; Munch, E. Strain-Induced Crystallization of Natural Rubber and Cross-Link Densities Heterogeneities. Macromolecules
**2014**, 47, 5815–5824. [Google Scholar] [CrossRef] - Middleton, W.I.; Davis, E.W. Skin effect in large stranded conductors at low frequencies. J. Am. Inst. Elect. Eng.
**1921**, 40, 757–763. [Google Scholar] [CrossRef] - Franco, S. Design with Operational Amplifiers and Analog Integrated Circuits, 4th ed.; McGraw-Hill: New York, NY, USA, 2015; 672p. [Google Scholar]
- Carter, B.; Mancini, R. Op Amps for Everyone, 5th ed.; Elsevier: Oxford, UK, 2018. [Google Scholar]
- Pilkey, W.D.; Pilkey, D.F. Peterson’s Stress Concentration Factors, 3rd ed.; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2008; pp. 180–184. [Google Scholar]
- Robertson, C.G.; Ned, J.; Hardman, N.J. Nature of Carbon Black Reinforcement of Rubber: Perspective on the Original Polymer Nanocomposite. Polymers
**2021**, 13, 538. [Google Scholar] [CrossRef] - Tang, Z.; Jia, S.; Zhou, C.; Li, B. 3D Printing of Highly Sensitive and Large-Measurement-Range Flexible Pressure Sensors with a Positive Piezoresistive Effect. ACS Appl. Mater. Interfaces
**2020**, 12, 28669–28680. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The geometry of studied samples: (

**a**) basic configuration, (

**b**) double side inhomogeneity, (

**c**) central inhomogeneity.

**Figure 5.**The evolution of elongation vs. contraction for sample containing 70 phr of CB (similar trends were observed for all tested CB concentrations).

**Figure 6.**(

**a**,

**c**,

**e**) force registered during the testing protocol; (

**b**,

**d**,

**f**) electrical resistance of samples.

**Figure 8.**(

**a**) Scheme of contact areas conductivity model; (

**b**) Ratio between parallel and perpendicular contact area vs. interparticle displacement in tensile direction.

**Figure 9.**(

**a**) Sketch defining the resistive domains with corresponding nodes for samples type b and (

**b**) for samples type c, (

**c**) The equivalent circuit for samples type b and (

**d**) equivalent circuit for samples type c.

**Figure 10.**Illustration for resistance calculation for samples of arbitrary shape: (

**a**) undeformed and (

**b**) deformed state.

**Figure 11.**The resistivity of reference samples containing 50 phr CB, 60 phr CB and 70 phr CB vs. the sample’s strain and corresponding DIC domain’s color.

**Figure 12.**The relative effective electric resistance for samples type b and c experimentally measured and calculated: (

**a**) samples containing 50 phr CB, (

**b**) 60 phr CB and (

**c**) 70 phr CB.

NR | Oil | Carbon Black | CBS | Sulfur | ZnO | Stearic Acid | |
---|---|---|---|---|---|---|---|

Content in phr * | |||||||

NR50_# | 100.00 | 10.00 | 50.00 | 1.00 | 2.50 | 5.00 | 2.00 |

NR60_# | 60.00 | ||||||

NR70_# | 70.00 |

50 phr | ρ, Ωm |
6.80 × 10^{4} |
8.10 × 10^{5} |
3.60 × 10^{6} |
9.20 × 10^{6} |

R, Ω |
9.50 × 10^{7} |
3.10 × 10^{8} |
3.10 × 10^{9} |
3.10 × 10^{10} | |

60 phr | ρ, Ωm |
1.40 × 10^{3} |
2.80 × 10^{3} |
6.00 × 10^{3} |
1.40 × 10^{4} |

R, Ω |
1.90 × 10^{6} |
1.00 × 10^{6} |
5.10 × 10^{6} |
4.60 × 10^{7} | |

70 phr | ρ, Ωm |
1.70 × 10^{2} |
1.40 × 10^{2} |
1.20 × 10^{2} |
1.10 × 10^{2} |

R, Ω |
2.40 × 10^{2} |
5.30 × 10^{1} |
1.10 × 10^{2} |
3.60 × 10^{2} |

50 phr | ρ, Ωm |
6.80 × 10^{4} |
8.10 × 10^{5} |
3.60 × 10^{6} |
9.20 × 10^{6} |
2.10 × 10^{7} |

R, Ω |
3.40 × 10^{7} |
1.80 × 10^{9} |
6.30 × 10^{9} |
5.20 × 10^{10} |
4.90 × 10^{10} | |

60 phr | ρ, Ωm |
1.40 × 10^{3} |
2.80 × 10^{3} |
6.00 × 10^{3} |
1.40 × 10^{4} |
3.90 × 10^{4} |

R, Ω |
7.00 × 10^{5} |
6.10 × 10^{6} |
1.10 × 10^{7} |
7.90 × 10^{7} |
9.30 × 10^{7} | |

70 phr | ρ, Ωm |
1.70 × 10^{2} |
1.40 × 10^{2} |
1.20 × 10^{2} |
1.10 × 10^{2} |
9.80 × 10^{1} |

R, Ω |
8.80 × 10^{4} |
3.10 × 10^{5} |
2.10 × 10^{5} |
6.30 × 10^{5} |
2.30 × 10^{5} |

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**MDPI and ACS Style**

Harea, E.; Datta, S.; Stěnička, M.; Maloch, J.; Stoček, R.
The Influence of Local Strain Distribution on the Effective Electrical Resistance of Carbon Black Filled Natural Rubber. *Polymers* **2021**, *13*, 2411.
https://doi.org/10.3390/polym13152411

**AMA Style**

Harea E, Datta S, Stěnička M, Maloch J, Stoček R.
The Influence of Local Strain Distribution on the Effective Electrical Resistance of Carbon Black Filled Natural Rubber. *Polymers*. 2021; 13(15):2411.
https://doi.org/10.3390/polym13152411

**Chicago/Turabian Style**

Harea, E., S. Datta, M. Stěnička, J. Maloch, and R. Stoček.
2021. "The Influence of Local Strain Distribution on the Effective Electrical Resistance of Carbon Black Filled Natural Rubber" *Polymers* 13, no. 15: 2411.
https://doi.org/10.3390/polym13152411