Activation energy for diffusion and welding of PLA films.
Molecular diffusion (interfacial healing) occurs during the welding of plastics when two polymer surfaces are brought into contact above their glass transition temperature (T g) . At this temperature, the polymer chains diffuse across the contact area (faying surface) until the interface is indistinguishable from the bulk material and the polymer-polymer interface disappears, while the mechanical strength of the bond increases until it reaches the bulk material strength. Such healing is important not only in polymer welding, but also in many other polymer processing areas such as extrusion, casting, or molding, where individual polymer pellets are formed into larger shapes. It is also important during the impregnation of polymer-coated fibers with a thermoplastic resin as well as the consolidation of individual plies of thermoplastic resin composites and during the crack healing in polymeric materials (1-3). Interfacial healing was also observed below [T.sub.g] for material combinations such as polystyrene (PS)/poly(phenylene oxide) and PS/PS (4).
This healing process can be subdivided into several stages. These are surface rearrangement, surface approach, wetting, diffusion, and randomization (mixing). The driving force for such a mixing of polymer chain segments is the increase in entropy of the polymer chains at the interface. The parameters of such autohesion processes are contact time, temperature, and pressure at the interfaces (5).
In this study, the diffusion behavior during welding of polylactic acid (PLA) films was studied. PLA is a biodegradable polymer derived from lactic acid and made from 100% renewable resources such as sugar beets, wheat, corn, or other starch-rich crops. The starch from the feed-stock is extracted by a range of mechanisms and hydrolyzed into fermentable sugars to produce lactic acid. The lactic acid is polymerized either by a condensation reaction or through a ring-opening process (6). PLA is typically transparent, rigid, and exhibits mechanical properties similar to some petroleum-based polymers. For example, PLA has a tensile strength of ~70 MPa and a modulus of elasticity of 3.6 GPa; similar to polyethylene terephthalate (PET) that has a tensile strength of 60-80 MPa (7) and a modulus of elasticity of 2.1-3.1 GPa. In addition, PLA has a density of 1.25 g/[cm.sup.3], which is ~8% lower than PET. PLA is relatively resistant to moisture and solvents such as oil and grease and has high-vapor barrier properties. Depending on its formulation, it can be made rigid or flexible and can have a melting point between 130 and 220[degrees]C. The molecular weight of PLA typically varies between 100,000 and 300,000 Da. As with other polymers, the strength, viscosity, melt temperature, and glass transition temperature of PLA all increase with molecular weight due to increased molecular entanglement and decreased molecular movement. However, with increasing molecular weight, the ease of processing is expected to decrease (8).
Effect of Contact Time on Autohesion
To re-establish the bulk material strength at polymer interfaces or the virgin material strength during crack healing or welding, the polymer chains need to diffuse a distance to cross the physical separation. The time to achieve a significant level of interpenetration of the polymer chains during a welding process with a sufficiently high strength can be called the welding time [t.sub.w] (9).
The mechanism by which autohesion occurs can be attributed to the two characteristics of high-molecular weight polymers, which are a random chain network and flexible macromolecules that are able to move within the bulk material. Figure 1 shows a schematic of the autohesion process for an amorphous polymer, assuming that no flow is needed to match the surfaces to each other. At the initial contact in Fig. la, the surfaces are brought in intimate contact but no adhesion is seen. Figure lb shows the autohesion represented by DAt, (degree of autohesion), which ranges between 0 and 1, after some contact time above [T.sub.g]. Some of the polymer chains have penetrated the surfaces of the touching parts, which is due to free chain movement that results from increased molecular free volume at the elevated temperatures. With increasing contact time, the penetration depth of the diffusing polymer chains, as well as the number of chains crossing the interface increases, resulting in more entanglements and higher interfacial strength. In Fig. lc, at a contact time of t = [t.sub.8], the healing process can be considered complete as the interface disappears, and strength as well as morphology is not distinguishable from the bulk material (9). Some argue that it is sufficient when the center of mass of a molecule has diffused a distance of approximately equal to its radius of gyration to achieve full adhesion or complete healing (10).
Two methods are reported to test the autohesion phenomenon (9), which are (1) radioactive doping of the polymer chains and measuring the rate of diffusion of the tracers and (2) mechanical testing of the strength of the interface. The later will be the method used in this work.
Using the mechanical strength-testing method, it is assumed that the interface structure at any time during the welding is related to the bulk material structure by the fourth root of the weld time [t.sub.w]. (5). According to de Gennes (4), the average interpenetration depth (X) of a polymer at a symmetric polymer interface is given by
(X([t.sub.w])) n [X.sub.0](t/[t.sub.[infinity]])1/4 (1)
where [X.sub.0] is the distance after a reptation time [t.sub.8]. In the reptation model, the polymer motion is described as a serpentine movement of the chains within a tangle of the surrounding chains. Each chain is considered to he embodied in a tubelike surrounding and executes movements along the tube axis by thermal energy (11).
This model of polymer chain diffusion across interfaces predicts the cohesive strength of the interface and is proportional to the fourth root of the weld time ([t.sub.w]) 1/4. assuming that no flow and the interfaces are in intimate contact. The time dependency of the interface strength during welding can be expressed as the ratio of a cohesive strength at a weld time t, and the cohesive strength when the strength becomes independent of the processing time at [t.sub.8] (9) as it is defined in Eq. 2, where DAB, is the degree of autohesion.
[D.sub.Au] = [A.sub.u](t, T)/[A.sub.u], [t.sub.[infinity], T) (2)
The strength of autohesion [A.sub.u] depends on the contact time of the interfaces [t.sub.w] and the temperature of the material. Jud (12) reported that the diffusion constant can be represented by an Arrhenius law of diffusion as defined in Eq. 3.
D (T) = [D.sub.0]exp(- [E.sub.a]/RT) (3)
where R is the universal gas constant (8.3144 J [mol.sup.1] [K.sup.1]), T is the absolute temperature, [E.sub.a] is the activation energy of diffusion, and [D.sub.0] is a diffusion constant. The degree of autohesion in Eq. 4 can be written as
[D.sub.Au] (T, t) = [D.sub.Au0] + K (T)[t.sup.1/4] (4)
where [D.sub.Au0] is the initial degree of autohesion due to surface attraction of the interfaces. In, Eq. 4, it is important to note that variable K(T) is a temperature-dependent parameter that represents the temperature-dependent slope of [D.sub.Au] when it is plotted as a function of the fourth root of the weld time and No is the intersection with the ordinate. In more detail, K(T) is the "product of a proportionality constant Ko times the self-diffusion coefficient" (9) of the Arrhenius as seen in Eq. 5:
K (T) = [K.sup.*.sub.0]exp (- [E.sub.a]/RT]) (5)
With Eqs. 4 and 5, the degree of autohesion [D.sub.Au], can be written as
[D.sub.Au] = [D.sub.Au0] + [K.sup.*.sub.0]exp (- [E.sub.a]/RT * [t.sup.1/4] (6)
By plotting the natural logarithm of K(T) as a function of the inverse temperature, the slope of the graph is proportional to the activation energy of diffusion, and the intersection of the graph with the ordinate is the proportionality factor [K.sub.0] (Eq. 7)
ln (K(T)) = ln ([K.sub.0] - [E.sub.a]/RT (7)
Because the degree of autohesion [D.sub.Au] is temperature-dependant and assuming that there is no autohesion before the weld process starts ([D.sub.Au0] = 0), the degree of autohesion can be calculated as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
where [E.sub.a] and [K.sub.0] can be determined experimentally (9), (13) and At is discretized time interval. For example, it has been previously shown that it is possible to discretize a varying temperature profile (history) into small time intervals (13), such that the temperature during each time interval is relatively constant, so that the final degree of autohesion is the summation of all the autohesions of each time internal.
Effect of Contact Pressure on Autohesion
The degree of autohesion increases asymptotic with pressure until it reaches a maximum at the so-called saturation pressure for a given temperature and time. Further increase in pressure does not result in any increase of autohesive strength. This behavior can be explained with an increasing contact area for increasing pressure. Once the interfaces are in full intimate contact after the surfaces have rearranged and deformed, a higher pressure will have less of an effect on the autohesive strength (9). In addition, excessive can reduce molecular mobility because of the loss of free volume. Thus, excessive pressure can reduce molecular diffusion.
Time Temperature Superposition
The time temperature superposition method can be used to characterize a material's behavior under certain conditions at fixed temperatures and different times. This information Can be gained from the so-called shift factor [a.sub.T]. In case of a linear time temperature relationship, the shift factor can be considered as the ratio of a time [t.sub.r] for the material to reach a certain state at a reference temperature [T.sub.r] to the time t to reach the same state at a different temperature T (14), as seen in Eq. 9.
log [a.sub.T] = log(t/[t.sub.r] (9)
In more detail, the time temperature superposition theory defines that a material property E at any time and temperature is the same as it is at a reference time and temperature as seen in Eq. 10.
E(t, T) = E ([t.sub.r], [T.sub.r]) (10)
The shift factor depends on the temperature and can be estimated with an Arrhenius-type equation assuming a constant activation energy (15), (16):
log [a.sub.T] = [E.sub.a] (1/T - 1/[T.sub.ref]) (11)
where [E.sub.a] is the activation energy of diffusion, R is the ideal gas constant, T is the given temperature, and [T.sub.ref] is is a reference temperature. Although we consider the Arrhenius form in detail, which generally applied to conditions where the temperature is above [T.sub.g] (glass transition temperature), we will also consider exponential form as defined in Eq. 9.
PLA films with a thickness of 100 [micro]m from Evlon[R] produced by Nature Works (Minnetonka, MN), were used in this study. The glass transition temperature and melting temperature ([T.sub.m) were determined with a TA Instruments Q20 DSC using a heating rate of 5[degrees]C/min. The results are shown in Fig. 2 where it is seen that the [T.sub.g], and [T.sub.m] are ~62 and 150[degrees]C, respectively.
The impulse welding system and the dimensions of the samples used in this study for welding the PLA films are detailed in Fig. 3. When the welding head is, which is shown in Fig. 3a, closed, a DC constant current is applied to the heating element, which are held at a relatively constant temperature during the welding cycle. A thermocouple (TC-type J) connected to a digital data acquisition system (sample rate of 1000 Hz) was placed in an auxiliary sample between the heating elements close to the overlap area of the tested samples to measure the temperature during the welding cycles. At a time of [t.sub.w], - 1 s, 1 s before the heating cycle (c) was completed, and a heat exchanger circulated chilled water (~20[degrees]C) through water channels in the welding head to reduce the temperature back to ambient temperature as quickly as possible. The water channels were placed closely to the heating element to assure rapid cooling of the welded samples. Films with a lap shear configuration with an overlap of 4.8 mm were welded and tested using an Instron (Norwood, MA) model 4500 load frame at a crosshead rate of 10 mm/min. Six duplicate samples were produced and tested for each set of experimental parameters. The failure load was used to estimate the average shear stress in the weld, which was calculated as the applied force at failure divided by the overlap area. The welded samples were cut to a modified standard tensile test bar according to ASTM D 638.
The effect of pressure was studied (17) as it was assumed that it would have a significant impact on the diffusion process. The initial pressure chosen to be studied was the lowest possible by the equipment, which was ~0.33 MPa and was increased to 0.66 and 0.99 MPa, respectively.
RESULTS AND DISCUSSION
Effect of Contact Time and Temperature
Figure 4 shows the temperature profiles during welding for the data used for the calculation of the activation energy (relatively constant temperatures). Only those temperature profiles that reached a relatively steady-state condition (similar to a top hat profile) were used to estimate the coefficients in Eq. 8. It is seen that a relatively steady-state temperatures were reached at welding times greater than 10 s at 135[degrees]C and for welding times greater than 15-20 s at 120 and 130[degrees]C, respectively. In more detail, referring to the temperature profiles for 135[degrees]C, the thermal cycle for the 5-8 weld varied too much to be considered a steady temperature. It is important to note that at lower temperatures, the measured degree of autohesion was not linearly increasing with the fourth root of the weld time, while at higher temperature, the degree of autohesion was [D.sub.Au] = 1 before a steady state in the temperature could be achieved. In these experiments, the pressure was held constant at 0.33 MPa.
Because the temperature history does not show a perfect "top hat" profile, the weld time was calculated as schematically shown in Fig. 5 (no units are detailed). The beginning of the weld time is the intersection of the linearly increasing temperature profile with the extension of the constant weld temperature, and the end of the weld time is the point when the weld temperature starts decreasing.
With the maximum shear strength of 1.83 MPa, the degree of autohesion ([D.sub.Au]) as a function of the fourth root of the weld time was calculated as detailed in Eq. 2 and from the slope shown in Fig. 6. In more detail, slopes of the trend lines can be used to determine the temperature-dependent K(T) of Eq. 5. Because the shear strength develops linearly with the fourth root of the weld time, it can be considered as diffusion limited (4), (9), (10).
By plotting the natural logarithms of the slopes of the trend lines from Fig. 6 as a function of the inverse of the welding temperature at steady state, it is possible to calculate the activation energy [E.sub.a] of Eq. 8, and the extension of this plot to the intersection with the ordinate corresponds to the natural logarithm of the material parameter [K.sub.0].
With the slope and intersection shown in Fig. 7, the activation energy Ea is calculated as 74.8 kJ [mol.sup.-1] and [K.sub.0] = 3.37 x [10.sup.9].
Using these experimental values of [E.sub.a] and [K.sub.0], the degree of autohesion was calculated from [E.sub.q]. 9 for welds with varying temperatures (nontop hat) for increasing weld times of 1.25, 1.875, and 2.5 s. Figure 8 compares the degree of autohesion based on theoretical calculations using the previously mentioned approach and experimental values. The experimental and calculated data ([DELTA]t = 0.105 s) are in good agreement when the degree of autohesion was calculated with an estimated ln([K.sub.0]) = 20.35 ([K.sub.0] = 6.88 x [10.sup.8] [s.sup.-1/4]) instead of the value of ln([K.sub.0]) =-21.94 ([K.sub.0] = 3.37 X [10.sup.9] [s.sup.-1/4]). When the original material parameter ([K.sub.0] = 3.37 X [10.sup.9] [s.sup.-1/4]) was used to estimate the degree of autohesion, the calculated and experimental values were not in good agreement, and only through experimental varying this value was it possible to generate good agreement. There is no decisive understanding for this apparent shift, but it may be related to a temperature-dependent activation energy as reported in previous work (13). It is important to note that through statistical analysis of the data, it was found that the adjusted value of the [K.sub.0] was within one standard deviation of the predicted value, and so it is believed that the experimental error may have been a significant factor in the need to adjust the final value.
Effect of Contact Pressure
The effect of pressure on the degree of autohesion was examined by doubling (0.66 MPa) and tripling (0.99 MPa) the initial pressure value that was used in the previous experiments. The slopes of the degree of autohesion measurements were taken to plot the ln(K(T)) as a function of [T.sup.-1] to calculate the activation energy at the various pressures.
Figure 9 shows the degree of autohesion that was measured for a pressure of 0.66 MPa. As expected with the higher pressure, the weld time to achieve a measurable [D.sub.Au] is relatively short compared to the initial lower pressure. In addition, the lowest possible temperature at which welding can be achieved is lower compared to the lower pressure (0.33 MPa). It is seen from Figs. 6 and 9 that the rate of degree of autohesion development (rate of welding) is proportional to weld pressure and time, as expected. For example, at 120[degrees]C and a pressure of 0.33 MPa, a degree of autohesion of 0.75 can be achieved in ~ 60 s, while the same degree of autohesion at 0.66 MPa only requires a weld time of ~10 s. This suggests that the weld pressure is below the value where the effects of pressure on free volume reduce molecular movement and diffusion.
Figure 10 shows the degree of autohesion at a pressure of 0.99 MPa. It can be seen that at lower temperatures (90[degrees]C), the weld time is excessively long (~1000 s) to achieve a measurable degree of autohesion, while it is relatively short (13 s) at elevated temperatures. In summary, the degree of autohesion at the lower temperature (90[degrees]C) appears to be disproportionally low compared to the other temperatures. This large dependence on temperature suggests that there is a critical temperature value at which the diffusion rate increases suddenly and that this critical temperature is approximately slightly above 90[degrees]C, because below this temperature, welding was not possible. In addition, this may be related to the effect of pressure reducing free volume and limiting diffusion, so that until enough energy (temperature) is applied, the healing of the interface is limited.
Figure 11 shows the In(K(T)) as a function of [T.sup.-1] plots for all three pressures. With increasing pressure, the slope of the graphs decreases, which suggests an increase in the activation energy for diffusion. This is most likely the result of reduced free volume and reduced molecular freedom of movement after surface matching.
Table 1 details the activation energy from the slopes and [K.sub.0] values and intersections with the ordinate of the graphs in Fig. 11. The calculations show that for increasing pressure, the activation energy increases too.
TABLE 1. Activation energy and [K.sub.0] values for increasing contact pressure. Pressure Slope Conf. [R.sup.2] ln([K.sub.0]) Conf. (MPa) interval slope interval 0.33 -8,986 -17,973 - 0.91 21.94 0.5 - (-302) 43.3 0.66 -10,484 -21,592 - 0.92 25.95 -1.57 - 624 53.47 0.99 -14,362 -28,733 - 0.89 36.75 0.06 - 9 73.44 Pressure [E.sub.a] [K.sub.0] (MPa) (kJ/mol) (s.sup.l/4) 0.33 74.71 3.37E + 9 0.66 87.16 1.86E+ 11 0.99 145.69 4.64E + 14
A statistical analysis of the measured values for the slope and ln([K.sub.0]) was done to show the confidence interval for these two numbers. The confidence interval shows a large magnitude, which is almost twice the measured value. This is probably due to the low number of data points. However, the true value of the slope and in([K.sub.0]) lies within the low and high.
Figure 12 shows an asymptotic increase in degree of autohesion ([D.sub.Au]) with increasing contact pressure measured after a contact time of 25 s at a contact temperature of 115[degrees]C. The asymptotic characteristic indicates that the [D.sub.Au] will reach a saturation pressure above which a further increase in pressure has little effect on the diffusion rate.
Time Teniperature Superposition
Figure 13 shows the temperature shift factor [a.sub.T] for all three pressures calculated according to Eqs 9 and 11. The reference temperature for each pressure was assumed to be lowest temperature at which welding could be achieved and was 108, 105, and 90T for weld pressures of 0.33, 0.66, and 0.99 MPa, respectively. Although both shift factors are in relatively good agreement for a pressure of 0.33 MPa in a temperature range between 108 and 135[degrees]C, it can be seen that the two shift factors diverge slightly at the 0.66 and 0.99 MPa pressures. This suggests that the activation energy for diffusion is constant at the lower pressure and given temperature range, but at higher pressure becomes temperature dependent as previously mentioned.
The activation energy for diffusion was calculated based on constant temperature profiles and used to accurately predict weld strength for welds made with varying temperature profiles. In more detail, it was possible to divide a varying temperature profile into discreet time intervals that had a relatively constant temperature and summing the various degrees of adhesion to predict the final total level of adhesion (degree of welding).
In addition, the time temperature superposition reasonably predicted the weld/diffusion behavior for temperature a range between 108 and 135[degrees]C at a contact pressure of 0.33 MPa. However, higher temperatures or pressures the time temperature superposition did not allow For the development of accurate shift values.
Future work should focus on developing models for temperature-dependent activation energy.
The authors thank "Branson Ultrasonics Corporation" for the donation of equipments, "Bi-Ax International Inc." for the donation of PLA samples, the Iowa State University Center for Crops Utilization Research (C.C.U.R.), and the USDA Bio-Preferred Program.
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Correspondence to: David Grewell; e-mail: firstname.lastname@example.org
Published online in Wiley Online Library (wileyonlinelibrary.com).
[c] 2012 Society of Plastics. Engineers
Julius Vogel, (1) Michael R. Kessler, (2) Sriram Sundararajarill David Grewell (3)
(1.) Department of Mechanical Engineering, Ames 50011, Iowa State University, Iowa
(2.) Department of Materials Science and Engineering, Ames 50011, Iowa State University, Iowa
(3.) Department of Agricultural and Biosystems Engineering, Ames 50011, Iowa State University, Iowa
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|Author:||Vogel, Julius; Kessler, Michael R.; Sundararajan, Sriram; Grewell, David|
|Publication:||Polymer Engineering and Science|
|Date:||Aug 1, 2012|
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