Determining the rheological parameters of polyvinyl chloride, with change in temperature taken into account

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1 Plasticheskie Massy, No. 1-2, 2016, pp Determining the rheological parameters of polyvinyl chloride, with change in temperature taken into account A.E. Dudnik, A.S. Chepurnenko, and S.V. Litvinov Rostov State Civil Engineering University, Rostov-on-Don Selected from International Polymer Science and Technology, 43, No. 8, 2016, reference PM 16/1-2/30; transl. serial no Translated by P. Curtis Summary A procedure for determining the relaxation constants of material that enter the Maxwell Gurevich non-linear equation was examined. For recycled polyvinyl chloride, relaxation curves plotted at different temperatures were processed, and the influence of temperature on the relaxation parameters was investigated. For each parameter, an empirical formula was selected to describe its temperature dependence. Today, PVC materials account for 69% [1] of all plastic structural materials on the market. They are used in the production of pipes, windows, wall facings, decorative finishing materials, wire and cable insulation, and so on. Such broad application of polyvinyl chloride is due both to its high physicomechanical properties and to its fire resistance and resistance to environmental effects. The mechanical behaviour of polymers and their service qualities are inextricably linked to the relaxation processes that occur in them. Therefore, it is important to be able to predict the rheological behaviour of polymeric materials, taking into account the time factor, and for this it is necessary to know the equation describing the relationship between the high-elastic strains of the polymer and the stresses arising within it, and also all the relaxation parameters that enter this equation. Unfortunately, for polyvinyl chloride, in spite of its very broad application, data on creep and relaxation are fairly rarely encountered in the literature. Furthermore, all the elastic and relaxation parameters of PVC, as for other polymers, depend very greatly on temperature. From recent studies in which these circumstances would have been taken into account, two stand out [2, 3]. In these two studies, the relaxation of virgin and recycled PVC at different temperatures is investigated. For recycled polyvinyl chloride, Figure 1 shows the stress relaxation curves at a relative strain ε of 3% that were obtained by Solov eva et al. [2, 3]. Table 1 presents the values of the stresses at different moments of time that correspond to Figure 1. To describe the obtained curves, Solov eva et al. [2, 3] use integral equations with a fairly complex kernel. The integral form of the equations is convenient for the processing of experimental data; however, in solving problems of the mechanics of polymers, its use presents considerable mathematical difficulties. Therefore, Figure 1. The relaxation curves of recycled PVC, obtained by Solov ev et al. [2, 3] 2017 Smithers Information Ltd. T/43

2 Table 1. The change in stresses in time at different temperatures Temperature, σ, MPa C 0 min 3 min 6 min 15 min 30 min 45 min 60 min 90 min 120 min 180 min where possible, a switch is made from integral form to differential form. In the literature, the Maxwell Gurevich non-linear equation is well known, and it is in good agreement with experimental data for polymers of both linear and three-dimensional structure. In the case of a uniaxial stress state, it has the form [4]: where ε* is the high-elastic strain, f* is the function of stresses, and η* is the relaxation viscosity. The function of stresses f* and the relaxation viscosity are calculated in the following way: where σ is the stress, E is the rubbery modulus, η * 0 is the initial relaxation viscosity, and m* is the rate modulus. Let us examine the procedure for determining the rheological parameters from the relaxation curves, and establish the temperature dependence of each parameter for the case of recycled PVC, using the curves shown in Figure 1. The full strain ε, remaining constant during relaxation, comprises the sum of elastic and high-elastic strain: (1) (2) Differentiating (5) with respect to time, we will obtain: Then the function σ(t) must be differentiated numerically with respect to t in order to find the quantity n ε = [ ε*(t)]/ t at each moment in time. Given that stress measurement was carried out not after equal time intervals, we will use the method of indefinite coefficients. We will denote by a prime a derivative with respect to t. For the first point, at t 1 = 0, we will write the derivative σ in the form of the linear combination of stresses at the given point and at two located alongside: (6) (7) If the approximation (7) is valid for the arbitrary function σ(t), it should be valid also for the following functions: We will calculate the derivatives of functions (8): (8) (9) Having substituted each of the functions (8) into (7), we will obtain a system of linear equations: where E is the instantaneous elastic modulus. As there is no high-elastic strain at the initial moment in time, we can immediately find the value of E: (3) Solving this system, we will obtain: (10) Knowing the elastic modulus, it is possible to determine the magnitude of the high-elastic strain at each moment in time: (4) (5) (11) T/44 International Polymer Science and Technology, Vol. 44, No. 1, 2017

3 In the particular case when t 2 t 1 = t 3 t 2 = Dt, the expression for σ i will take the form: At the end of the relaxation process, the growth rate of rubber strain is zero. From this condition it is possible to obtain the magnitude of E : (12) For points with the numbers i = 2,, n 1, we write the derivative σ i in the form: (13) To determine the coefficients c 1, c 2, and c 3, we substitute into (13) the following functions: (18) Knowing the magnitude of E, it is possible to find at each moment in time the values of f* and η* = f*/n ε. We will take the logarithm of the expression for η* in (2): (14) As a result, we will obtain a system of linear algebraic equations: (19) From expression (19) it can be seen that the quantity y = ln η* should depend linearly on the quantity x = f*, i.e.: The solution of this system has the form: where Dt 1 = t i t i 1 and Dt 2 = t i + 1 t i. (15) (16) Having several values of ln η* corresponding to different magnitudes of f*, it is possible to select by the leastsquares method the coefficients a and b, and then to find the rate modulus m* and the initial relaxation viscosity η * 0. The curve of change in the quantity ln η* as a function of f*, obtained by processing data in Table 1 at T = 20 C, is presented in Figure 2. Deviation from a straight line in Figure 2 is due to a number of factors, chief among which is the small In the particular case of an even time step (Dt 1 = Dt 2 = Dt), we will obtain the well-known formula: (17) At the final point we do not determine σ n, assuming that σ n 0. Figure 2. The change in ln η* as a function of f* Table 2. The elastic and relaxation parameters of recycled PVC at different temperatures T, C E, MPa E, MPa m*, MPa η* , MPa min Smithers Information Ltd. T/45

4 number of time points, which affects the degree of error of numerical differentiation, and also the error in measuring stresses in the experiment. From the stress relaxation experiment it is possible to determine all the parameters entering the Maxwell Gurevich equation. The results of processing the relaxation curves presented in Figure 1 are listed in Table 2. Graphs of change as a function of temperature in the elastic modulus E, the rubbery modulus E, the initial relaxation viscosity η* 0, and the rate modulus m* are presented in Figures 3 to 6 respectively. From Figures 3 to 5 it can be seen that the elastic modulus, rubbery modulus, and initial relaxation viscosity of PVC are very temperature dependent. The dashed lines in each of the graphs show the approximating curves. The temperature dependence of the elastic modulus has the form: (22) Note that, for polymers EDT and PMMA, the temperature dependence of the initial relaxation viscosity is also exponential [5]. The rate modulus at a temperature of C remains roughly constant, and at 70 C is decreases a little. It is proposed to approximate m*(t) with the linear function: (23) The continuous lines in Figure 7 are relaxation curves plotted on the basis of the Maxwell Gurevich equation (20) where T is the temperature ( C). The reliability of approximation by formula (20) R 2 = The temperature dependence of the rubbery modulus with reliability R 2 = is approximated in the following way: (21) The relationship between the initial relaxation viscosity and temperature is approximated fairly accurately (R 2 = 0.989) by the exponential dependence: Figure 5. The temperature dependence of the initial relaxation viscosity Figure 6. Change in the rate modulus m* as a function of temperature Figure 3. The temperature dependence of the elastic modulus of recycled PVC Figure 4. The temperature dependence of the rubbery modulus Figure 7. Theoretical and experimental relaxation curves T/46 International Polymer Science and Technology, Vol. 44, No. 1, 2017

5 and dependences (20) to (23). The triangles mark the experimental stress values. At T = 20, 30, and 60 C, the agreement with experimental data is very good, and at the other temperatures it is satisfactory. REFERENCES 1. Wilkes C. et al., PVC Handbook. Hanser Verlag, Munich, Germany (2005). 2. Solov eva E.V. et al., Investigating the relaxation properties of virgin and recycled polyvinyl chloride. Plast. Massy, (2):54 62 (2013). 3. Solov eva E.V., Experimental investigations of the stress relaxation of polyvinyl chloride. Nauka Tekhn. Obrazov., (8):26 28 (2015). 4. Rabinovich A.L., An Introduction to the Mechanics of Reinforced Polymers. Nauka, Moscow, 283 pp. (1970). 5. Babich V.F., Investigation of the effect of temperature on the mechanical characteristics of polymers. Cand. Tech. Sci. dissertation, Moscow, 140 pp. (1966) Smithers Information Ltd. T/47

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