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Appearance of Perfect Amorphous Linear Bulk Polyethylene under Applied Electric Field and the Analysis by Radial Distribution Function and Direct Tunneling Effect Rong Zhang, Yuezhen Bin, Wenxiao Yang, Shaoyan Fan, and Masaru Matsuo* Department of Polymer Material Science, Faculty of Chemical, Environment and Biological Science, Dalian University of Technology, Dalian 116024, China S Supporting Information *

ABSTRACT: Without melting flow, linear ultrahigh molecular weight polyethylene (UHMWPE) provided X-ray intensity curve from only amorphous halo at 129.0 °C (surface temperature, Ts arisen by Joule heat) lower than the conventionally known melting point 145.5 °C on applying electric field to UHMWPE-nickel-coated carbon fiber (NiCF) composite. Such surprising phenomenon was analyzed by simultaneous measurements of X-ray intensity, electric current, and Ts as a function of time. The calculated radial distribution function revealed the amorphous structure with disordered chain arrangement. The appearance of such amorphous phase was arisen by the phenomenon that the transferring electrons between overlapped adjacent NiCFs by tunneling effect struck together with X-ray photons and some of the transferring electron flown out from the gap to UHMWPE matrix collided against carbon atoms of UHMWPE. The impact by the collision caused disordering chain arrangement in crystal grains.

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electric field. The same phenomenon was also confirmed by the repeated self-heating (by Joule heat) and cooling processes. In comparison with the UHMWPE-NiCF composite with a 10 vol % content, the composite with a 4 vol % content close to critical volume content was adopted as the test specimen. For the composite with a 4 vol % NiCF content close to the percolation threshold, the Ts was not beyond 92.0 °C at 5.5 V/ cm and further increase of electric field provoked the drastic decreases of current (I) and Ts with elapsing time indicating the breakdown of electric channels. The limited electric field to promote direct tunneling effect is 5.5 V/cm. Because of no increase in Ts beyond 92.0 °C, no change of X-ray diffraction profile was confirmed at any electric field 0.55 V/cm, the gradual descent of I in the initial stage was probably due to a gradual increase in resistivity by the delay thermal expansion of UHMWPE matrix. Figure 3b,c shows X-ray diffraction peaks measured in the interval from 0 to 100 s and from 100 to 200 s, respectively, under the indicated applied fields. The corresponding WAXD intensity distribution provided extremely unusual profiles. At the initial stage ( 0 (where E denotes the electron energy in the tunneling direction) to facilitate numerical calculation. Following Sheng,21 the conductivity σ (= 1/ρ) was calculated as a function of temperature as follows

disordering. The crystallinity decreased drastically in the range 115 to 129 °C and the diffraction peaks from the (110) and (200) planes disappeared at 129.0 °C as shown in Figure 4. Of course, the behavior by self-heating is thought to be limited on the X-ray transmission place in the composite. Figure 6b shows the temperature dependence of the lattice spacing of the (110) and (200) planes and amorphous chain spacing estimated for the composite under self-heating and external heating. The thermal expansion for the (200) plane was more considerable than that for the (110) plane but the difference between self-heating and external heating was almost negligible up to 115 °C, although the peaks from the (110) and (200) planes disappeared beyond 129.0 °C in the case of selfheating (Joule heat). On the other hand, the amorphous chain spacing beyond 100 °C by self-heating is longer than that by external heating. As discussed before, the results for self-heating arisen by Joule heat is limited in the transmission part of X-ray, because electron transfer between Ni surfaces of overlapped adjacent NiCFs by tunneling effect is thought to be strongly affected by the collision against irradiated characteristic X-ray photons. In contrast, the results for external heating obviously represent the thermal behavior of the whole samples, because the structure change by external heating is hardly affected by X-ray. Figure 7a shows the time (t) dependence of surface temperature (Ts) and current (I) for the composite with a 4

Figure 7. (a) Time (t) dependence of surface temperature (Ts) and current (I) measured for the composite with 4 vol % NiCF content. (b) WAXD intensity curves measured at the indicated electric field (V/cm) in the period from 300 to 400 s.

vol % NiCF content. The behaviors of Ts and I at the indicated fields were different from those of the composite with a 10 vol % NiCF content. In comparison with the behavior of the composite with a 10 vol % NiCF content shown in Figure 3, much higher field was needed to elevate Ts but Ts did not arise beyond 92.0 °C. At electric fields beyond 5.5 V/cm, the current and Ts decreased with elapsing time. The behavior shown in Figure 7a supports the same X-ray profiles that are independent of electric field as shown in Figure 7b. As an example, WAXD intensity curves were measured under the indicated electric

σ=

⎛ 4T1 ⎞1/2 ⎡ ⎜ ⎟ ⎢ ⎝ πT ⎠ ⎢⎣ +

∫1

∞

∫0

1

⎧ T ⎫ T Σ0(εT)exp⎨− 1 εT2 − 1 φ(εT)⎬dεT T0 ⎩ T ⎭

⎛ T ⎞ ⎤ Σ1(εT)exp⎜ − 1 εT2⎟dεT⎥ ⎝ T ⎠ ⎥⎦

(5)

where εT is a dimensionless field parameter given by εT = δT/δ0, in which δT is thermal fluctuating field and δ0 is δ̅0U0/eD. D is 2232

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Figure 8. (a) Logarithmic plots (open circles) of resistivity versus temperature by external heating measured for a 4 vol % NiCF content by digital multimeter. The red solid curve lower than 20 °C was calculated by using parameters in (b). The blue solid curve beyond 20 °C was obtained by using eq 5 and 6. (b) Logarithmic plots of conductivity versus the reciprocal absolute temperature and the red solid curve showing the best fitting at D = 1.24 nm, A = 1.17 nm2, and λ = 0.080 (T < 20 °C). (c) Black and red solid curves for D and λ versus temperature ≤20 °C and the corresponding dashed curves for D and λ versus temperature >20 °C, which were calculated at A = 1.17 nm2. (d,e) U versus u (and D) plot calculated using eq 6 at the indicated values of ε in the temperature range from −146 to 20 °C and at 130 °C. (f) Logarithmic plots (open circles) of the resistivity versus temperature for a 10 vol % NiCF content in the range from −146 to 140 °C. (g) Black and red solid curves by D versus temperature for K = 2.3 and K = 1.0, by using parameters shown in (f). (h) Arrangement model (I) of NiCFs through junction gaps and (II) enlargement of the gap between neighboring NiCFs with distance D and A in model (I).

where u = x/D is the reduced spatial variable in which x is the distance from the left surface of the junction and U0 is height of the rectangular potential barrier in the absence of an imageforce correction. λ, which is given by 0.795e2/4DKU0 (K is the permittivity of insulating barrier = 2.3 for UHMWPE36), is an important dimensionless parameter. The potential U(u,δ̅) is a peaked function of u and with a maximum Um = U(u*,δ̅), where u* satisfies the condition (∂U/(∂u) = 0. Defining δ̅ = δ̅0 at Um = 0, δ̅0/λ has a physical meaning for λ ≤ 0.25. The parameters T1 and T0 are defined as DAδ02/8πK and T1/ 2χDξ(0), respectively, where A is the surface area over which the most of tunneling occurs (see Figure 8h) and k is the Boltzmann constant. The parameter χ = (8π2mU0/h2) (h is

the gap distance between overlapped adjacent NiCFs (see Figure 8h) and e denotes the electron charge (which equals 1.602 × 10−12 erg). ∑0(εT) and ∑1(εT) are formulated as differential tunneling current by εT. δ̅0 (= δ̅/ε) and U0 are parameters to form a nonparabolic potential function as follows ⎡ ⎤ λ U (u , δ ̅ ) = Uo⎢1 − − δ ̅u⎥ ⎣ ⎦ u(1 − u) ⎡ ⎤ λ = Uo⎢1 − − δo̅ εu⎥ ⎣ ⎦ u(1 − u) = U (u , ε )

(6) 2233

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Figure 8d shows the potential barrier calculated by eq 6 at ε = 0, 0.2, and 0.5 using D = 1.24 nm, A = 1.17 nm2, and λ = 0.080 at temperature ≤20 °C; therefore the potential barrier U(u,ε) was negative near u = 0 and unity. Increasing ε caused the barrier to become asymmetrical and shifted the maximum peak (Um) to lower u values, thereby decreasing Um. The barrier width at U(u,ε) = 0 corresponding Fermi energy level is shorter as ε increases. This indicates that in the case of the same gaps between adjacent NiCFs, the tunneling effect becomes more pronounced with increasing ε. Figure 8e shows nonoparabolic potential barrier calculated by using D = 7.92 nm and λ = 0.012 at 130 °C at the indicated ε values. The potential barrier width at Fermi energy level calculated at ε = 0.5 was much shorter than the virtually fixed distance 7.92 nm between adjacent NiCFs because of very steep of potential barrier under high applied field. Considering from the profile of potential barrier in Figure 8e, the slope of potential barrier at Fermi level calculated at ε = 0.2 and 0.5 became also much steeper in comparison with D between adjacent NiCFs except at ε = 0 denoting no electric field. This means that the potential barrier width at Fermi energy level became much shorter than the gap distance between overlapped adjacent NiCFs. This indicates the possibility of current flow at large distance D. If parabolic function or rectangular function used in many papers13,21,37,39−41 is adopted as the potential barrier, the barrier width at Fermi level must be equal to the real distance between adjacent NiCFs, which is described in Supporting Information III. That is, however, few current flows at such high temperature as 130 °C cannot be explained by the parabolic function. Figure 8f shows temperature dependence of the resistivity for the composite with a 10 vol % NiCF content in the temperature range from −146 to 140 °C. The resistivity increased slightly from −146 to 125 °C like metal. The resistivity increased suddenly in the temperature range from 125 to 140 °C like PTC phenomenon but the maximum was ca. 0.12 Ωcm. Such behavior is quite different from the thermal behavior of the composite with a 4 vol % NiCF content beyond 106 Ωcm, which was due to high conductivity of the composite with a 10 vol % NiCF content. The best fitting up to 120 °C can be obtained at D = 0.185 nm, A = 500 nm2, and λ = 0.235. Sheng’s equation assures the good agreement for resitivity increase with elevating temperature at very narrow gap distance D. The problem is that the gap, 0.185 nm, is too narrow to inhabit UHMWPE between overlapped adjacent NiCFs. The electron transfer between the narrow gap by tunneling effect is similar to the current flow along continuous Ni layer. This means that many conjunctions of NiCFs are thought to be continuous polygonal line for electric current. However, the current transfer between very narrow Ni surfaces by tunneling effect is not equal to usual electric transfer along continuous surface. The value of D to give the best fitting as a function of temperature was given at A = 500 nm2 and λ = 0.235, which is shown in Figure 8g. Despite the increasing electron transfer power between adjacent NiCFs by tunneling effect with elevating temperature, the increasing power is thought to be less effective than the growth of collision frequency of free electrons against Ni atoms along NiCF axis associated with an increase in resistivity with elevating temperature. Accordingly, the tunneling effect associated with the decrease of resistivity with temperature was not observed in Figure 8f. Even so, it is obvious that the movement of free electrons along fiber direction by tunneling

Plank constant) is the tunneling constant and φ(εT) in eq 5 given by ξ(εT)/ ξ(0) has the property that φ(0) = 1 and φ(1) = 0. The complicated derivation of ξ(εT) was described elsewhere21,22 to determine the barrier distance D and area A. The logarithmic plots of conductivity σ against the reciprocal absolute temperature were shown in Figure 8b in the temperature range −146 to 20 °C. The fairly good agreement between experimental and calculated results could be obtained by choosing suitable values of D, A, and λ parameters. The fitting was done by computer simulation. The values can be obtained as a function of temperature up to 20 °C, which are shown as solid curves for D = 1.24 nm, A = 1.17 nm2 and λ = 0.080 in Figure 8c. The resistivity up to 20 °C calculated by using the above fixed values of D, A, and λ is shown as a solid red curve in Figure 8a. The calculated solid curve up to 20 °C is in good agreement with the experimental results. As discussed previously, λ must be below 0.25 from the theoretical considerations.21,22 The curve fitting results by eqs 5 and 6 in the temperature range from 20 to 140 °C are shown as a blue solid curve. The good agreement between experimental and calculated results can be achieved by computer simulation, which was found to be realized by changing D and λ at a fixed value of A = 1.17 nm2. This indicates that the number of electrons pass through the expanded gap (UHMWPE matrix) by tunneling effect became fewer with increasing D with elevating temperature up to 140 °C. The values of D and λ at A = 1.17 nm2 to give the best fittings are added in Figure 8c as dashed black and red curves, respectively. Judging from the insulator of PE (ca. 1012 Ωcm) at 150 °C, it is obvious that the electric channel is maintained up to 133 °C because of electric current to give resistivity 107 Ωcm. In this case, the distance 7.92 nm at 130 °C and 16.2 nm at 133 °C can be obtained by Sheng’s equation. However, the calculated distance is too long in consideration with specific volume PE with increasing temperature.37 This is because NiCFs in the original specimens are oriented randomly in the threedimensional space but the NiCFs tended to orient parallel to the film surface predominantly because of the heavy weight of NiCFs to keep their steady state, because UHMWPE matrix softens with increasing temperature, which is shown later in Figure 9. Accordingly, the gap between overlapped adjacent NiCFs becomes wider.

Figure 9. The diagram of adjacent NiCFs against temperature.

It was already demonstrated that the resistivity at room temperature becomes higher with increasing cycle number by repeated experiments of PTC effect, when the filler content is slightly higher than the critical volume content at the percolation threshold.(see Figure 2 in ref 38). The tunneling effect at long distance beyond 10 nm can be explained by the nonparabolic function of eq 6, because the potential barrier width at Fermi level is sensitive to the value of ε associated with electric field (see Figure 8e). 2234

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adjacent NiCFs. This indicates that the gap difference between overlapped adjacent NiCFs for the composites with 4 and 10 vol % contents provided quite different tunneling effect power. Actually, as shown in Figure 7a for the composite with a 4 vol % content with low conductivity, the maximum Ts (92.0 °C) decreased with elapsing time under 6 V/cm and the corresponding current decreased. Of course, the corresponding X-ray profiles did not change up to 6 V/cm. As discussed before, drastic decrease of Ts and current (I) beyond 6 V/cm were due to the chain scission of adhesive UHMWPE between overlapped adjacent NiCFs as breakdown by high electric field as well as appearance of airspaces (void) by the breakdown. In contrast, the composite with a 10 vol % content has a number of contact point between overlapped CFs. The narrow gaps between overlapped adjacent CFs contain air (no UHMWPE). Hence, even at low electric field such as 0.6 V/ cm, the electron transfer between the narrow gap by tunneling effect is similar to the current flow along continuous Ni layer as discussed before. Because many conjunctions of short NiCFs are thought to be continuous polygonal line for electric current, a number of electrons transfer between the narrow gaps by tunneling effect. To facilitate understanding a series of analysis, the appearance of amorphous phase under electric field is represented as schematic drawing. Figure 11 shows the appearance mechanism of amorphous phase on the X-ray transmission part for the composite with a 10 vol % NiCF content. Here the important factor is because there are a number of gaps between overlapped adjacent NiCFs in the Xray transmission part of the composite. When X-ray beam is irradiated to the composite under electric field, photons (whose energy of a X-ray photon is hc/λ = 8.05 KeV) collided against transferring electrons between overlapped Ni surfaces by tunneling effect and some of the transferring electrons were flown out from the gap to UHMWPE medium. The flown electrons were collided against carbon atoms of UHMWPE and the disarrangement of UHMWPE chains by the impulsion provoked disruption of crystallites because of lighter weight of carbon atom than Ni atom. In this case, Ts arisen by Joule heat must be higher than the temperature of α-dispersion. Accordingly, the amorphous phase disappeared when the electric field was cut off, because of no collision between photons and transferring electrons. Following Zhang and Luo,20 as described before, the disruption of crystallites is attributed to segmental rotation associated with weak dihedral angles along UHMWPE chains that can vary for a large range. Of course, the appearance of amorphous phase in the present case is thought to be the segmental rotation by collision between the flown electrons and the carbon atoms. Of course, there is no transfer of electrons in conductive materials without applying electric field and amorphous phase appeared only on X-ray transmission place. The present fundamental work indicates the characteristic moving behavior of electrons by tunneling effect, when beating the transferring electrons by X-ray photons. Of course, amorphous phase as shown in Figures 4a and 5a cannot appear in the composite with a 10 vol % under electric field without irradiating X-ray. The electron emission at low electric field in the present system can be expected to many systems instead of Fowler−Nordheim tunneling effect23 under high electric field (close to 108 V/cm) on metal surface field emission.

effect can be realized through the narrow gap between overlapped adjacent NiCFs ensuring high conductivity of the composite with a 10 vol % NiCF content. The generation of huge Joule heat is associated with drastic collision of free electrons against Ni atom (cation) along NiCF direction, because the conductivity of Ni layer was higher than that of CFs. Actually, current mainly flowed along Ni layers within NiCFs, which was demonstrated by the Hall Effect. The experiment was done under 5, 10, and 12 mV and corresponding Hall constants were −1.35 × 10−5, −3.45 × 10−4, −5.08 × 10−5, respectively, indicating that carriers are surely electrons of Ni. Figure 10a shows resistivity against surface temperature (Ts) arisen by Joule heat. The measurements were done in Ts range

Figure 10. (a) Open circle plots of resistivity versus surface temperature (Ts) arisen by Joule heat. Solid red and black curves show the theoretical calculation at K = 1.0 (air) and K = 2.3 (PE), respectively by eqs 5 and 6. (b) Calculated parameters to give the best fit between experimental and theoretical results at K = 1 (air) and 2.3 (PE).

from 25 to 129 °C. Ts were measured at position 4 as shown in figure in Supporting Information II. The resistivity increased with elevating Ts. The red curve calculated by eqs 5 and 6 shows the best agreement with experimental results at the indicated parameters, D, A, and λ in Figure 10b. The values of D increase slightly with elevating Ts. The values of D were 0.155−0.172 nm, which is similar to the 0.185 nm estimated by external heating as discussed before in Figure 8f. Anyway, such narrow gap between overlapped adjacent NiCFs indicates no existence of UHMWPE. Accordingly, the numerical calculation was done again by using λ, which is given by 0.795e2/4DKU0 (K is the permittivity of insulating barrier = 1 for air). The red curve in Figure 10a shows the best fitting for self-heating, which was calculated by using the parameters in Figure 10b. The values of D become 0.220−0.260 nm, which is longer than the values (0.155−0.172 nm) calculated at K = 2.3. Anyway, the gap 0.220−0.260 nm are too narrow to inhabit UHMWPE between overlapped 2235

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Figure 11. (a) Usual PE chain arrangement within crystal units. (b) PE chain arrangement disturbed by striking of transferring electrons flown out from the gap under applied field. (c) Collisions between the transferring electrons flown out from the gap between adjacent NiCFs and UHMWPEcrystallites. (d) The diagram representing X-ray photons and transferring electrons.

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CONCLUSION The structural changes of UHMWPE under self-heating were mainly investigated in terms of temperature dependence of WAXD intensity. For the composite with a 4 vol %, the surface temperature (Ts) increased with electric field but the maximum Ts was 92.0 °C at 5.5 V/cm and the further increase of electric filed caused drastic decrease of Ts and current by chain scission of UHMWPE chains indicating breakdown by high electric field and then airspaces resulted between NICFs. In contrast, surprising phenomenon was observed for the composite with a 10 vol %. The diffraction peaks from (110) and (200) planes disappeared perfectly when the surface temperature (Ts) reached 129.0 °C at 1.2 V for the composite with a 10 vol % NiCF content. This temperature (129.0 °C) was lower than the endothermic peak (139.1 °C) and the foot (149 °C) at higher temperature side in DSC curve and also lower than the equilibrium temperature (145.5 °C). The radial distribution function calculated from the intensity curve from the amorphous phase indicated the crystallite disappearance was due to expansion distance of ca. 0.140 nm (from 0.444 to 0.584 nm) between UHMWPE chains. To make clear the different behaviors between 4 and 10 vol % contents, the gaps between overlapped adjacent NiCFs were evaluated by using nonparabolic potential barrier in terms of thermal fluctuationinduced tunneling effect. As the results, the gaps (ca. 1.24 nm) of the composite with a 4 vol % contain UHMWPE, while the gaps (ca. 0.220 nm) of the composite with a 10 vol % contain no UHMWPE because of very narrow gap distance (0.220 to 0.260 nm). Accordingly, for 10 vol % composite it is thought to be because when the X-ray beam is irradiated to the composite under electric field, X-ray photons collided against transferring electrons between overlapped Ni surfaces by tunneling effect and some of the transferring electrons were flown out from the gap to UHMWPE medium. The flown electrons collided against carbon atoms of UHMWPE and the collision impact caused disarrangement of UHMWPE chains in crystal grains. Accordingly, the amorphous phase disappeared when the electric field was cut off, because of no collision between photons and transferring electrons.

ASSOCIATED CONTENT

S Supporting Information *

The experimental description in detail and the derivation of eq 5 are shown. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel/Fax: +86 411 84986093. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the financial support from the Natural Science Foundation of China (NSFC) program (No. 21074016) and (No. 21374014). The authors are indebted to Professor P. Sheng of Hong Kong University of Science and Technology who has developed tunneling effect in many material fields. He kindly taught us how to derive eq 14 from eq 10 that appeared in ref 21.

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dx.doi.org/10.1021/jp4112734 | J. Phys. Chem. B 2014, 118, 2226−2237