Ultrasonic Devulcanization of n Sulfur Vulcanized Natural Rubber by Shantha Maduwage library' ravFBsiTY cf raruwA, sri m m This thesis was submitted to the Department of Chemical and Process Engineering of the University of Moratuwa in partial fulfillment of the requirements for the Degree of Master of Philosophy 6 6 " 0 2 Tyt Department of Chemical and Process Engineering University of Moratuwa Sri Lanka December, 2002 University of Moratuwa 7 6 8 ^ 7 76827 76827 Declaration I hereby declare that this submission is my own work and that, to the best of my knowledge and behalf, it contains no material previously published or written by another person nor material which to substantial extent, has been accepted for the award of any other academic qualification of a university or other institute of higher learning except where an acknowledgment is made in the text. Shantha Maduwage December • 2002 Acknowledgements I am very grateful to my supervisors Dr. A.D.U.S. Amarasinghe and Dr. D.A.I. Munidradasa for their guidance, patience, time, encouragement and commitment throughout the project. I wish to express my gratitude to the Head of the Department of Chemical and Process Engineering and all other academic staff members for their assistance and encouragement. Special note to mention Dr. K.G.P. Dharmawardana of Department of Electronics Engineering for his guidance in programming Matlab software. I appreciate the valuable contribution extended to me by Mrs. N. Rathnayake, Director Post Graduate Studies, University of Moratuwa for the completion of the research project. Asian Development Bank helped me to drive this project by way of granting all financial assistance required for the same. My special thanks are to Technical and Technical Assistant staff members at Chemical and Process Engineering, and Electronics Engineering departments for their great support for me to carryout experimental works successfully. Finally, my gratitude must go to my husband, daughter and family for their whole­ hearted support for the completion of this thesis. Shantha Maduwage December , 2 0 0 2 Abstract The high-energy ultrasound could be used to devulcanize rubber as it can focus energy into localized sites for selective bond rupture. The research work reported to- date suggests that the ultrasonic technology is more suited to convert rubber waste to a usable material efficiently, effectively and environmental friendly. The ultrasonic devulcanization reactor consisted of three main sections, namely a power source, ultrasonic transducer with sample holding unit, and a monitoring system to measure the amplitude, frequency and power. N-cyclohexyl-2-benzthiazyl sulfanamide (CBS) accelerated unfilled natural rubber vulcanized with conventional sulfur vulcanizing system and with efficient sulfur vulcanizing system were used as the model rubber compounds in these experiments. 2 mm thick vulcanized rubber sheets were directly kept on the vibrating diaphragm of the ultrasonic transducer. The frequency of ultrasonic wave was varied in a range of 20 to 50 kHz and the power level was varied up to 800 watt. The treatment time was limited to 10 minutes when treated at high power levels. The vibrating amplitudes were measured at different power levels with the variation of ultrasonic frequency. Curing behaviour, gel content and cross-link density were studied for rubber samples devulcanized at different process conditions. The increase in cross-link density and gel content of the samples treated at lower amplitudes indicated the formation of additional cross-links. However, the higher vibrational energies associated with high amplitudes resulted in lower cross-link densities and gel contents indicating a breakdown of bonds. Cure curves of virgin and devulcanized NR samples suggested that the fast initial curing of devulcanized NR was due to the presence of active sufidized rubber molecules formed due to break down of some cross-links during devulcanization. The lower maximum torque values observed in the devulcanized samples were due to the partial breakdown of C-C bonds in the main chain. The tensile properties of the revulcanized samples gave comparable results with that of virgin rubber. A theoretical process model was developed to express the extent of devulcanization in terms of cross-link density. It was based on the vibrational energy transfer mechanism. The model treated the vulcanized rubber as a pure elastic solid containing void regions. Experimental and theoretical values lied within ± 10% error limits. The model showed that the media effect on the nature of void excitation was significant and the viscoelasticity was also considerable. However, the effect due to surface tension was negligible. Contents Chapter 1 Introduction 1 1.1 Devulcanization 1 1.2 Ultrasonic Devulcanization 2 1.3 Theoretical Model for Ultrasonic Devulcanization 3 1.4 Aim and Scope of the Project 4 1.5 Approach 5 1.6 Outline of the Thesis 5 Chapter 2 Literature Review 7 2.1 Disposal of Rubber Waste 7 2.2 Conventional Rubber Recycling Methods 2.2.1 Chemical Methods 2.2.1.1 Reclaiming 9 2.2.1.2 Chemical Probes 10 2.2.1.3 Other Chemical Methods 10 2.2.2 Mechanical Methods 12 2.3 Wave energy transfer for rubber reprocessing 2.3.1 Microwave Devulcanization 14 2.3.2 Ultrasonic Devulcanization 14 2.4 Devulcanization Model 16 2.5 Suitability of present work for selective bond rupture 17 Chapter 3 Reactor for Ultrasonic Devulcanization 20 3.1 Ultrasound 20 3.2 Ultrasonic Transducers 22 3.3 Ultrasonic Reactor 23 Chapter 4 Devulcanization of Accelerated Sulfur Vulcanized Natural rubber 25 4.1 Vulcanization 4.1.1 Formation of Initial Polysulfidic Cross-links 25 4.1.2 Network Maturing Process 27 4.2 Ultrasonic Devulcanization 31 4.3 Kinetics of Devulcanization 33 Chapter 5 Devulcanization at Low Power Levels 36 5.1 Experimental Techniques 5.1.1 Compounding Materials 37 5.1.2 Experimental Methods 38 5.2 Characterization methods 5.2.1 Mooney viscosity 39 5.2.2 Cure Characteristics 40 5.2.3 Gel Content 41 5.2.4 Cross-link density 42 5.2.5 Tensile properties 43 5.3 Vulcanization Experiments 5.3.1 Mooney Viscosity 45 5.3.2 Cure Characteristics 46 5.3.3 Structural Characteristics 47 5.3.4 Tensile Properties 49 5.3.5 Correlation of Structural properties with Tensile properties 51 5.4 Devulcanization Experiments 5.4.1 Capacitive Ultrasonic Transducer 54 5.4.2 Inductive Ultrasonic Transducer 57 5.5 Approach to Future Work 5.5.1 Transducer Performance 61 5.5.2 Methodology Developments 63 Chapter 6 Devulcanization at High Power Levels 65 6.1 Structural Characteristics 6.1.1 Gel content 6.1.1.1 CV system 65 6.1.1.2 EV system 68 6.1.1.3 Comparison of Gel Content between CV system and EV system 71 6.1.2 Cross-link density 6.1.2.1 CV system 73 6.1.2.2 EV system 76 6.1.2.3 Comparison of Cross-link Density between CV system and EV system 79 6.1.3 Relationship between Cross-link density and Gel content 80 6.2 Cure Characteristics 81 6.3 Tensile Properties 84 6.4 Conclusion 86 Chapter 7 Process Model for Ultrasonic Devulcanization 88 7.1 Theoretical Modeling 7.1.1 Devulcanization Model 89 7.1.2 Vibrational Energy Transfer Model 95 7.1.2.1 Acoustic Pressure Amplitude and Void Fraction 98 7.1.2.2 Vibrational Energy Transfer in an Elastic Medium 99 7.2 Simulation of the Model 7.2.1 Acoustic Pressure Amplitude 101 7.2.2 Void radii at Critical Events 104 7.2.3 Cross-link Density 112 7.3 Comparison of Theoretical and Experiment Results 120 Chapter 8 Effect of Viscoelasticiry on Dynamics of Vibrational Energy Transfer 124 8.1 Vibrational Energy Transfer in a Viscoelastic Medium 124 8.2 Viscoelastic Models 127 8.3 Comparison of Elastic Model with Viscoelastic Models 131 Chapter 9 Conclusions and Recommendations for Future Work 139 9.1 Conclusions 139 9.2 Recommendations for Future Work 142 References Appendix - I Appendix - II Appendix - III Appendix - IV Appendix - V Appendix - VI List of Tables and Figures Chapter 3 Figure 3.1 Two types of wave motions Figure 3.2 Ultrasonic Power Source Chapter 4 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Different positions that sulfur attack on the polyisoprcnc backbone Dcsulfuration and decomposition of sulfidic cross-links The probable course of sulfur vulcanization of NR in the presence of accelerators and activators Typical chemical groupings present in a sulfur-vulcanized NR network Energy levels of different states of atoms Chapter 5 Table 5.1 Dry Formulations for Rubber Compounding Figure 5.1 Dumb-bell test specimens Figure 5.2 Mooney viscosity of NR with CV system and EV system Figure 5.3 Cure curves of CV system and EV system Figure 5.4 Variation of Gel content with Vulcanizing time for CV system and EV system Figure 5.5 Variation of Cross-link densities with Vulcanizing time for CV system and EV system Figure. 5.6 Variation of Tensile properties with Vulcanizing time for CV system and EV system (a) % Elongation at break (b) Tensile strength (c) Modulus@300%clongation Figure 5.7 Variation of Cross-link density with % Gel content for CV system and EV system Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Figure 5.16 Figure 5.17 Chapter 6 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Variation of Tensile properties with Cross-link density for CV system and EV system (a) % Elongation at break (b) Tensile strength (c) Modulus @300%elongation Variation of Cross-link density with Treatment time for CV system at different power levels, /=20 kHz Variation of %Gel content with Treatment time for CV system at different, power levels, /=2() kHz Variation of Cross-link density with Treatment time for CV system al different frequencies, P=25 W Variation of Cross-link density with Supplied power for CV system, at treatment times of 30 and 60 minutes,/=20 kHz Variation of Cross-link density with Ultrasonic frequency for CV system at constant power levels, /= 30 minutes Variation of %Gel content with Ultrasonic frequency for CV system at constant power levels, /=30 minutes Variation of Tensile properties with Ultrasonic frequency for revulcanized CV system at constant power levels, /= 30 minutes (a) tensile strength (b) modulus @300% elongation Experimental set up to measure vibrating amplitude Variation of Vibrating amplitude (peak to peak amplitude) with Ultrasonic frequency at constant power levels Variation of % Gel content of devulcanized CV system with ultrasonic frequency at constant power levels, /=10 minutes (a) power levels up to 500W (b) power levels of 700 W and 800 W Variation of % Gel content of devulcanized CV system with treatment time at frequencies of 30, 35 and 40 kHz, / J=800 W Variation of % Gel content of devulcanized CV system with vibrating amplitude at constant frequencies, r=\0 minutes Variation of % Gel content of devulcanized and revulcanized CV system with vibrating amplitude, f= 35 kHz, r=\0 minutes Figure 6.5 Variation of % Gel content of dcvulcanizcd EV system with ultrasonic frequency at constant power levels (a) 50 W to 200 W (b) 300 W to 800 W, /= 10 minutes Figure 6.6 Variation of % Gel content of devulcanized EV system with treatment time at. frequencies of 30, 35 and 40 kHz, /*=8()0W Figure 6.7 Variation of % Gel content of devulcanized EV system with vibrating amplitude at constant ultrasonic frequencies, /=1() minutes Figure 6.8 Variation of % Gel content of devulcanized and revulcanized EV system with vibrating amplitude, /= 35 kHz, /= 10 minutes Figure 6." Variation of % Gel content with vibrating amplitude for devulcanized CV system and EV system, /=35 kHz, t= 10 minutes Figure 6.10 Variation of % Gel content with treatment time for devulcanized CV system and EV system, /=35 kHz, /'=800 W Figure 6.1 1 Variation of % Gel content with vibrating amplitude for revulcanized CV system and EV system,/=35 kHz, /=10 minutes Figure 6.12 Variation of cross-link density of devulcanized CV system with ultrasonic frequency at constant power levels (a) 50 W to 200 W (b) 500 W to 800 W, /= 10 minutes Figure 6.13 Variation of cross-link density of devulcanized CV system with treatment time at frequencies of 30, 35 and 40 kHz, /^ SOOVV Figure 6.14 Variation of cross-link density of devulcanized CV system with vibrating amplitude at constant ultrasonic frequencies. Figure 6.15 Variation of cross-link density of devulcanized CV system with ultrasonic frequency at vibrating amplitudes of 40 and 60 um, /=10 minutes Figure 6.16 Variation of cross-link density of devulcanized EV system with ultrasonic frequency at constant power levels (a) 50 W to 200 W (b) 300 W to 800 W, t= 10 minutes Figure. 6.17 Variation of cross-link density of devulcanized EV system with treatment time at frequencies of 30, 35 and 40 kHz, / , = 800W Figure 6.18 Variation of cross-link density of dcvulcanizcd EV system with vibra.ting amplitude at constant frequencies f=IO minutes Figure 0.19 Variation of cross-link density of devulcanized EV system with ultrasonic frequency at vibrating amplitudes of 40 and 60 um, /=I0 minutes Figure 6.20 Variation of cross-link density with vibrating amplitude for devulcanized CV and EV systems, /=35 kHz, /=I0 minutes Figure 0.21 Variation of cross-link density with treatment time for devulcanized CV system and EV system,/=35 kHz, P=H00 W Figure 6.22 Variation of %Gel content with cross-link density for devulcanized CV system and EV system, /=35 kHz, / J=800 W Figure. 6.23 Cure curves of virgin and devulcanized rubber treated at 35 kHz ultrasonic frequency and at vibrating amplitudes of 78 um and 90 um (a) CV system (b) EV system Figure 6.24 Variation of tensile properties of revulcanized rubber, devulcanized at 35 kHz ultrasonic frequency, with vibrating amplitude (a) Tensile strength (b) Modulus @300% elongation (c) % Elongation at break Chapter 7 Tabic 7.1 Model parameters for devulcanization model Table 7.2 Properties of natural rubber utilized in the simulation process Figure 7.1 Overstressed network fragment around an excited region Figure 7.2 Inflation of a cavity contained in a large body Figure 7.3 Variation of acoustic pressure amplitude with vibrating amplitude at the frequencies of 25,35 and 45 kHz Figure 7.4 Variation of ycoustic pressure amplitude with ultrasonic frequency at the vibrating amplitudes of 40 and 60 um Figure 7.5 Variation of interactive void concentration with vibrating amplitude at the ultrasonic frequencies of 25, 35 and 45 kHz Figure 7.6 Variation of interactive void concentration with ultrasonic frequency at vibrating amplitudes of 40 and 60 um figure 7.7 Variation of relative radius with time for CV system at ultrasonic frequency of 35 kHz and at amplitudes (um) of (a)4 (b)2() (c) 53 (d) 90 Figure 7.8 Variation of relative radius with time1 for CV system at the frequency of 20, kHz and at amplitudes of (a) 2 um (b) 4 urn Figure 7.0 Variation of relative radius with time for CV system at the frequency of 25 kHz and at amplitudes of (a) 2 um (b) 4 pm Figure 7. JO Variation of maximum relative radius with vibrating amplitude for CV system at the frequency of 30, 35 and 40 kHz Figure 7.1 I Variation of maximum relative radius with vibrating amplitude for CV systeiyi at the frequency of 20, 25, 45 and 50 kHz Figure 7.12 Variation of maximum relative radius with vibrating amplitude for CV and EV systems at the frequency of 35 kHz Figure 7.13 Variation of maximum relative radius with ultrasonic frequency for CV and EV systems at the vibrating amplitude of 40 um Figure 7.14 Relative radius with time for CV system without considering the media response, at frequency of 35 kHz and at amplitudes of (a) 4 pm (b)9Qpm Figure 7.15 Relative radius with time for CV system without considering the surface tension at the frequency of 35 kHz and at amplitudes of (a) 4 pin ,(b) 90 um Figure 7.16 Relative radius with time for CV system without considering the media response, at die frequency of 20 kHz and at amplitudes of (a) 2 urn (b) 4 um Figure 7.17 Variation of rnaximum relative radius with vibrating amplitude for CV systems at the ultrasonic frequency of 35 kHz considering with and without the media elastic response Figure 7.18 Variation of maximum relative radius with vibrating amplitude for CV systems at 35 kHz ultrasonic frequency, considering with and without surface tension Figure 7.19 Variation of different types of cross-link densities with treatment time for CV system with considered reactions that take place during devulcanization, /= 35 kHz, A= 90pm. Figure 7.20 Variation of cross-link density with treatment time at constant amplitudes of (a) 4, 10, 20, 33 and 53 um (b) 53, 78 and 90 um (CV system),/=35 kHz Figure 7.21 Variation of cross-link density with treatment time at constant amplitudes of _(a) 4, 10, 20, 33 and 53 um (b) 53, 78 and 90 um (EV system}, /=35 kHz Figure 7.22 Variation of crqss-link density with treatment time for CV system and EV system, at Jhc vibrating amplitude of 90 um and al the ultrasonic frequency of 35 kHz Figure 7.23 Variation of cross-link density with vibrating amplitude for CV system and EV system, at the ultrasonic frequency of 35 kHz and at the treatment time of 10 minutes. Figure 7.24 Variation of cross-link density with ultrasonic frequency for (a) CV system and (b) EV system, at constant vibrating amplitudes of 40 and 60 um and at the treatment time of 10 minutes Figure 7.25 Variation of crpss-link density with treatment time for (a) CV system (b) EV system at the vibrating amplitudes of 90 um and the ultrasonic frequency of 35 kHz Figure 7.26 Variation of cross-link density (theoretical and experimental values) with vibrating amplitude for (a) CV system (b) EV system at the ultrasonic frequency of 35 kHz under the treatment time of 10 minutes Figure 7.27 Variation of cross-link density (theoretical and experimental values) with ultrasonic frequency for (a) CV system (b) EV system at the vibrajting amplitudes of 40 and 60 um under the treatment time of 10 minuses Figure 7.28 Variation of cross-link density (theoretical and experimental values) with treatment time for (a) CV system (b) EV system at the vibrating amplitudes of 90 um and the ultrasonic frequency of 35 kHz Chapter 8 Table 8.1 Additional parameters used in viscoelastic model Figure S.I Relative radius versus time for one pressure cycle (a), (b) Elastic model, (c), (d) Zener model and (e), (t) Rouse model, /= 20 kHz Figure 8.2 Rcjatiyc radius versus time for one' pressure cycle (a), (b) Elastic model, (c), (d).Zener model and (e), (t) Rouse model, /= 35 kHz Figure 8.3 Relative radius versus time for four pressure cycles (a) Elastic model (c) Zcncr model and (c) Rouse model, /= 20 kHz, A=2 um Figure 8.4 Relative radius versus time for four pressure cycle (a), (b) Elastic model, (c), (d) Zcner model and (e), (t) Rouse model, /= 35 kHz Figure 8.5 Relative radius versus time for ten pressure cycles for the Rouse model, /= 35 kHz, A=9() um Appendix - 1 Figure A.I.I Sigma (a ) and Pi (n) bonds in an ethylene molecule Figure A. 1.2 Hybrid resonance structures of polyisoprene Appendix - III Figure A.3.1 Mooney-Rivlin plot for simple extension of both vulcanizing systems (symbols are experimental data and lines are fitting based on equation A..3.1) Appendix - V Figure A.5.1 Apparatus for determination of total sulfur Figure A.5.2 Apparatus for determination of sulfide sulfur Appendix - VI Figure A.6.1 IR spectra of CBS accelerated sulfur vulcanized NR with CV system and with EV system 4 Nomenclature Vr A deformation gradient s Deviatoric part of the stress tensor Radius vector of a point in the initial configuration G' Storage modulus V fr(/) A relative deformation gradient V,F(s) A relative deformation gradient G" Loss modulus A deformation gradient d Cauchy stress tensor Dual vector in the initial configuration Finger strain tensor Radius vector of a point in the actual configurations Dual vector in the actual configuration %Eb Percentage elongation at break X Huggins interaction constant P Density of the material Poisson's ratio 9 Void fraction C(T) Stress history o(0 Longitudinal stress 6 ( 0 Strain at instant t Co Friction coefficient per monomer unit AH Enthalpy difference Ma Active volume thickness Pr Dry density of rubber ^Rp Relaxation time for Rouse model T]s Viscosity of solvent Density of solvent €((") Infinitesimal strain tensor AVa Active volume around a single excited region t:. Relaxation time for Zener model /_(/) A domain in the actual configuration at instant / Q0 A domain in the initial configuration A Initial cross-section area Vibrating amplitude a Root-mean-square end-to-end distance F.(/,.*) Finger deformation tensor AiS- A sulfidic group A2S- A sulfidic group co Angular frequency ASTM American Society of Testing and Materials B1S- A sulfidic group B2S- A sulfidic group c Velocity of sound in rubber C=C Carbon-carbon double bond C / A Mooney-Rivlin constant C2 A Mooney-Rivlin constant CBS N-cyclohexyl-2-benzthiazyl sulfenamide C-C Carbon-carbon bond C-H Carbon-hydrogen bond c,u Velocity defined by Mallock's formula. cs Modified velocity of sound in rubber with voids C-S Carbon-sulfur bond CV Conventional sulfur vulcanizing system E Young's modulus E(t) Elastic modulus Eo Reference bond strength Ec Carbon-carbon bond strength Ej Bond strength of specific type of the bond EPDM Ethylene propylene diene terpolymer Es Bond strength of monosulfidic cross-links Esx Bond strength of polysulfidic cross-links EV Efficient sulfur vulcanizing system EVA Ethylene vinyl acetate / Force at the required elongation Ultrasonic frequency F Breaking force G,G(t) Shear modulus Gr Relaxed shear modulus GTR Ground tyre rubber G„ Unrelaxed modulus Shear modulus of rubber in glassy state H Allylic hydrogen atom Hz Herts / Specific bond // First strain invariant I2 Second strain invariant IPPD N-isopropyl-N'-phenyl-p-phenylene diamine k Boltzmann constant An integer Kj A fitting parameter kj Rate constant of the breakage of specific molecular bonds Kr A fitting parameter kr Rate constant of the reaction of destruction of rubber bound intermediates ksi Rate constant of reaction of breakage of monosulfidic cross-links Ksi A fitting parameter Ksc A fitting parameter ksc Rate constant of reaction of conversion of polysulfidic cross-links to monosulfidic cross-links ksx Rate constant of reaction of breakage of polysulfidic cross-links Ksx A fitting parameter KSXf A fitting parameter ksxf Rate constant of the reaction of formation of additional cross-links by reacting rubber bound intermediates with zinc complexes L Length between gauge marks at break L Large rotor Lo Initial length between gauge marks ^ Lagrangain curvilinear coordinates, M Mooney viscosity number Mo Molecular weight of initial network chain Molecular weight of monomer unit Mc Molecular weight between cross-links MHz Megahertz * mm millimeter MPa Megapascal N Newton n Number of moles of the ideal network chains Number of molecules per unit volume of solution No Initial void concentration NA Avogadro's number NBR Nitrile rubber Nc Number of network chains per unit volume « nc Cross-link density Nc(t) Total concentration of cross-links remain at time t Ncs(t) Concentration of monosulfidic cross-links formed by conversion of polysulfidic cross-links to monosulfidic cross-links at time t Nj(t), Nj Number of specific molecular bonds per unit volume at time t Nja(t) Active molecular bonds per unit volume at time t Nia, Number of active molecular bonds per unit volume NMR Nuclear magnetic resonance NR Natural rubber » Nr(t) Concentration of rubber intermediates remaining at time / Ns(0) Initial concentration of monosulfidic cross-links Ns(t) Concentration of monosulfidic cross-links remaining at time / Nsx(0) Initial concentration of polysulfidic cross-links Nsx(t) Concentration of polysulfidic cross-links remaining at time t Nsxf(0 Concentration of polysulfidic cross-links formed due to reaction of rubber bound intermediates with zinc complexes during time / Nsxr(t) Concentration of polysulfidic cross-links remains at time /, after breaking into non cross-links Nv* Number of k t h cross-links per unit volume ODCB Dissolution by o-dichlorobenzene P Power level p Sound pressure An integer Poc, P~(t) Pressure at infinity in rubber P, P(r,t) Local pressure in rubber P0 Ambient pressure PA Ultrasonic pressure amplitude Pc Critical value of the pressure difference Pg Gas pressure phr Parts per hundreds rubber Pi, P,(t) Pressure in rubber at the wall. Pm Inflation pressure Pm(elastic) Critical pressure for elastic model P„,(viscoelastic) Critical pressure for viscoelastic model q Number of monomeric units between cross-links Q(t,x) Relaxation measure r Radial distance from center R Universal gas constant R Velocity R Rate of change of velocity R(t,i) Relaxation kernal. R, R(t) Radius of a void region at time t Ro Initial radius of a void region y Ratio of the specific heats of gas RCOOH An organic activator RH Rubber hydrocarbon Rmax Radius of a void region at peak excitation R-S2z-R Initial polysulfide cross-links RSR Monosulfidic cross-link RSS Ribbed smoke sheet RSXR Polysulfidic cross-link s Actual configuration Sg Elementary sulfur SBR Styrene butadiene rubber p Sensitivity of void region formation Si-C Silicon-carbon bond Si-0 Silicon oxygen bond S-S Sulfur-sulfur bond STP Standard test procedure a Surface tension S x Polysulfides t Treatment time T Temperature. 775 Tensile strength at break T g Glass transition temperature T,•,, Radial stress on the wall due to motion U Velocity of the wall u Radial velocity in rubber relative to center v Total number of moles of various cross-links per unit volume Vi Molecular volume of the solvent rj Viscosity of the dash pot k t h type sulfur cross-link Number of moles of k t h cross-link per unit volume Vr Volume fraction of rubber in the swollen vulcanizate W Stored-energy function W Watt A. Wave length Wr Weight of dry rubber Ws Weight of solvent x An integer X Accelerator residue x(t) Relative radius of a void region Extension ratio X}(t) An auxiliary function \4(t) An auxiliary function Xmax Relative void radius at peak excitation XSS xZnS xSX Zinc per-thio-salt Active sulfurating agent XSXSR Rubber bound intermediate Accelerator-terminated polysulfidic groups Initial cross-link precursors XSZnSX Zinc accelerator complexes (zinc complexes) y An integer z An integer ZnO Zinc oxide ZnS Zinc sulfide bp Pressure difference (J> Velocity potential Jim microns