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Kinetic finite element model to optimize sulfur vulcanization: application to extruded EPDM weather-strips.

INTRODUCTION

A weather-strip is typically an extruded elastomers bulb with complex geometry and curved shape. Typically, weather-strips (1) are installed in automotive industry and civil engineering, in order to prevent water leakage, block exterior noises, minimize body and window vibration, and provide some shock absorbing capacity. Initial use of weather-strip seals in automotive applications was aimed at accommodating for manufacturing variations. Lately, the isolation issue became important, and the key design was to better isolate the passenger compartment from dust, air, and water leakage. Nowadays, the general trend in seal design is the isolation of the passenger compartment from noise and vibration. Isolation is again the main role played by weather-strips installed in windows and doors. Here, stiffness, strength, isolation capacity, and aging resistance are key issues to be considered, since the average life of a building is typically longer than that of a car.

Weather-strips are realized by elastomers, which typically are viscoelastic materials. Their mechanical properties are strain, frequency, and temperature dependent. In addition, being the geometry of weather-strip often rather complex, this generally implies that a numerical model has to be performed with a high level of detail. Unlike plastic and other materials, elastomers have characteristics such as high flexibility, high elasticity, and high elongation. Their major drawback is that they require vulcanization during the manufacturing process, and (2) they cannot be reshaped after curing. Ethylene-propylene-diene monomers (EPDM) are elastomers utilized in a wide range of applications, including weather-strips. The advantage of EPDM is its outstanding resistance to heat, ozone, and weather. Usually, EPDM vulcanizates are composed of rubber, filler (carbon black and calcium carbonate), curatives (fatty acids, zinc oxide, accelerators, and sulfur), antidegradants, and processing aids.

As a matter of fact, weather-strip producers are interested in the improvement of cured EPDM final mechanical performance, at the same time limiting the production costs. For instance, sulfur is normally preferred to peroxides merely for economic reasons, despite the fact that the performance of rubber cured with peroxides--in terms of final mechanical properties--may be sensibly higher than that of rubber vulcanized with sulfur. Indeed, from a chemical point of view, sulfur vulcanization determines transversal chains constituted by more than one sulfur atom (link energy 270-272 kJ/mol) whereas for peroxides the link is created between two back-bone carbons belonging to contiguous chains (energy 346 kJ/mol). In addition, it has to be emphasized that, when peroxides cure method is used, the rubber base could exhibit a peculiar smell, which is obviously inacceptable for the production of weather-strips.

Despite the wide diffusion of sulfur vulcanization and the fact that its discovery and utilization go back to Goodyear (3-6), the chemistry of vulcanization remains an open issue. In this field, among the others, historical contributions by Ding and Leonov (7) and Ding et al. (8) are worth noting. Globally, it can be stated that the majority of the available approaches are models enforced to resemble to peroxidic laws.

Another important issue to consider is reversion, which occurs quite frequently in practice. From a macroscopic point of view, it consists of remarkable decrease of rubber vulcanized properties at the end of the curing process. Chen et al. (9) have shown that this phenomenon seems to appear when two reactions are competing during vulcanization. Reversion is often associated with high-temperature curing. For instance, Loo (10) demonstrated that, as the cure temperature rises, the crosslink density drops, thus increasing the degree of reversion. Morrison and Porter (11) confirmed that the observed reduction in vulcani-zate properties is caused by two reactions proceeding in parallel, i.e., desulfuration and decomposition, see Table 1. Generally speaking, this could give rise to the items with considerable thickness and undergoing different temperatures gradients during curing a strongly inhomogeneous final level of vulcanization.

TABLE 1. Products and schematic reaction mechanisms of accelerated
sulfur vulcanization of polydiene and EPDM elastomers.

Reaction  Compounds        Process/reaction    Kinetic    Model
ID                                             constant   constants

NA        [S.sub.8] +      Mechanical mixing   NA         NA
          accelerators +   by open roll mill,
          ZnO + stearic    internal mixers,
          acid [right      and/or extruders
          arrow] soluble   ai T <
          sulphurate zinc  100[degrees] C
          complex (A) +
          polydiene
          elastomers (P)

(a)       [MATHEMATICAL    Allylic             [K.sub.1]  [K.sub.1]
          EXPRESSION NOT   substitution
          REPRODUCIBLE IN
          ASCII]

(b)       [MATHEMATICAL    Disproportionate    [K.sub.2]  [K.sub.2]
          EXPRESSION NOT
          REPRODUCIBLE IN
          ASCII]

(c)       [MATHEMATICAL    Oxidation           [K.sub.3]  [~.K]
          EXPRESSION NOT
          REPRODUCIBLE IN
          ASCII]

(d)       [MATHEMATICAL    Desulfuration       [K.sub.4]
          EXPRESSION NOT
          REPRODUCIBLE IN
          ASCII]

(e)       [MATHEMATICAL    De-vulcanization    [K.sub.5]
          EXPRESSION NOT
          REPRODUCIBLE IN
          ASCII]


Very recently Milani and Milani (12) have proposed a simple kinetic numerical model to predict EPDM reticulation level, which may also take into account reversion. The model is a simplified one and relies into the derivation of a single second order nonhomogeneous differential equation, representing the degree of reticulation (or conversely the torque resistance) of rubber in dependence of curing time. Kinetic parameters to set in the kinetic model are only three and they can be evaluated by at least two torque curves performed on the same blend at two different vulcanization temperatures. Cure tests have to be maintained at fixed vulcanization temperature and may be performed by means of both traditional oscillating disc (ODR) and rotor-less (13), (14) (RPA2000) cure-meters.

In the present article a RPA2000 cure-meter is utilized to perform the experimentation. Such device has a test chamber with controlled stable vulcanization temperature allowing the storage of 5 gm of product, with diameter 20 mm and height 12.5 mm (total neat volume 8 [cm.sup.3]). Basically, there are no perceivable geometric effects related to the torque measure because (1) the test is fully standardized and (2) in case of RPA2000 devices the ODR is missing. Quite small secondary torques may be present as a consequence of material viscosity and plates friction, which are obviously negligible in practice. Also in case of experimentations conducted with ODRs, it is worth remembering that the dimension of the disc is relatively small, allowing to disregard inertia forces, always present when mechanical elements move and responsible of secondary geometric effects.

After experimental data reduction, the aforementioned kinetic model is adopted to predict rubber degree of vulcanization during the industrial curing process of a thick weather-strip used in civil engineering applications. Once evaluated the kinetic constants involved in the reticulation process, the second phase relies in implementing kinetic model parameters within a nonstandard finite element (FEM) software for a thermal analysis of the item. Such approach follows a relatively long tradition regarding FEMs applied to then-no--mechanical problems of weather-strips installed within devices subjected, after thermal curing, for static and dynamic loads (15-19). FEM is, indeed, recognized as the most suitable technique to interpret the thermo--mechanical behavior of vulcanized rubber items with complex geometries, giving the possibility to quickly study combined nonlinear 3D problems, with error estimates and error reduction upon mesh refinements. The software developed allows obtaining, element by element, temperature profiles at increasing curing times. In addition, it is possible to evaluate output mechanical properties (tensile strength, tear resistance, and elongation) increase as a function of curing time. In this case, the numerical database collected in the first phase (reticulation kinetic model) is used, allowing a point by point estimation of any output mechanical property.

The blend studied to realize the weather-strip is a mix of two different EPDMs (Dutral TER 4049 and 9046) with a medium amount of propylene content (ca. 31% or 40% in weight) and 9% or 4.5% in weight on ENB (5-ethylidene-2-norbornene), vulcanized through accelerated sulfur, as described in detail next. Once evaluated the final mechanical properties of the item point by point, a compression test is numerically simulated, assuming that the rubber behaves as a Mooney-Rivlin material under large deformations and using contact elements between the compression device and the item. From an industrial point of view, the numerical approach may be useful to optimize (especially in economic terms) (i) vulcanization time, (ii) energy utilization, (iii) temperature of vulcanization, and (iv) accelerators quantities. The procedure is quite general and can be used in presence of any rubber blend, provided that suitable experimental data are at disposal to characterize crosslinking reactions at different temperatures.

THE KINETIC NUMERICAL MODEL: A REVIEW

The recently presented kinetic model (20) is utilized to evaluate the degree of vulcanization reached by a rubber specimen subject to thermal predefined conditions (constant temperature) and vulcanized with sulfur. The model relies into a second order differential equation with solution evaluable in closed form, having only three kinetic constants to be determined. Constants are usually evaluated by means of rheometer experimental tests realized following the ASTM D 2084 and D 5289 methods (13).

Focusing exclusively on EPDM rubber, the commonly accepted basic reactions involved-see also Refs. [5. 2124 and Table 1, are:

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C.) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(d) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(e) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where P and A are the polymer (EPDM) and soluble sulfureted zinc complex ([S.sub.8] + accelerators + ZnO + stearic acid) respectively, [P*.sub.1] is the pendent sulfur (crosslink precursor), [P.sub.v] is the reticulated EPDM. [K.sub.1,...,5] are kinetic reaction constants, which depend only on reaction temperature, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [Q.sub.x]and [D.sub.e] are the matured crosslink, the oxidation product and diaryl-disultide, respectively. Reaction (a) in Eq. / represents the allylic substitution in Table 1, reaction (b) is the disproportionation, whereas reactions (c-e) occurring in parallel are respectively the oxidation, the de-sulfuration and the devulcanization.

Chemical reactions occurring during sulfur vulcanization reported in Eq. I obey the following rate equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The set of differential Eq. 2 may be obviously solved numerically by means of standard Runge--Kutta procedure (25), (26). However, this approach may become tedious both from a numerical and practical point of view. Closed form solutions are obviously preferable.

In (20) it is shown how the concentration of vulcanized polymer [P.sub.v](t) within the material during the vulcanization time range is ruled by the following single second order non-homogeneous differential equation with constant coefficients:

[d.sup.2][P.sub.v]/d[t.sup.2] + [K.sub.2]d[P.sub.v]/dt + [[~.K].sup.2][P.sub.v] = [K.sub.1][K.sub.2][P.sub.0.sup.2]/[([P.sub.0][K.sub.1]t + 1).sup.2] (3)

Having indicated with [[~.K].sub.2] the following constant:

[[~.K].sub.2] = [K.sub.2]([K.sub.3] + [K.sub.4] + [K.sub.5]) + [K.sub.3.sup.2] + [K.sub.4.sup.2] + [K.sub.5.sup.2] (4)

It can be shown that Eq. 3, after some reasonable simplifications on the nonhomogeneous term fully explained in Ref. (20), may be solved in closed form and the solution is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

As it is possible to notice, in Eq. 5 kinetic constants to be determined are only three, namely [K.sub.1], [K.sub.2], and [[bar.K].sub.2]. The most straightforward method to provide a numerical estimation of kinetic constants is to fit Eq. 5 on experimental cure-curves, normalized scaling the peak value to Po (for instance equal to 1) and translating the initial torque to zero, as suggested by Ding and Leonov (7). As a rule, variables [K.sub.1], [K.sub.2], and [[~.K].sup.2] are estimated through a standard nonlinear least square routine. The initial part of the curve, before scorch point, which is typically linked to viscosity exhibited by a fluid, cannot enter into the optimization process, because it is obviously ruled by other physical mechanisms.

EPDM Blends Under Consideration

Two different EPDMs (Dutral TER 4049 and Dutral TER 9046) experimentally tested in [20.1 are reanalyzed in this work. The characteristics of such EPDM blends, in terms of Mooney viscosity and compositions are summarized in Table 2. In the same table, a hypothetic product derived ad hoc mixing the two experimentally tested blends is also indicated. Generally speaking, from a practical point of view, an elastomeric blend is an interesting method commonly used in the rubber industry in order to increase final properties of rubber items. It has been shown in many applications that, while two polymers may be virtually mutually insoluble, blends may be industrially produced, which are macroscopically homogeneous and have improved properties, provided that mechanical mixing is suitable and viscosities after mixing are sufficiently high to prevent gross phase separation. In this situation a master-batch process would be desirable, i.e., the component polymers should be precompounded with vulcanizing agents and additives, and then individual stocks should be blended in desired proportions. In our specific case, the two rubber types are quite similar in terms of molecular weight, molecular weight distribution, chemical properties, and structure. They exhibit different cure rates because different amounts of ENB are present, namely 1% and 2% in terms of % moles in the polymers. The mix under consideration deserves to be studied for the following reasons: (1) it may be produced by means of the same catalyst system and the same process used for Dutra! TER 4049 and 9046; (2) when dealing with Mooney viscosity number, differences do not exceed 30 points between the two products, meaning that the mix between them do not modify the MWD extensively; (3) the composition in terms of ethylene of both principal polymers is about the same and in any case in the range in which the polymers are perfectly amorphous.

TABLE 2. Composition of Dutral 9046, Dulral 4049 and Duiral
9046-4049 blend.

                                                         Mix 70%
Product type             Dutral 9046  Dutral 4049  4049-30% 9046

Propylene (wt%)                   31           40           37.3
ENB (wt%)                        9.0          4.5           5.85
Ethylene (wt%)                    60         55.5          56.85
ML(1 + 4) [Degrees]C              67           93           85.2
ML(1 + 4) 125[Degrees]C           49           76           67.9


For all types of EPDM and the hypothetical mix considered, the same formulation was used, with the aim of comparing the experimental data with the data derived numerically from the solution of the ordinary differential equation system Eq. 2--hereafter labeled as ODEs system model for the sake of clearness--and the single second order differential Eq. 5--hereafter labeled as single EQ-DIFF model. Rheometer curves at 160 and 180[degrees]C are shown in Fig. 1. It is interesting to notice that Dutral TER 9046 exhibits more visible reversion at 180[degrees]C, whereas the behavior of Dutral TER 4049 is less critical. Being the mix composed by 70% Dutral TER 4049 and only 30% Dutral TER 9046, its reversion is expected to be quite little and in any case limited to the final vulcanization range at 180[degrees]C. Reversion is totally absent at 160[degrees]C. In Table 3, the compounds formulation adopted (in parts per hundred resins) is summarized. The experimental compounds were generally prepared in an internal mixer and by adding both curing and accelerating agents on the roll mixer, maintaining the temperature of mixing lower than 100[degrees]C.

TABLE 3. Compounds formulation adopted (in phr).

Ingredients   Description                   phr

Polymer       Dutral 4049, 9046             100
Zinc oxide    Activator                       5
Stearic acid  Coagent                         1
HAFN 330      Carbon black                   80
Paraffin oil  Wax                            50
Sulfur        Vulcanization agent           1.5
TMTD          Tetramethylthiuram disulfide  1.0
MBT           Mercaptobenzothiazole         0.5


A comparison among experimental data, ODEs system and single EQ-DIFF at temperatures equal to 180 and 160[degrees]C is represented in Fig. 2 for Dutral TER 4049, in Fig. 3 for Dutral TER 9046 and in Fig. 4 for the mix between 4049 and 9046, respectively. Convergence curves are also represented for both models. Each curve represents the difference between experimental and numerical normalized torque at a given iteration number. The mix of products here presented will be adopted for studying the production phase of whether-strips analyzed in the following sections. Here it is worth noting that, for these particular items, it is crucial to have at disposal a rubber compound with a rheometer test curve, which reaches asymptotically the maximum Mooney viscosity, without exhibiting reversion. As it has been demonstrated in Ref. (20) in a different technical problem, the overall vulcanization level of an item may be numerically predicted by means of the determination of temperature profiles and the knowledge of rheometer curves at different temperatures. In general, vulcanization density depends on curing temperature and time as well as on thickness of the items, obviously at given recipe used. Compounds exhibiting reversion at temperatures lower than 220[degrees]C should be avoided, since may result into items over-vulcanized near the skin. Conversely, from an economical point of view, it is important to reduce vulcanization time and hence to use high temperatures, blends that vulcanize more quickly (as Dutral TER 9046), but without reversion (as Dutral TER 4049): for these reasons, the polymer mix here proposed appears to be a good compromise between quality of final reticulation and reduction of production time needed.

Cured Rubber Mechanical Properties

To evaluate numerically the degree of vulcanization of a given rubber item, it is necessary to link degree of vulcanization with output mechanical properties. There are many important parameters that producers are interested to maximize at the end of the vulcanization process, as for instance tear resistance, tensile strength, tension set, etc. (27), (28). Unfortunately, in almost the totality of the cases, such parameters cannot be maximized at the same time. In this article, we will focus, for the sake of simplicity, only on a single output parameter (tensile strength).

As experimental evidences show, there is a precise relationship between torque values obtained in the vulcanization curve and output mechanical properties of a real cured item (see for instance Refs. (27), (29-40). In Fig. 5, several experimental data available in the literature concerning torque values M and tensile strength exhibited by vulcanized commercial products are represented, along with a possible numerical experimental data fitting. The numerical relation between torque and tensile strength found for Dutral blends and represented in Fig. 5 will be used in the following section to perform some simulations on a real weather-strip. An interesting aspect of the experimental relation existing from torque and tensile strength is that it is not necessarily mono-tonic, i.e., tensile strength may slightly decrease increasing torque resistance. In the same figure, the relation between hardness and percentage vulcanization degree is also depicted. The curve is taken from Coran (21) and depends on the blend used. Having at disposal such relationship, from the thermal model proposed hereafter it is possible to evaluate point by point the vulcanization degree at the end of the thermal process, and hence the local hardness of the item. From well-known numerical relationships (usually based on empirical formulas or equations based on theoretical formulations, e.g., Gent, BS 903, ASTM 1415, Qi et al. (41-43) it is finally possible to estimate resultant Young Modulus E of the rubber after vulcanization, to be used within the mechanical compression simulations reported at the end of the article.

Another important issue to underline is that, obviously, the representation of Fig. 5 is able to give correct information on output parameters only at constant temperatures (note that T [not equal to] [T.sub.n] being T rubber temperature and [T.sub.n] curing agent temperature), i.e., results cannot be applied directly to a rubber infinitesimal element subjected to curing. Indeed, the actual temperature of a point inside an item subjected to heating is rather variable and such variability can be estimated only resorting to numerical methods. In general, each point P of the item has its own temperature profile T = T(P,t). At a fixed time during the curing process, each point has also its own torque value, which can be calculated from the database collected previously. For each value of the torque reached, a corresponding value of the output parameter can be identically evaluated. In this way, torque-exposure time and output property-exposure time (or alternatively output property-temperature) profiles can be numerically estimated. Nonetheless. Fig. 5 gives a precise (although approximate) idea of the complex behavior of rubber during vulcanization, addressing that a strong variability of output mechanical properties is possible when changing curing time and vulcanization temperature.

Vulcanization of Thick Weather-Strips with Accelerated Sulfur

In a weather-strip subjected to any vulcanization process, an inhomogeneous distribution of temperatures between internal (cool) zone and (hot) skin occurs. With the aim of optimizing a production line, many parameters have to be chosen carefully but, among the others, the following variables play a crucial role: exposure time and temperature of the heating phase. The software presented here--fully developed in Matlab (44) language--is basically an assemblage of two elemental blocks, further subdivided into sub-blocks.

BLOCK 1: for each node of the item (already discretized by means of FEMs) at fixed values of the input parameters [T.sup.n] (curing agent temperature) and [t.sup.c] (curing time), temperature profiles are evaluated by solving numerically a Fourier's heat transmission problem in two-dimensions (45-51). Since, closed form solutions are rarely available and usually refer to simple geometries, a FEM (52) discretization is needed in the most general case. For each node of the item and for each value of exposure time and curing temperature inspected numerically, local final mechanical properties are evaluated. The average uniaxial tensile strength is considered as objective function. To maximize this quantity, lour sub-block instructions are repeated node by node.

SUB-BLOCK 1.1: at increasing times, the temperature of each node varies (increasing until the end of the vulcanization phase and then quickly decreasing during cooling). Each point is thus characterized by a couple of values representing temperature and time.

SUB-BLOCK 1.2: for each temperature value collected in the previous sub-block, the corresponding cure curve (evaluated numerically with the model described previously and having at disposal kinetic constants of the partial reactions) is uploaded. Here, it is worth noting that each node of the item to vulcanize undergoes continuously different temperatures, thus passing from a cure curve to a contiguous one (more precisely to a contiguous curve associated to a higher temperature in the heating phase).

SUB-BLOCK 1.3: for each point of the item, the torque-time diagram is evaluated making use of all the numerical cure curves provided by the single differential equation model described previously.

SUB-BLOCK 1.4: tensile strength and hardness (or elastic modulus) to be used in mechanical simulations may be finally evaluated (e.g., tensile strength--time and tensile strength--temperature profiles) for each node or point as a function of the torque reached by the point at successive time steps. This is possible only having at disposal a relation between final torque and tensile strength (or hardness) as those provided in Fig, 5. Usually experimental data may be useful in this step.

BLOCK 2: the procedure summarized in BLOCK 1 should be repeated changing input parameters (i.e., total curing time and vulcanization agent temperature). A very direct way to evaluate the best input couple ([t.sup.c] [T.sup.n]) is probably to subdivide the input domain in a regular grid of points, but for practical purposes random attempts and the experience of the user should be sufficient. Very advanced algorithms based on Genetic Algorithms or bisection has been already presented in Refs. (51), (53), (54). They are not used here, because the knowledge of the blend behavior drives almost directly to the choice of the most indicated production parameters to adopt.

The Modified Heat Transmission Problem

A modified heat transmission problem has to be solved to evaluate item temperature profiles, accounting for the chemical reactions occurring during vulcanization. We schematically subdivide the curing process into two phases: heating and cooling. In the first phase, elastomers are exposed to high temperatures in order to activate crosslinking and thus vulcanization, whereas in the second phase rubber is kept to ambient temperature through air and/or water. In the most general case of 2D items, temperature profiles for each point of the element are obtained solving numerically Fourier's heat equation law (45-51):

[[rho].sub.p][c.sub.p.sup.p]([partial derivative]T/[partial derivative]t) - [[lambda].sub.p][[gradient].sup.2]T - [r.sub.p][DELTA][H.sub.t] = 0 (5)

where [[rho].sub.p], [c.sub.p.sup.p] and [[lambda].sub.p] are EPDM density, specific heat capacity, and heat conductivity, respectively; [DELTA][H.sub.r](KJ/mol mol) is rubber specific heat (enthalpy) of reaction and [r.sub.p] [mol/([m.sup.3] sec)] is the rate of crosslinking.

When heat transmission at the external boundary is due to convection and radiation (extrusion process), the following boundary conditions must be applied in combination with the field problem:

[[lambda].sub.p][partial derivative]T(P, t)/[partial derivative]n(P) + h(T(P, t) - [T.sub.n]) + [q.sub.rad] = 0 (7)

where h is the heat transfer coefficient between EPDM and vulcanizing agent at fixed temperature T, [T.sub.n] is vulcanizing agent (e.g., nitrogen) temperature, P is a point on the object surface, n is the outward unitary vector on P. and [q.sub.rad] is the heat flux transferred by radiation. Radiation contribution for the vulcanization of complex 2D geometries may not be determined precisely, but the following formula may be utilized without excessive inaccuracies:

[q.sub.rad] = [sigma]([T.sub.n.sup.4] - T[([R.sub.p], t).sup.4])/[1/[[epsilon].sub.p] + [A.sub.p]/[A.sub.n](1/[[epsilon].sub.n] - 1)]

where [sigma] = 5.67 x [10.sup.8] W/([m.sup.2][K.sup.4]) is the Stefan--Boltz-mann constant, s are emissivity coefficients, [A.sub.p,n] are the areas of heat exchange (p: rubber item; n: curing agent).

The same considerations hold for the cooling phase, with the only differences that (a) the actual cooling agent temperature has to be used and (b) that heat exchange for radiation is negligible, i.e.:

[[lambda].sub.p][partial derivative]T(P, t)/[partial derivative]n(P) + [h.sub.w](T(P, t) - [T.sub.w]) = 0 (9)

where [h.sub.w], is the water (air) heat transfer coefficient, [T.sub.w] is the water (air) cooling temperature, and all the other symbols have been already introduced.

Initial conditions on temperatures at each point at the beginning of the curing process are identically equal to the room temperature. Initial conditions at the beginning of the cooling phase are obtained from the temperature profiles evaluated at the last step of the cooling zone, i.e., at T(P,[alpha][t.sub.c]), where ate is the heating interval, and P is a generic point belonging to f2. To solve partial differential equations system (6)-(9), standard elements (46), (47), (52) are used. The procedure has been completely implemented in Matlab (44) language and interfaced with the first phase of the procedure. In this way, resultant FEM temperature profiles at each time step are directly collected from the numerical analysis and utilized for the evaluation of output rubber mechanical properties by means of an integrated tool.

For the discretization of the weather-strip, which is schematically represented by its cross-section, 4-and 3-noded 2D elements are utilized. Temperature field interpolation is assumed linear inside each element, i.e., = [N.sup.e][T.sup.e], where, for 4-noded elements, [T.sup.e] = [[[T.sup.1], [T.sup.2], ... [T.sup.4]].sup.T] is the vector of nodal temperatures, [N.sup.e] = [[N.sup.1], [N.sup.2], ... [N.sup.4]] is the vector of so-called shape functions [N.sup.i]( i = 1, ... 4) and P is a point of coordinates [x.sub.p], [y.sub.p], and [z.sub.p] inside the element. Following consolidated literature in this field, e.g., see Ref. (55), in the numerical simulations reported, the following parameters have been used: EPM/EPDM density [[rho].sub.p] = 922 kg/[m.sup.3], rubber specific heat capacity [c.sub.p.sup.p], = 2700 J/(kg [degrees]C), [[lambda].sub.p] = 0.335 W/(m [degrees]C), [DELTA][H.sub.r] = 180 kJ/mol, water heat transfer coefficient [h.sub.w] = 1490.70 W/([m.sup.2] [degrees]C), curing agent heat transfer coefficient h = 900 W/([m.sup.2] [degrees]C) (for air heat transfer coefficient we assume h = 5 W/([m.sup.2] [degrees]C), [[epsilon].sub.p] = 0.60, [[epsilon].sub.c] = 0.70, water cooling temperature [T.sub.w] = 25[degrees]C.

The FEM discretization and the geometry of weatherstrip considered are sketched in Fig. 6 (top subfigure). In the same Fig. 6 (bottom subfigure), temperature contours obtained during a FEM simulation at increasing instants are represented (nitrogen temperature [T.sub.n] = 160[degrees]C, curing interval [alpha][t.sub.c], = 300 sec). As it is possible to notice, internal points heating is sensibly slower with respect to the skin, as also schematically represented in Fig. 7a, where temperature profiles of three different points of the item are depicted. This may influence the final quality of the vulcanized rubber. Points are labeled as A, B, and C. B and C are near the external boundary of the item. whereas A is an internal point. It is therefore expected that B and C thermal behavior is similar and at the same time quite different to point A behavior. Data adopted in the thermal simulations are in agreement with literature indications (56), (57). For the sake of simplicity, specific heat capacity [c.sub.p.sup.p] and conductivity [[lambda].sub.p] are assumed constants in the simulations. In reality, such parameters reasonably exhibit certain variability in the temperature range inspected. In particular heat capacity [56] ranges from 1700 J/(kg [degrees]C) at 80[degrees]C to 2900 J/(kg [degrees]C) at 200[degrees]C, with an almost linear behavior in such temperature range, whereas conductivity decreases asymptotically from 0.355 to 0.335 W/(m [degrees]C). In Fig. 7b and c, part of the huge amount of results obtained from a sensitivity analysis conducted varying material heat capacity and conductivity is summarized. In particular, temperature profiles for points A (b) and B (c) are represented, assuming three different constant values for the heat capacity ([c.sub.p.sup.p], equal to 1700, 2200, and 2700 J/ (kg [degrees]C), respectively) at fixed conductivity [0.335 W/(m [degrees]C)], a linear behavior of [c.sub.p.sup.p] with either [[lambda].sub.p] constant or 'P asymptotically variable following (58). As it is possible to notice, the variability of the temperature profiles is quite small and in any case the error introduced assuming [c.sub.p.sup.p] equal to 2700 J/(kg [degrees]C) is in the most unfavorable case within 10%, fully acceptable from an engineering standpoint. The greatest variability is obviously experienced for internal points, whereas for points near the core, temperature profiles almost coincide. Finally, authors experienced very little variability with [[lambda].sub.p].

The thermal behavior during vulcanization of the core and the skin, i.e., points A and B, subjected to the vulcanization conditions previously discussed and also with a different [T.sub.n], = 200[degrees]C, are represented from Figs. 8-11. The relationship used to evaluate the final tensile strength of the point is depicted in Fig. 5a, which is a reasonable hypothesis of the blend behavior under mechanical tests. As already pointed out, there is very little or absent reversion even at high temperatures when a Dutral TER 4049 70% and Dutral TER 9046 30% mix is utilized. In particular, the temperature profile of point A for both [T.sub.n] = 160[degrees]C and [T.sub.n] = 200[degrees]C, Fig. 8 (2nd and 4th row, 2nd column sub figures), shows a gradual initial temperature increase, meaning that Point A passes very slowly from rheometric curves at low temperatures to rheometric curves at high temperatures. The vulcanization phase stops at 300 sec and in both cases ([T.sub.n] = 160[degrees]C and 200[degrees]C) the point is not able to reach the maximum possible temperature, i.e., external nitrogen temperature. This was expected, because A is positioned in the core of the item. Such curves are numerically interpolated, as discussed previously, from experimental data using the kinetic model proposed, see Fig. 9, once that constants [K.sub.1], [K.sub.2], and [[~.K].sup.2] are at disposal. When the maximum temperature is reached, point A follows the rheometric curve associated to such temperature, stabilizing its behavior at increasing times on that rheometer curve. In other words, point A follows almost completely the rheometer curve at around 150[degrees]C and 190[degrees]C, respectively. When curing finishes, temperature decreases quickly. Here, two paths should be followed numerically for point A, represented in Fig. 9 with circles and crosses. The numerical behavior represented by crosses is not real, because it would be referred to a material without any memory of the vulcanization. Therefore, it relies on a material that ideally jumps from a rheometer curve at high temperature to the successive at low temperature (reversible process). On the contrary, the real behavior of the material under consideration is represented by the circles, indicating that the level of vulcanization reached at the end of the curing process cannot be inverted reducing temperature, but can only be stopped.

Once that the curing time-torque diagram of the rheo-metric curve is known numerically (2nd and 4th row, 1st column diagrams) for the point under consideration, tensile strength reached increasing exposure times can be easily determined through the relation sketched in Fig. 5. For point A, the exact vulcanization history in terms of curing time-[[sigma].sub.t], and temperature-[[sigma].sub.t] diagrams is represented in Fig. 8 (1st and 3rd row, 1st sub-figures). The same considerations can be repeated for points B (or similarly C) of the item, thus giving the possibility to evaluate the average final tensile strength. In particular, in Figs. 10 and 11 the curing behavior of point B is represented, whereas point C data are omitted for the sake of brevity. From a comparative analysis, it is rather evident that points B and C reach maximum vulcanization temperature more quickly with respect to point A. since the cure curves followed by such points are very near with each other. Also in this case, the reversion range is very little even at 160[degrees] and, as expected, it occurs after that the peak strength has been reached.

From a detailed analysis of points A, B, and C behavior, it can be argued that at 160[degrees] the item is slightly undervulcanized, because the thickness of the core would require either a higher vulcanization external temperature or a longer exposure time. The technical usefulness of the approach proposed seems rather clear, since manufacturers could calibrate (with a few numerical attempts) both curing time and vulcanization temperature of the production phase, to avoid reversion and under-vulcanization, thus minimizing the industrial process cost.

Mechanical Compression Test on the Weather-Strip

After a detailed numerical analysis of the industrial vulcanization process, a FE mechanical analysis of weather-strip installed in the body of a door (civil engineering application) is finally conducted using Strand7 commercial software package. The mechanical simulations reproduce a typical experimental compression test, where a part of the boundary (shown in Fig. 6) is supposed clamped and a rigid punch compresses the opposed boundary. Displacements in the x- and y-directions are restricted only in such region, to simulate the actual constraints acting where the weather-strip is installed in a door. Contact elements between the punch (Fig. 6) and the rubber material are utilized to simulate mono-lateral contact. Zero gap elements available in Strand 7 are used to model contact.

A two constants Mooney--Rivlin model under large deformation hypothesis is used for rubber, with parameters [C.sub.10] and [C.sub.20], respectively equal to 0.270 and 0.160 MPa. Such constants are suitably chosen in order to fit as close as possible, the experimental uniaxial stretch--strain curve available for the blend used. Here it is worth noting that, from the vulcanization model proposed in the previous section, only the initial Young Modulus is indirectly known, once that the average vulcanization level is evaluated and hence the final hardness S is at disposal. For a vulcanization executed at [T.sub.n], = 185[degrees]C and [t.sub.e] = 600 sec a very good vulcanization level with average hardness equal to 56 is obtained. The corresponding Young modulus is evaluated by means of BS 903 as 2.58 MPa. The elastic shear modulus is always E/3 and is linked with Mooney--Rivlin constants by means of the following formula: G = 2([C.sub.10] + [C.sub.20]). A further condition is necessary to set univocally [C.sub.10] and [C.sub.20], meaning that the uniaxial stretch--stress curve test is needed. In absence of specific experimental data available, from authors experience, the typical uniaxial behavior of the compound is well approximated, at least for stretches lower than 3, assuming [C.sub.20] roughly equal to 0.6 [C.sub.10].

In such a situation, defined the stretch as the ration between the length in the deformed configuration divided by the length in the undeformed state, let [[lambda].sub.1] = [lambda] be the stretch ratio in the direction of elongation and [[sigma].sub.l] = [sigma] the corresponding stress. In uniaxial stretching, the other two principal stresses are zero, since no lateral forces are applied ([[sigma].sub.2] = [[sigma].sub.3] = 0). For constancy of volume, the incompressibility condition [[lambda].sub.1] [[lambda].sub.2] [[lambda].sub.3] = 1 gives:

[[lambda].sub.2] = [[lambda].sub.3] = 1/[square root of [lambda]]. (10)

In the most general case, the strain energy function for a two-constant Mooney--Riv lin model is given by:

W = [C.sub.1]([I.sub.1] - 3) + [C.sub.2]([I.sub.2] - 3) (6)

where [I.sub.1.sup.2] + [[[lambda].sub.2.sup.2] + [[lambda].sub.2.sup.3] and [t.sub.2] = [[lambda].sub.1.sup.-2] + [[[lambda]sub.2.sup.-2] + [[[lambda].sub.3.sup.-2].

The FE model, as mentioned earlier, is constituted by two-dimensional elements, which are supposed here to undergo a plane strain mechanical condition under large displacement and contact elements. Quadrilateral 4-noded elements with bubble-shaped extra-functions and triangular linear elements are utilized to mesh the rubber weather-strip. To speed up computations, only half of the device is meshed for symmetry, applying suitable boundary conditions on displacements on the symmetry axis. The compression device is assumed to be a rigid body, since it is supposed to exhibit very little deformation when compared to rubber. The compression device is loaded by means of an increasing horizontal displacement up to 4 mm from its unloaded position. Displacement sub-steps equal to 0.2 mm are imposed on the FE model to facilitate convergence.

Contact forces distributions at successive time-steps are represented in Fig. 12a. It is particularly evident that the compressed contact zone, as expected, spreads at successively increased displacements of the compression device. A direct integration of contact forces allows the evaluation of the force to apply to the compression device at successive displacements, as represented in Fig. 12b. The loading condition obviously justifies the increase of first derivative of the load at increased displacements, since the contact surface becomes larger. Figure 12b shows the overall behavior is very important for producers, because it indicates the overall expected stiffness of the device under design loads. Such stiffness should fall within some lower and upper bounds to be acceptable. The global behavior of the device depends not only on the rubber compound used but also on the vulcanization process used to cure it, since elastic properties are Shore A dependent. It is therefore crucial to have at disposal of an integrated thermo--chemo--me-chanical software as that used here to predict such behavior.

For the sake of completeness in Fig. 12b compression curves obtained assuming a hardness S equal to 65 and 45, respectively are represented. For the blend and the specific item under consideration, such hardness values are obtained respectively when the following vulcanization conditions are assumed: [T.sub.n] = 195[degrees]C with [t.sub.e] = 500 and [T.sub.n] = 175[degrees]C with [t.sub.e] = 650. The same hypotheses assumed previously are adopted for Mooney--Rivlin constants [C.sub.10] and [C.sub.20]. From the sensitivity analysis conducted, it is very straightforward to conclude that the evaluation of the overall stiffness of the item in presence of different vulcanization conditions is crucial, especially when the item is installed as weather-strip in a real door. As a matter of fact, from an engineering point of view, the knowledge of such stiffness may be very useful, especially when a maximum displacement threshold must be respected, as for instance for large windows subjected to wind.

CONCLUSIONS

In the present article, the possibility to utilize, in weather-strip for industrial production processes, two different ter-polymers has been investigated by means of a comprehensive numerical model. The first ter-polymer has a low amount of ENB, whereas the second exhibits the maximum amount of ENB nowadays possible in EPDM production plants. For both polymers. experimental rheometer curves were at disposal at two different temperatures. The behavior of a mix of both polymers, 70-30% in weight, has also been investigated numerically. In particular, rheometer curves of the mix at two different temperatures have been numerically extrapolated from components experimental data available, taking into account their presence in the mix in terms of relative percentage in weight. The vulcanization behavior of the mix has been then numerically studied within the production line of a real weather-strip. Generally, weather-strips are produced from ter-polymers with an ENB content equal to around 6. but this custom is almost always empirical. In this framework, this article gives a numerical insight into the most suitable blend to be used in the production of such kind of items.

From a production point of view, it is well known that a huge amount of different EPM--EPDM commercial products are available. Differences are mainly due to the ENB content, the distribution of molecular weights, and the Mooney viscosity. The choices to mix two different products with low and high ENB content, respectively, appears interesting, since blends with an intermediate behavior could potentially increase the final mechanical quality of the vulcanized item, as well as reduce production time and hence costs. From the producer point of view, costs reduction is derived not only from the reduction of materials to stock, but also from the possibility to modulate time and temperature used to obtain an optimal vulcanization of the final item. It has been shown, indeed, that a ter-polymer with low ENB content exhibits a rheo-metric curve monotonically increasing, but with low rate of vulcanization. Conversely, a ter-polymer with high ENB content shows a very good vulcanization rate but exhibiting also reversion, especially at high temperatures. The present article shows how an intermediate blend, obtained ad hoc mixing the aforementioned products, may have both good vulcanization rate (intermediate between the products with low and high ENB content) and no or very little reversion. Furthermore, it has to be emphasized that the possibility to mix ter-polymers with different molecular weights and Mooney viscosity higher than 60, should modify the molecular weight distribution of the mix, with the important advantage to obtain an ENB uniform distribution in the back bone, which should allow a higher crosslinking homogeneity.

From the numerical model proposed, it seems that, at least theoretically, it is possible to obtain mixtures whose vulcanization can be tuned as a function for industrial production requirements, with an obvious increase of the flexibility. A comparison with experimental data on the production line, unavailable at this stage, will be obviously rather useful to have an insight into the actual predictive capabilities of the comprehensive numerical approach proposed.

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G. Milani, (1) F. Milani (2)

(1.) Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

(2.) CHEM.00 Consultant, Via J.F. Kennedy 2, 45030 Occhiobello, Rovigo, Italy

Correspondence to: G. Milani; e-mail: gabriele.milani@polimi.it

Published online in Wiley Online Library .(wileyonlinelibrary.com).

[c] 2012 Society of Plastics Engineers

DOI 10.1002/pen .23270
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