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Polymer weld strength predictions using a thermal and polymer chain diffusion analysis.


As the use of plastic components has increased, so has the need for secure and cost effective Joining methods. Optimal performance is expected with molecular bonding, which may be subdivided into two categories: adhesion and welding. Adhesion bonding describes chemical processes where a third substance is used to create a bond between two materials. Welding describes a thermally agitated phase change process where the two materials are molecularly mixed. The suitability of either process is often determined by material properties, such as thermal stability or molecular architecture surface energy. For example, thermoset plastics have favorable surface energetics for adhesive bonding but degrade during attempts to weld, while nonpolar plastics such as polyethylene (PE) are best joined by welding and generally require extensive surface modification for adhesion. Since polyethylene is used in a range of piping applications, improvements in the fundamental understanding of polymer welding processes would greatly benefit these applications.

Relative to metal welding, where a considerable body of literature exists on techniques and fundamentals, little information exists on plastic welding (2). In particular, the weld strength is not adequately described in terms of the weld parameters. Several types of plastic welding processes are currently used, and with increasing plastics use, new techniques are constantly being introduced. Modifications to the process design are brought about by best-guess adjustments of process parameters. It is generally known that the weld zone temperature, time of contact, joining pressure, and material properties influence the weld strength. The fundamental interactions between these parameters are not well understood, and a theoretical basis for weld predictions as a function of weld parameters is needed for plastics. Characterizing welding processes is approached by two quite different methods in the literature. Several studies have correlated empirical results for weld strength with thermal conditions without significant focus on the polymer properties which influence this strength. Other studies have focused on the interactions of the polymer chains without examining the resulting bond.

Nakashiba et al. (3) use a two-dimensional cylindrical model to simulate the weld process in electro-fusion of polyethylene. They solved only the energy equation and use variable property data for the PE. They found that temperatures measured with a thermocouple were in good agreement with those calculated using their finite element model. However, there is no explicit relationship in their computations between the thermal effects and the polymer weld results. Experiments confirmed that the strength of the welds was related to the input power. Two limiting cases were noted: very high power densities caused material degradation, which compromised the weld strength, and sufficiently low power densities could never create welds.

Butt fusion is analyzed by Pimputkar (4) using a quasi-analytical method. Pimputkar uses uncoupled integral solutions for the energy and momentum equations. Noting that no consensus method exists to determine weld quality, he defines a nondimensional "joining parameter" through which he correlates experimental results for fracture impact. The joining parameter itself is a function of the pipe thickness, location of the melt front after the heating phase, and the axial deformation during the weld. Weld strength is predicted using the joining parameter and information on the heating time, pressure, and interface temperature. Very interesting nomographs, which may be used to calculate weld impact strength from said parameters, are presented for use by practitioners. Welds are also categorized as "acceptable," "unacceptable," and "marginal."

In both studies discussed above, it is important to note that a fundamental aspect of the weld process, polymer inter-diffusion, is excluded from the analysis. On the other end of the spectrum are studies that exclude the implications of the thermal field on the eventual weld strength and which primarily investigate the polymer Inter-diffusion processes. As an example, Wool and co-workers present various microscale scaling relationships that specify the dependence of the individual polymer chain characteristics in terms of time to heal and the molecular weight of the polymer (5, 6). These relationships do not, however, address nonisothermal effects. Our study is intended to be a bridge between the two types of studies (macroscale and microscale) that have been described. In this work, a simplified weld geometry is investigated with the aim of presenting a rational approach to incorporating the essential physical characteristics of the weld process into a model. By considering the underlying material behavior, this model can predict the weld strength as a function of easily measured process parameters such as the power input into the weld zone, the contact time, and the heating rate.


Two methods currently exist for characterizing welding processes: reptation theory and empirical thermal studies. Reptation theory, conceived by DeGennes in the 1970s, examines the thermally influenced interactions of polymer chains. The basic premise of this theory is that polymer chains are confined to an imaginary tube by the entanglements of surrounding chains. When a polymer interface becomes molten, the ends of each chain become free and begin to assume new orientations. These free ends are called "minor chains." Figure 1 (adapted from reference 5) provides a schematic of this behavior. Eventually, at a time called the reptation time, the entire chain forgets its initial conformation, and center of mass diffusion takes place. The reptation time may be expressed by a scaling relationship of the form:

[Mathematical Expression Omitted] (1)

where [R.sub.g] is the radius of gyration and [D.sub.s] is the center of mass diffusion coefficient. Equation 1 represents a characteristic diffusion time for a polymer chain to diffuse a distance equal to its radius of gyration. Various references to the above scaling exist in related literature, such as Adolf et al. (7) and Pecorini and Swo (8).

At times less than the reptation time, the diffusion of the free ends cannot be described by Fick's law. This presents a challenge when attempting to describe the average diffusion distance of the polymer chains. Once the reptation time is reached and center of mass diffusion takes place, Fick's law is then an acceptable method to describe the diffusion.

Reptation theory explains strength formation by examining the average inter-penetration distance of polymer chains across an interface. Strength develops as polymer chains diffuse across the interface, and scales as [Mathematical Expression Omitted], where [t.sub.m] is time measured after the interface has reached its melt temperature. A detailed discussion on the various mechanisms responsible for the strength scaling on melt time is given in Wool et al. (6). For times less than the reptation time, Wool and Zhang (5) define a curvilinear diffusion coefficient, which scribes polymer chain diffusion. Using a statistical analysis of the non-Fickian concentration profile, they derive the following relation for time-dependent average interpenetration depth:

X(t) = (3[[Pi].sup.1/4]/[32.sup.1/2])[a.sup.1/2][(Dt).sup.1/4] (2)

where a is the average monomer length, and D is the curvilinear diffusion coefficient. While this relation provides insight into the non-Fickian behavior of molten polymers, the ambiguous and poorly defined nature of the curvilinear diffusion coefficient makes it impractical for analysis of polymer chain diffusion.

The power law dependence of strength on melt time is evident in Equation 2 as well. At the reptation time, the average inter-penetration distance is equal to the radius of gyration, the material has re-organized to erase the interface, and the weld achieves a strength equal to that of the virgin material. By describing what happens to the actual polymer chains, reptation theory shows that strength development depends on keeping a weld interface above the melt temperature for a time equal to or greater than the reptation time. Keeping an interface molten for times longer than the reptation time is unnecessary, as it will not result in additional strength formation. These relationships are shown graphically in Figure 2. Strength development is graphed with respect to [Mathematical Expression Omitted] for several different diffusion coefficients.

As healing temperatures, and thus diffusion rates, increase, the time required to achieve a virgin strength weld decreases. If the weld strength were plotted with respect to X(t), the average diffusion distance, all the graphs would collapse to the same line, and the virgin strength would be achieved at X(t) equal to the radius of gyration. This provides much insight into the feasibility of describing weld strength formation by average polymer chain diffusion distance. The non-Fickian diffusion is difficult to characterize, and depends upon unconventional polymer properties such as curvilinear diffusivity.

Wool et al. (6) attempt to describe strength formation in a different manner. They note a molecular weight relationship that governs the time dependence of most descriptors of polymer displacement, such as number of chains crossing interface and average diffusion distance. These parameters exhibit a power law dependence on the time and molecular weight. Like Wool and Zhang, these relationships are correct, but incomplete. They fall to address the influence of temperature, and though useful for comparisons of similar polymers, such as the various grades of polyethylene, thus are impractical for describing an actual welding process.

While reptation theory can be used to describe weld strength, it is more typical in actual conditions to simply measure the bond strength and correlate it to process parameters such as input power, heating time, and interface temperature. However, such correlations do not provide a physical understanding of the polymer chain interactions, and cannot provide a physical explanation for the correlations. The situation arises, then, where the full empirical relationship must be determined for every combination of material, heat source, and system configuration.


In the model problem to be considered, two sections of polyethylene pipe are to be welded together usIng an intermediate doped polyethylene coupling section for heat generation. Welding configurations using intermediate doped sections are typical of Radio Frequency (RF) coupled weld systems, which we use as a convenient representation. A schematic of the model is shown in Fig. 3. The pipe wall thickness is 1 cm and the length of pipe simulated is 10 cm. A one-dimensional treatment of the problem is appropriate when the temperature distribution is lumped in the transverse direction. The pipe wall thickness (1 cm) is small relative to the inner radius of the pipe (13.2 cm) and curvature effects are neglected. A control volume finite difference code is then used to model a one-dimensional welding process (9), The validity of lumping the temperature distribution in the radial direction may be examined by considering the Blot number for the pipe, which compares convective and conductive resistance:

[Bi.sub.L] = hL/k (3)

The heat transfer coefficient is specified to be 5 W/[m.sup.2]K, using an empirical relation for natural convection (10), and the characteristic size (L) is half the wall thickness of the pipe section. The lowest value of the PE thermal conductivity is 0.25 W/mK (11), which gives a conservative estimate of the Biot number to be 0.1, indicating that the primary resistance is convective and a one dimensional analysis is appropriate.

The boundary conditions are defined as follows:

T(x = 0, t) = [T.sub.[infinity]]

[Mathematical Expression Omitted] (4)

In all welding processes the fundamental physical mechanism for generating a weld zone is the application of heat to an interface between two materials. Although the mechanistic description of this heating varies between weld types (e.g., ultrasonic, Induction, butt, spin, etc.) the basic effect is the addition of heat. Thus, to a in-st approximation, the most important conservation equation for describing the evolution of a weld is the energy equation.

[Mathematical Expression Omitted] (5)

The boundary conditions become very important when the type of weld process is specified. For induction/implant welding where a filler material is placed between the two surfaces to be welded together, the volumetric source, which is a general function of space, is located only within the implant. in order to produce a melt region in the adjacent materials, it is necessary to thermally communicate the effect of the heat source across any gap to the surrounding materials. As a result, it is important to describe the contact resistance between the adjacent materials in terms of the joining pressure.

Contact Resistance:

Contact resistance in solids arises from the relative roughness of real interfaces as compared with ideal surfaces. With increases in the plastic deformation of the surface the contact resistance decreases. Contact resistance expressions typically show a strong dependence on incident or contact pressure. An expression of the form:

[h.sub.c] = 956.82 + 0.25625 x P (6)

is taken from Marotta and Fletcher (12) to specify the variation of contact conductance (W/[m.sup.2]K) with respect to interface pressure (kPa) in our model. The function displays a slightly increasing linear behavior. For the model, an average contact pressure of 950 kPa is assumed, which has a corresponding contact conductance of 1200 W/[m.sup.2]K.

As heating occurs, the plastic transitions from a semicrystalline state to an amorphous state as its temperature increases. The viscosity is a strong function of temperature and with additional heating the plastic begins to flow. As the plastic flows, the contact resistance disappears, and it may no longer be appropriate to solve the energy equation without considering the effects of the convective terms. However, in the approximation used in this study we will initially assume that the bulk polymer does not flow, eliminating advective effects.

Material Thermal Properties:

For the temperature range that is considered, it is important to use a variable specific heat capacity since the heat capacity of high-density polyethylene varies between the melt and solid phases. More important, it is possible to define an effective heat capacity that accounts for the heat of fusion. An expression given by Woo et al. (11) qualitatively incorporates fusion effects into the heat capacity.

[c.sub.p] = 2250[1 + 5.5exp(-a[(T - 135).sup.2])] [J / kg [degrees] C] (7)

a = 0.005 (T [less than or equal to] 135 deg C)

a = 0.05 (T [greater than] 135 deg C)

This relationship is graphed in Fig. 4. Woo et al. also provide an expression for the thermal conductivity both in the melt and solid phases.

[[Kappa].sub.p] = 0.17 + 5([[Rho].sub.p] - 0.9) - 0.001 x T (T [less than or equal to] 135 [degrees] C) [W/mC]

[[Kappa].sub.p] = 0.25 (T [greater than] 135 [degrees] C) (8)

The thermal conductivity is also graphed with respect to temperature in Fig. 4.

A related issue deserving attention is that of thermal stability. Excessive temperatures cause thermal degradation of polymer chains which can either lower the final weld strength or entirely destroy the weld. Every polymer has a maximum permissible temperature, above which it cannot be handled without incur~ ring damage to the polymer chains. Polyethylene has a maximum temperature of approximately 260 [degrees] C (13).

In general, welding processes involving PE should not exceed this temperature at any place in the welding apparatus. Welding processes that involve rapid heating or high temperature elements, such as RF or electrofusion, run the greatest risk of violating the maximum temperature constraints and causing thermal degradation.

Weld Strength:

Wool et al. discuss two modes of failure for a polymer interface: chain pullout and chain fracture. They develop a scaling law for the pullout stress, which has a power law dependence on the heal time and the molecular weight of the polymer. This dependence is further clarified by Figure 30 in Wool et al., which shows the shear stress dependence on heal time for three different heal temperatures for polystyrene. A similar graph for polyethylene is shown in Fig. 2 for predicted values. As expected, with increasing weld temperature there is greater weld strength at the same heal time. For a fixed temperature, increasing the heal time increases the shear stress until the material reaches the strength associated with the virgin material.

The goal of the current endeavor is to combine the empirical and theoretical models and create a single, more generally useful model. A method for strength prediction is created that relates the strength to important welding parameters, from the perspective of the polymer chains. As previously noted, the main weakness of reptation theory-based studies is that they do not describe realistic welding processes, where the weld zone temperature evolves in time. Instead, analyses of this type generally assume steady, low healing temperature interfaces, which permit detailed examination of molecular weight effects. The diffusion of polymer chains across the weld interface is accelerated by higher temperatures. For isothermal healing of an interface, the diffusion rate is constant, but for a real process the weld zone sees a dynamic range of rates over the healing time. Green (14) defines the temperature dependence of the diffusion coefficient as:

log [[D.sub.rep](T)/[D.sub.rep]([T.sub.ref]) x [T.sub.ref]/T] = B/([T.sub.ref] - [T.sub.0]) - B/(T - [T.sub.0]) (9)

Equation 9 uses the Vogel-Fulcher constants and a reference center-of-mass diffusion coefficient to account for temperature effects. The Vogel-Fulcher constants describe the temperature dependent behavior of polymer melt viscosity. For the model, B = 1225 [degrees] C and [T.sub.0] = -112 [degrees] C (15). The reference serf diffusion coefficient is equal to 8 x [10.sup.-16] [m.sup.2]/s at 176 [degrees] C for polyethylene of molecular weight 250,000 (16). For a given polymer, the diffusion coefficient at a temperature different from that of the reference value can be calculated. Diffusion distances required for a complete or virgin heal to occur are of similar magnitude as the chain radius of gyration, [R.sub.g]. Typical values of [R.sub.g] for polyethylene are on the order of 100-1000 Angstroms. For a molecular weight of 250,000, [R.sub.g] for polyethylene is 1000[Angstrom] for the cases considered (17, 18). For such relatively small diffusion length scales, the temperature over which chain diffusion occurs is effectively constant and the interface temperature is used as the appropriate diffusion temperature. For a given weld interface temperature response, the corresponding diffusion coefficient is time averaged to yield an equivalent diffusion coefficient for the weld process.

[Mathematical Expression Omitted] (10)

This equivalent coefficient can then be used to calculate an equivalent reptation time using Eq 1. The percentage of maximum strength at a heal time of [t.sub.m,max] is then specified to be:

[Mathematical Expression Omitted] (11)

Finally, it is easily seen from the above relationship that if the heal time is greater than the reptation time (based on the equivalent diffusion coefficient), then the strength is equal to the virgin maximum strength. This approach to predicting weld strength accounts for the temporal changes in interface temperature in a simple manner. As the temperature changes, so does the diffusion rate of polymer chains, which necessarily influences the reptation time.


Baseline Model

For all the results that are presented, the polymer is high-density polyethylene. Five source power level conditions are considered: 5 x [10.sup.6], 7.5 x [10.sup.6], 11.25 x [10.sup.6], 20 x [10.sup.6] and 30 x [10.sup.6] W/[m.sup.3]. These power levels are similar to those examined by Nishimura, et al. (19, 20) and Nakashiba (3). For each case the heating is active until the temperature in the heat zone reaches 260 [degrees] C, at which time power is cut off and the pipe cools through convective losses. This cutoff temperature is selected based on thermal stability of the polyethylene in the heat zones. For the baseline case with a source power of 11.25 x [10.sup.6] W/[m.sup.3] the evolution of the temperature profile across the length of the model is shown in Fig. 5. The low thermal conductivity of polyethylene does not allow much energy to diffuse through the system, creating a large temperature increase near the weld interface. For short times (e.g., 20 sec), the contact resistance can be seen as a sharp temperature gradient at the interface (x = 9.8 cm). The temperature response of the weld interface is shown in Fig. 6. This plot is qualitatively similar to Fig. 4 in Nishimura et al. (19). The decrease in slope halfway through the heating process is caused by the heat of fusion at the melt temperature of 135 [degrees] C. After the interface reaches this temperature, it is assumed that a wetted interface exists and the contact resistance is terminated. The heating continues past the melt temperature until the maximum temperature constraint [TABULAR DATA FOR TABLE 1 OMITTED] is reached, at which point the heating is terminated, and the model cools until the interface falls below 135 [degrees] C.

For each power density the following properties are calculated and presented in Table 1: applied power, interface surface flux, equivalent diffusion coefficient, maximum temperature of the weld interface, healing time, and reptation time. The equivalent diffusion coefficients are calculated according to Eq 10, and maximum interface temperatures are determined by how much energy diffuses into the weld zone before the maximum temperature constraint is reached at any point in the model. This procedure involves inputting the calculated temperature profile into Eq 10, and integrating over the time frame that the interface is molten. The two lowest power densities do not violate the maximum temperature constraint. The lowest, 5 x [10.sup.6] W/[m.sup.3] does not, however, reach the melt temperature of 135 [degrees] C, and no healing takes place. The next highest power density, 7.5 x [10.sup.6] W/[m.sup.3], reaches a steady temperature above 135 [degrees] C but below the maximum temperature constraint (260 [degrees] C). This condition results in a virgin-strength weld, but does not conform with the trends found in the last three power densities, all of which reach the maximum temperature constraint, terminating the heating.

For increasing power densities, there is an accompanying decrease in equivalent diffusion coefficient and maximum interfacial temperature. While at first this may seem counter-intuitive, it becomes more reasonable when one considers reputation theory and thermal degradation effects. During any polymer welding process a maximum temperature constraint exists. Higher power densities cause this constraint to be reached in a shorter time, decreasing the total amount of energy that is added to the weld zone, and the time an interface has to heal. Diffusion of energy out of the heat zone, across the interface, and into the surrounding material is slow, owing to the low conductivity of polyethylene. Decreasing the heating time, the result of higher power densities, limits the amount of energy available, decreasing the maximum temperature the interface sees, as well as the total amount of time it is above the melt temperature.

Lower power densities take longer for the maximum temperature constraint to be reached, allowing more energy to diffuse across the weld zone. Thus, higher temperatures and longer healing times are achievable for lower power densities. The equivalent diffusion coefficient increases for smaller power densities, and is coupled by a decrease in the reputation time. At the same time the healing times are increasing. Recall that healing of polymer interfaces is dependent upon a critical healing time, the reputation time. Where these two times meet is a threshold value for the power density; below this value a virgin strength weld is achievable, and above it weaker welds are formed. The data show that a lower bound also exists for power density. The applied power must be large enough to overcome the convective losses from the heated section. If the power density is too low, the interface will never reach the melt temperature to begin polymer chain diffusion (5 x [10.sup.6] W/[m.sup.3]). Slightly higher power densities reach a steady state temperature above 135 [degrees] C, allowing a virgin bond to heal for long contact times. This is the case for a power density of 7.5 x [10.sup.6] W/[m.sup.3]. A steady state temperature of 170 [degrees] C is reached at 650 sec. The bond achieves virgin strength after healing for 61 sec. Healing times longer than this also produce virgin welds, but are unnecessary. Higher power densities, such as 11.25 x [10.sup.6] W/[m.sup.3], cause the maximum temperature constraint to be reached, and a virgin weld is formed.

For the five power densities considered, the accompanying strengths are graphed in Fig. 7. Recall that the lowest power density case never reached the melt temperature and thus formed no strength. The graph clearly demonstrates the importance of allowing a weld zone sufficient time to heal. The highest and lowest power levels did not produce virgin strength welds, while the other three densities did. The case of power density = 20 x [10.sup.6] W/[m.sup.3] came closest to an ideal heating rate with its healing time of 46 sec and reputation time of 31 sec (an ideal heating rate would be one in which the reputation lime is equal to the healing time, minimizing the required input energy). Overall, the three middle densities (7.5 x [10.sup.6], 11.25 x [10.sup.6], 20 x [10.sup.6] W/[m.sup.3]) bad proper coupling of heal time and reputation time to produce virgin strength welds.

Comparison with Experimental Results 1

The data suggest that an ideal range exists for power levels in polymer welding. Above the range thermal stability of the polymer is an issue, and heating must be terminated before enough healing bas taken place. Below the range, heating is not sufficient to overcome convective losses. A similar conclusion was reached empirically by Nishimura et al. (19), who experimentally tested welds for a range of energy levels and found an ideal range that produced virgin strength welds. One difference between the current work and Nishimura (19) is that the latter presented results in terms of the total amount of energy added to a weld zone, instead of power densities. Viewing the influence of heat transfer on weld strength formation in terms of total energy may not be a sound method, because it does not account for heating rate. Power density is important in any heat transfer problem because of characteristic diffusion times for different materials. For different power densities, a material's thermal constraint will be attained at different limes, which does not necessarily agree with a total energy approach. Reputation theory, as well as intuition, suggest that if a weld interface is heated to a temperature below 135 [degrees] C, and held there so that the total amount of energy input into the weld zone falls into Nishimura's "ideal" zone, no weld will form even though the total energy approach dictates that one should. Moreover, if a weld interface is heated to a steady temperature above 135 [degrees] C, and held for an indefinite amount of time, an amount of energy win be input into the weld interface that falls well beyond the ideal zone. Clearly, the rate of energy input is more important than a total energy perspective, and provides more insight into the problem of polymer weld healing.

In later works (3, 20), a similar range of "energy" input is observed, but this time is viewed as surface flux (W/[m.sup.2]) into the weld interface. Electrofusion weld specimens were created and creep tested to failure. The results showed that a range of energy fluxes (W/[m.sup.2]) exists that provide acceptable welds. In addition, the measured temperature profiles were in good agreement with the current model [ILLUSTRATION FOR FIGURE 6 OMITTED]. For each flux that provides adequate welds, a range of heating times exist. Below this range, a weld is produced that has less strength than the virgin material. Above this range, thermal degradation occurs in the material.

To examine agreement with Nishimura's work, our model was adapted to run for similar conditions. The maximum temperature constraint was changed to 350 [degrees] C to agree with that of Nishimura. Other assumptions of the original model were also changed. To test for similar conditions as Nishimura's work, simply attaining the maximum temperature constraint was considered to be the upper bound of acceptable welds. If the temperature of the polyethylene reaches 350 [degrees] C, the weld is assumed to be compromised. This differs from our original model, where the heating was shut off at the maximum temperature constraint and allowed to cool. For these cases healing still took place during the cooling phase as long as the temperature of the weld was above 135 [degrees] C. In the present adaptation, no cooling phase exists: the upper constraint on weld quality due to thermal degradation is established as soon as the maximum temperature of 350 [degrees] C is reached. Furthermore, the thermal constraint is examined at two points: in the weld zone and at the weld interface.

The new model was run for a range of energy fluxes similar to Nakashiba and Nishimura (3, 20). The power densities used in the original model were converted to power fluxes. This conversion provided a surface flux applied to the weld interface. The data from the adapted model are graphed on top of the Nishimura data in Fig. 8. Any heating condition that falls within the two experimental lines of Fig. 8 causes a weld to form whose strength is equal to the virgin material. The Figure covers a broad range of power densities, but does not show the absolute boundary of the highest and lowest power densities that provide viable welds. However, the trends in Nishimura's data suggest that the two curves meet at some point to establish an upper bound on power densities, and a lower bound is established by the lower constraint curve.

While no theoretical explanation was offered for the observed trends, the previously mentioned reputation theory based explanation still holds. Higher power densities reach the upper thermal constraint in a shorter amount of time, decreasing the available heal time. At the same time, the required heal time, as dictated by the reputation time, is increasing. Clearly, a power density exists where the two curves meet, establishing an absolute upper bound, above which no viable welds are possible. A similar argument describes the determination of minimum heal time for the experimental virgin heal curve. This curve is determined by the healing time equaling the reputation time, the latter being based on the temperature profile up to that particular point in time. A lower bound for power densities exists as well, below which no welds are feasible because the temperature of the interface never becomes molten.

The model's virgin heal curve is graphed showing where a virgin heal is attained, as well as the thermal degradation data from the two points of interest (Thermal deg1 = interface, Thermal deg2 = heat zone). The data generated using reputation theory agree reasonably well with that of Nishimura. The overall trends are very similar, and appear to be shifted slightly down and left of Nishimura's data. The thermal deg2 line is almost the same as the thermal degradation line of Nishimura. The new model shows a maximum permissible power flux of 8 x [10.sup.4] W/[m.sup.2] (equivalent to 40 x [10.sup.6] W/[m.sup.3] power density). For this heating rate, a virgin strength weld is formed just as the maximum temperature constraint is reached. Power levels above this value do not form virgin strength welds. Lower energy fluxes, such as 6 x [10.sub.4], 4 x [10.sup.4], and 3 x [10.sup.4] W/[m.sup.2], provide acceptable welds for certain time ranges before thermal degradation occurs. Finally, even lower power fluxes such as 2.2 x [10.sup.4] and 1.5 x [10.sup.4] W/[m.sup.2] provide virgin strength welds at relatively long heal times, but reach steady temperatures before the maximum temperature constraint is reached.

As mentioned previously, even lower power densities do not allow any healing to take place. This lower constraint, which determines where no healing occurs, and where the interface reaches a steady temperature between 135 [degrees] C and 350 [degrees] C, is dictated by the convective losses assumed in the heat diffusion equation. Environmental and geometry effects play a large role in the magnitude of the convective losses.

The data strongly suggest the existence of four distinct energy flux ranges: one that does not provide enough energy for a weld, another that provides too much, and two in which welding is feasible but exhibit different heat transfer behavior. Fig. 9 shows these ranges. The data from Fig. 8 was extended to a longer time frame. The intent of this modification is to provide better resolution of the boundaries between the four power density ranges.

Certain differences exist between the adapted model and Nishimura's work that may explain discrepancies in the two sets of data. Nishimura's work dealt with a specific kind of welding process: electrofusion. The model, even when adapted to test for similar conditions, is a generic model that simulates a global weld. In a realistic process like electrofusion, geometry and environmental effects play a more prominent role than In the model. The presence of solid objects, like the coils of the electrofusion wire, hinder polymer diffusion and the strength of the resulting weld. Effects such as these are not addressed in the model, as its intent is to show how reputation theory can be coupled with heat transfer meters to explain polymer weld formation. Furthermore, the polyethylene in Nishimura's model is never specified, and is likely to have different properties, such as self diffusion coefficient and radius of gyration, than those used In the present model.

Examination of Strength Formation

The evolution of weld strength to the virgin value was examined for the original model for a power density of 11.25 x [10.sup.6] W/[m.sup.3]. The temperature profile was "stepped" through at different Increments. For each increment over which the Interface was molten, Eq 10 was used to calculate the equivalent diffusion coefficient, reputation time, and percentage strength formation (of the virgin value) for said increment. For example, the interface becomes molten after 98 sec of heating for a power density of 11.25 x [10.sub.6] W/[m.sup.3]. Extending the heating time to 108 sec allows one to look at the time increment 98-108 sec, a healing time of 10 sec. Equation lois integrated with the temperature profile from 98-108 sec to determine the equivalent diffusion coefficient and reputation time for this particular increment. The equivalent diffusion coefficient is essentially an "instantaneous' value, which describes the diffusion of the polymer chains over the time increment in question. Dividing the heal time (10 sec) by the reputation time provides the percent attainment of the virgin weld strength value. Next, the time increment of 98-118 sec is examined, and the same procedure is used to provide the same parameters. This procedure is performed for increasing increments for which the interface is molten. Stepping through the temperature profile in this manner provides much insight into the evolution of weld parameters.

Figure 10 shows the "instantaneous" equivalent diffusion coefficient graphed for different time increments. The graph shows a steadily increasing coefficient until the heating is terminated, at which point the diffusion coefficient decreases. Initial examination of this graph suggests that a lower equivalent coefficient indicates a lower strength. However, this is not the case, as the lower equivalent coefficients at long times have greater time frames over which healing takes place. Multiplying the equivalent coefficients by their respective time increments provides the area underneath the curve (Eq 10). Even though the equivalent coefficients decrease, the area always increases, showing that strength is a function of the time-integrated equivalent diffusion coefficient.

To test this observation, data for a power density of 20 x [10.sup.6] W/[m.sub.3] were examined in a similar manner to determine instantaneous diffusion coefficient, reputation time, and percent heal. The data are shown in Table 2. A virgin strength heal is reached just after the maximum temperature constraint is achieved. The equivalent diffusion coefficient decreases while the percent heal increases to 100%, supporting the notion that weld strength depends on the area under the curve of the diffusion coefficient time history.
Table 2. Data for Original Model, Power Density = 20 x [10.sup.6]
W/[m.sup.3]. The Table Shows How the Percent Heel of the Weld
Increases Even as the Equivalent Diffusion Coefficient Decreases.

Heating Healing
Time Time Ds, eq [t.sub.rep] Percent
sec sec [m.sup.2]/s sec Heal

51 10 2.41E - 16 41 24
56 15 3.46E - 16 28 53
61 20 4.11E - 16 24 82
63 22 4.16E - 16 23.98 91
63.5 22.5 4.18E - 16 23.97 93.8
64 23 4.17E - 16 23.96 96
65 24 4.16E - 16 24 100

Comparison With Experimental Results 2

In addition to Nishimura and Nakashiba, the current reputation model is also compared to the work of Pimputkar (4). That study analyzes butt fusion in polyethylene pipe, using a quasi-analytical method to model the heat transfer, and performing tensile tests to determine weld strength. The results of the strength tests are related to important heat transfer parameters through a joining parameter. As with the comparison with Nishimura et al., our model is adapted to run for similar conditions as Pimputkar. The heat transfer is coupled to the reputation model to determine strength formation, and the results are compared.

From previous empirical work, Pimputkar notes that certain heat transfer and processing parameters influence weld quality: heater temperature, heating time, and joining pressure, First a mathematical model is developed to calculate the temperature response of different locations in the pipe wall near the fusion zone. In the mathematical model, the uncoupled integral solutions for the energy and momentum equations are solved. These temperature profiles are then compared with actual measured profiles. As with Nishimura's work, the profiles exhibit a strong similarity to those calculated by our model [ILLUSTRATION FOR FIGURE 6 OMITTED].

The next phase of Pimputkar's work was to test the strength of welds created under different combinations of the aforementioned heat transfer parameters. Test welds were created for different values of heating time, heater temperature, and joining pressure. These specimens were both tensile (yield and elongation) and impact tested to assess strength. In analyzing trends in the strength test results, the impact data showed a stronger correlation with the joining conditions than the yield stress or elongation at break. This is not too surprising, since crack propagation will occur at the weakest location, while tensile elongation and yield will occur over a larger volume of the material than just the weld zone.

The impact data are organized into a correlation called the "joining parameter," a function of the axial deformation, [h.sub.[infinity]], the calculated location of the melt front after the heating phase, [h.sub.o], and the wall thickness, W.

Joining Parameter = 2[h.sub.[infinity]](8[h.sub.o] - 2[h.sub.[infinity]])/1000 [W.sup.2] (12)

It is important to note that the impact strength is in terms of total energy, joules, and not MPa or psi. Pimputkar noticed that tensile impact strength varies linearly with the Joining parameter up to a certain value, above which impact strength remains constant for increasing values of the joining parameter. The constant of proportionality in the linear range is noted to be a function of the type of polyethylene. The correlation between the joining parameter, weld strength, and weld conditions is summarized in a set of nomographs: a unique nomograph exists for each type of polyethylene tested.

The present model is used to simulate a butt welding apparatus. The intent is to calculate the temperature profile for similar conditions as Pimputkar's work. The area of focus changes to examine heating time and heater temperature, the same heat transfer parameters used in the nomographs, as compared with heat flux. The boundary condition on the right side of the model changes to a constant temperature condition, and the interface moves to this boundary. The boundary condition changes with respect to time, as described by the following relations.

[Mathematical Expression Omitted] (13)

The temperature profile of the interface is calculated for different combinations of heating time and heater temperature. At the end of the prescribed heating time, the constant temperature interface changes to an insulated boundary condition, and as the interface cools, the code calculates the temperature profile. This stage simulates the actual healing of a butt welding process, when the heating element is removed and the ends of the two pipes are brought into contact.

Inherent differences exist between the original model and the adapted butt welding model. The original model uses a volumetric source to simulate heating. A maximum temperature constraint is designated, and the heating turns off once this temperature is attained at any point in the model. Healing takes place once the interface temperature exceeds 135 [degrees] C during the heat-up phase, and continues until the interface cools below 135 [degrees] C. The interface is at the boundary between the heat zone and the surrounding pipe. The butt welding model uses a constant temperature boundary condition instead of a volumetric source to prescribe the heat addition process. The interface is located at this boundary, and healing takes place only during the cooling phase. During the heating phase, the heating element is still in contact with the ends of each pipe, and no polymer chain diffusion takes place. Healing may only occur once the heating element is removed and the ends of the pipes are pushed together. Thus, the equivalent diffusion coefficient is calculated for a cooling temperature profile. This is different from the original model, where the temperature profile had a heating and cooling section, and the adapted model for Nishimura's work, where only a heating temperature profile was used.

Thermal degradation is not an issue in the adapted butt welding model. The maximum possible temperature in the model occurs where the polymer contacts the heater plate. No thermal degradation takes place as long as the heating element does not exceed the maximum temperature. in the nomograph, the largest heater temperature is 275 [degrees] C, so it is assumed that the maximum temperature of the polyethylene in the adapted butt fusion model is greater than this temperature.

The model is run for heating times ranging from 30 to 120 sec and heater temperatures of 190-260 [degrees] C. For each combination, the temperature profile for the cooling phase is input into Eq 10 and integrated over the time range specified by the end of the heating time to when the interface drops below 135 [degrees] C. The main issue in devising an effective method for butt welding polymer pipes is to choose a good combination of heating time and heater temperature. If enough energy diffuses into the pipe, the interface will stay molten long enough once the two ends of pipe are mated to form a strong weld. The results are presented in Table 3 and Fig. 11. Recall that the no time is the time over which heating, by volumetric source or constant temperature boundary condition, takes place. In contrast is healing time, the lame for which the interface is molten. Heating is a factor, and healing is a response.

The data show that a heater temperature of 190 [degrees] C provides fully healed welds only for heating times greater than 120 sec. A heater temperature of 220 [degrees] C provides virgin strength welds for heating times greater than 60 sec. A heater temperature of 260 [degrees] C provides fully healed welds for virtually all heating times.
Table 3. Data for Second Modified Model. This Table Shows That the
Heater Temperature Has Larger Influence on the Quality of the
Resulting Weld Then the Heating Time.

 Heating Time (sec)
 Temp., [degrees] C 30 60 90 120


 Virgin heal? No No No Yes
 Healing time 15 20 24 26
 Rep. time N/A N/A N/A 24


 Virgin heal? No Yes Yes Yes
 Healing time 23 31 37 40
 Rep. time N/A 29 30 32


 Virgin heal? Yes Yes N/A N/A
 Healing time 32 45 N/A N/A
 Rep. time 2 1 N/A N/A

Inspection of the adapted model data gives rise to several important observations. The time it takes the interface to cool below 135 [degrees] C, the healing time, and thus time available for polymer chain diffusion, is a stronger function of the heater temperature than the heating time. Notice that the healing time for a heater temperature of 190 [degrees] C and 60 sec of heating is 20 sec. Doubling the heating time to 120 sec increases the healing time only by 6 sec to 26. It is, however, enough to produce a virgin strength weld. In contrast, increasing the heater temperature from 190 [degrees] C to 220 [degrees] C for a 90-sec heating time increases the healing time from 20 sec to 31 sec, producing a virgin strength weld. For a heater temperature of 220 [degrees] C and a heating time of 60 sec, a virgin strength weld is produced after healing for 29 sec. The differences between these cases are shown in Fig. 11, where they lie on different sides of the boundary between marginal and acceptable welds. Employing a heater temperature of 260 [degrees] C produces excellent healing for almost all heating times. With this heater temperature, a virgin strength weld is achieved in seconds, and available healing times are on the order of 45 to 60 sec.

The observation of almost "instantaneous" healing for a heater temperature of 260 [degrees] C deserves more attention. At high temperatures such as this, the diffusion rate is extremely fast. For a temperature of 260 [degrees] C, the rate of diffusion is so fast that the polymer chains diffuse the radius of gyration in seconds, or even fractions of seconds. One might then ask why there exist cases in the original model when the maximum temperature constraint of 260 [degrees] C was reached for certain power densities but virgin strength welds were not always produced. While it is true that certain spatial points in the original model reached a temperature of 260 [degrees] C, "instantaneous healing" did not occur. This is because the interface temperature did not necessarily reach this high temperature. The higher power densities, which violate the maximum temperature constraint very quickly, allow little time for enough energy to diffuse to the interface for it to share the same high temperature. A virgin strength weld does not always form. In other cases, the interface does in fact reach 260 [degrees] C. A virgin strength weld is always produced for this case. If one examines the strength formation, the weld becomes completely healed before the interface reaches the high temperatures. The extremely high diffusion rates caused by the high temperatures are not needed except for power densities on the high end of the absolute range.

Another point to consider with the high heater temperatures is that the adapted model for Pimputkar did not account for any cooling that takes place between the time the heating element is removed and the time the ends of the pipes come into contact. In a realistic butt welding process, this time would be at least 2 sec. Examination of the cooling temperature profile shows that the interface cools very quickly after the heating is terminated. Thus, the very high temperatures that cause "instantaneous" healing do not participate in healing in a realistic butt welding process.

These observations suggest that heater temperature is much more important than heating time in butt welding. The heater temperature has a strong influence over the available healing times, as more energy is stored at higher temperatures, and the interface remains molten for a longer period of time. Increasing heating times does slightly increase the healing times, but not to a significant degree. Recall that previous sections suggested that strength is a function of the area under the diffusion curve. Since the path of the diffusion curve follows the temperature profile, a logical conclusion is that strength is related to the area under the temperature profile as well, i.e., the average temperature.

[Mathematical Expression Omitted] (14)

Figure 12 shows the temperature profile for different heating times and a heater temperature of 190 [degrees] C. The Figure also shows the temperature profile for a heater temperature of 220 [degrees] C and a heating time of 90 sec. The area under the latter curve is larger than the three curves for a heater temperature of 190 [degrees] C, and thus has a higher average temperature. Reputation theory supports these conclusions. Higher temperatures greatly increase the rate of diffusion of polymer chains. Employing larger heater temperatures enhances the rate of polymer chain diffusion, as well as increases the available healing time. Thus, the required reputation time decreases while the healing time increases, allowing virgin strength welds to form.


Two methods exist for describing polymer fusion processes: reputation theory and empirical studies of weld strength. Both offer an incomplete picture of any realistic welding processes. Reputation theory has not taken into account the influence of thermal variation in the welding process, and empirical studies offer no physical explanation for their correlations. Knowledge of one welding system does not easily translate to knowledge of another system. This study blends the desirable aspects of reputation theory and empirical studies to create a model that relates material properties and engineering parameters. By taking into account the temperature dependence of polymer chain diffusion, an equivalent diffusion coefficient is derived, which shows that an ideal range of power densities exist that produce virgin strength welds. Above this range the polymer does not have enough time to sufficiently heal before degrading, and below the range not enough energy is input into the weld zone to produce a quality weld.

The model was adapted to run for similar conditions as two separate empirical studies. The adapted model showed good agreement with the empirical studies, displaying the same overall trends. A further analysis of the development of polymer weld strength showed that strength is related to the area under the diffusion-time curve. Coupling the temperature profile of a weld interface with the temperature dependent diffusion equation is a viable method for predicting polymer weld strength, and has a strong theoretical basis.


a - Monomer length - [m]

[A.sub.c] - Cross sectional area - [[m.sup.2]]

B - Vogel-Fulcher constant - [[degrees] C]

[B.sub.i] - Blot number - [ - ]

[c.sub.p] - Specific heat - [J/(kg [degrees] C)]

Dc - Curvilinear diffusion coefficient - [[m.sup.2]/s]

[D.sub.rep](T) - Temperature-dependent self diffusion coefficient (Eq 7) - [[m.sup.2]/s]

[D.sub.rep]([T.sub.ref]) - Reference self diffusion coefficient for Eq 7 - [[m.sup.2]/s]

[D.sub.s] - Self diffusion coefficient - [[m.sup.2]/s]

[H.sub.s] - Heaviside step function

h - Heat transfer coefficient - [W/([m.sup.2] [degrees] C)]

[h.sub.c] - Contact conductance - [W/[m.sup.2]K]

k - Thermal conductivity - [W/(m [degrees] C)]

[k.sub.p] - Thermal conductivity - [W/(m [degrees] C)]

L - Length - [m]

P - Pressure - [Pa]

[Q[triple prime] - Volumetric heat generation - [W/[m.sup.3])

[R.sub.g] - Radius of gyration - [ml

[[R.sub.g].sup.2] - Mean square radius of gyration - [[m.sup.2]]

t - Time - [sec]

[t.sub.m] - Time for which interface is molten - [sec]

[T[infinity] - Ambient temperature - [[degrees] C]

[T.sub.o] - Vogel-Fulcher constant - [[degrees] C]

[T.sub.ref] - Reference temperature for [D.sub.s] - [[degrees] C]

[U.sub.i] - Velocity - [m/s]

X - Displacement - [m]

X(t) - Time-dependent displacement - [m]

[alpha] - Thermal diffusivity - [[m.sup.2]/s]

[rho] - Density - [kg/[m.sup.3]]

[sigma] - Critical fracture stress - [MPa]

[[Tau].sub.char] - Characteristic diffusion time - [s]

[[Tau].sub.rep] - Reptation time - [s]


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Article Details
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Author:Ezekoye, O.A.; Lowman, C.D.; Fahey, M.T.; Hulme-Lowe, A.G.
Publication:Polymer Engineering and Science
Date:Jun 1, 1998
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