The best current LCE materials are based on the robust and error-proof &#;click&#; chemistry of thiol-acrylate Michael addition20, using the cheap commercially available starting materials and easily controlled crosslinking density (Supplementary Fig. S1). For practical applications, we need naturally non-aligned, polydomain LCEs21 with relatively low crosslinking density, which show the wide stress plateau reflecting the elastic softness on nematic director alignment (see Fig. 1a), comparing the basic LCEs with two crosslinking densities: 10% and 40% (as labelled in the plot). Figure 1b shows the typical oscillating dynamic-mechanical response of an LCE, scanning the temperature range at a fixed frequency, again comparing the crosslinking densities in the same materials labelled LCE10 and LCE40, respectively. The result illustrates the key regimes of the dynamic response: the rigid low-dissipation glass below a glass transition Tg, the low-modulus high-damping nematic range below the nematic-isotropic transition TNI, and the ordinary isotropic rubber above TNI. The anomalous damping window between the glass at low temperature and the isotropic phase at high temperature is roughly [5&#;60&#;°C] in these LCEs. There is a large literature on how one can control both these key transitions by chemical modifications, moving the transition temperatures up or down (see the discussion in Supplementary); this was not our focus in this study. While the test in Fig. 1b shows the components of complex tensile modulus E*(ω,T), the theory of LCE response11,12 makes it clear that only the shear deformation carries the anomalous dynamic soft elasticity; not surprisingly, the early published data showed much higher tan&#;δ values, being tested in the pure shear geometry10,14.
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Fig. 1: Mechanical characterisation of LCE materials.a Near-equilibrium stress&#;strain curves for LCE10 and LCE40 materials, highlighting the wide soft plateau through internal re-orientation of the polydomain LCE on stretching, as well as the ability to withstand large deformations. b Temperature scan of linear oscillating response (at fixed 1&#;Hz), highlighting the high tan&#;δ across all of the nematic range.
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Figure 2a presents the Master Curves of frequency dependence of the linear dynamic modulus (tensile modulus, as in Fig. 1b). Note that above the glass transition one has E&#;&#;&#;3&#;G, given the effective incompressibility of rubbers and the average isotropy of polydomain LCE, with the bulk modulus remaining high: K&#;>&#;2&#;GPa. These Master Curves have the frequency scaled by time&#;temperature superposition11,12, using the T&#;=&#;20°C as the reference temperature; the unusual behaviour of E&#;(ω) at low frequencies reflects the dynamic soft elasticity when the quasi-equilibrium modulus in the nematic LCE phase is much lower than in the isotropic phase above TNI. Unlike in ordinary polymers, where the Williams&#;Landel&#;Ferry (WLF) time&#;temperature superposition3 is based on the changing timescales during the glass transition, in nematic LCE (well below its Tg) the rubbery response adds the additional complexity: the lower modulus in the nematic phase due to the internal director relaxation modes (the origin of anomalous damping), and the increasing rubber modulus in the isotropic phase above TNI (due to the entropic rubber elasticity E&#;~&#;kBT). Supplementary Fig. S2 gives more detail on this procedure and the building of Master Curves in LCE.
Fig. 2: Master Curves of LCE materials.a The tensile storage modulus E&#;(ω) for LCE10 and LCE40 materials, obtained by time&#;temperature superposition of frequency-scan tests at different temperatures (labelled in the plot) with the frequency scaled for the reference T&#;=&#;20&#;°C. The nematic transition TNI for both materials is between 50 and 70&#;°C and indicates the end of dynamic softness (the isotropic rubber modulus increases on heating). b The corresponding Master Curve of the loss factor tan&#;δ for LCE10, with the frequency scaled for the same reference T&#;=&#;20&#;°C. Unlike the storage modulus E&#;, whose magnitude at high temperatures is affected by the nematic-isotropic phase transition and the entropic rubber elasticity, the time&#;temperature superposition for the loss factor works well at this high-temperature/low-frequency region.
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Figure 2b shows the same time&#;temperature superposition for the dynamic loss factor tan&#;δ, obtained by scaling the frequency of the tests carried out at different temperatures (as labelled in the plot, in the same format as in Fig. 2a). Here, unlike for the storage modulus in Fig. 2a, the frequency scaling works well for high temperatures, producing the predicted values at ultra-low frequencies (for the reference temperature of 20&#;°C). The frequency band where LCE materials show high damping is roughly between 0.1&#;Hz and 20&#;kHz at ambient temperature. As usual in time&#;temperature superposition, this window proportionally shifts to higher frequencies for a higher reference temperature, and vice-versa for a lower reference temperature. However, one should be careful with interpretation, since the nematic&#;isotropic phase transformation occurs around TNI&#;=&#;60&#;°C in these LCE materials: if the actual LCE is tested above 70&#;°C, it will be plain isotropic and its vibration damping will not be different from ordinary elastomers irrespective of the testing frequency. This is because the WLF time&#;temperature superposition is based on the (empirically validated) assumption that the internal dynamics are controlled by the &#;cage confinement&#; in the glass state, which gradually gets weaker and eventually gets fully released in the melt state. In the nematic LCE, the internal dynamics are slowed down by the orientational relaxation modes, but only at a temperature within the thermodynamic nematic phase: unlike glass, a different thermodynamic phase cannot be produced by changing the input frequency of probing oscillation.
The impact damping was tested in the Hopkinson split pressure bar experiment and also in a separate but closely related ball-impact test. The Hopkinson bar test is based on the impact of two parallel faces of metal bars, with the test sample inserted in the impact area; it is well described in the ASM Handbook volume22. However, we quickly established that inserting a flat elastomeric pad in such an impact only activates the compressional deformation modes, and therefore shows no significant effect of the LCE dynamic softness: the test probes the bulk modulus, which is approximately the same in all elastomers (see Supplementary for further detail and plotted data). LCE10, LCE40, as well as ordinary siloxane and thermoplastic polyurethane elastomers from the market-leading vibration damping suppliers Sylgard® and Sorbothane®, respectively, all showed the same effect of dissipating about 25&#;27% of impact energy in this geometry.
For this reason, we looked into exploring the impact geometry where the shear deformation modes are excited, the most straightforward being the impact of a flat elastomeric pad with a spherical projectile ball, which initially produces spherical waves from the point of impact. This was tested in two settings: Fig. 3a shows the results of impacting a flat elastomer pad with a spherical projectile, while Fig. 3b presents the impact of the flat face of the Hopkinson split pressure bar on a semi-spherical cap of the elastomer. In both cases, a significant shear deformation component is generated in the material, and the comparison between the ordinary elastomers and LCE is stark. Integrating the measured power over time allows us to estimate the full energy budget: the initial kinetic energy converting into the transmitted, reflected, and dissipated &#;lost&#; energy. For example, for the spherical projectile impact on an elastomer pad of 3&#;mm thick, with the initial energy of 3&#;J, we found that the PDMS had 25% of energy in the rebound, 1.2% of energy transmitted, with ca. 74% of impact energy dissipated. For the same 3&#;mm-thick LCE40 pad the results showed 6% of energy in the rebound, 0.7% of energy transmitted, with ca. 93% of impact energy dissipated, and for LCE10: 5% of energy in the rebound, 0.6% of energy transmitted, with ca. 94% of impact energy dissipated.
Fig. 3: Impact damping test.a Spherical ball impact on a flat elastomer pad. The incident kinetic energy of ca. 3&#;J was partially transmitted through the elastomer into the receiving bar, the plot showing the power transmitted after the impact at t&#;=&#;0&#;s. PDMS sample has 74% of impact energy dissipated, and clearly shows the under-damped oscillations after impact. Both LCE samples have dissipated over 90% of impact energy, and show the overdamped response both at the onset and after the force pulse. b Flat surface impact on a semi-spherical elastomer pad (of PDMS, LCE40 and LCE10), showing the low and heavily overdamped wave transmission: of the impact energy of 2&#;J, only 0.01&#;J (0.4%) was transmitted into the target rod by the LCEs while almost all energy was reflected by PDMS, with 28% of impact energy absorbed in the elastomers in this geometry. In both panels, the sketches illustrate the geometry of impact and the deformation field distribution in the elastomer.
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In the impact of the flat face of split Hopkinson bar on a semi-spherical cap of an elastomer, of 10&#;mm diameter (Fig. 3b), receiving a strike with the impact energy of 2&#;J, we found that the PDMS cap had almost no energy transmitted, but rebounded 79% of impact energy (with 21% energy loss). The LCE damping caps did transmit a very small fraction of impact energy: LCE40 transmitted 0.3%, rebound 72% (loss 27.5%), and LCE10 transmitted 0.4%, rebound 71% (loss 28.5% of impact energy). It is clear that the shape and dimensions of the damping pad play a significant role in this energy distribution. However, we note that even with spherical-cap shaped pads, the mechanical loss in the material was comparable between PDMS and LCE systems, and much lower than in the projectile-impact test (where very little energy was in the rebound). In both tests, the PDMS elastomer had a distinct elastic response with resonance bounces, in contrast to both LCE materials showing a strongly overdamped response.
It is important to establish a correspondence between the impact measurements, providing data in real time, and the frequency domain where we see both the material properties (such as Master Curves in Fig. 2) and the elastic waves. For this, we examined the spectral distribution of power transmission in the ball impact, (Fig. 3a, also see Supplementary Fig. S3). As expected for the sharp, 0.2&#;ms pulse in real time, its frequency distribution is a relatively flat-top until the cutoff frequency for PDMA at ca. 8&#;kHz, and ca. 10&#;kHz for both LCE40 and LCE10. This frequency range captures almost all of the enhanced-tan&#;δ region in the Master Curve in Fig. 2(b), which explains the almost complete dissipation of the impact energy in the material.
The study of transmission and attenuation of vibrations, particularly in the sonic range from 50 to &#;Hz, were carried out on a home-made device, described in Supplementary Fig. S6. To generate spherical waves23, and thus explore their shear deformation component, the cylindrical elastomer sample was held upright; this way, the elastic waves were initially radiating spherically before the resonant standing wave pattern was established in the cylindrical samples. Three sets of cylindrical samples were prepared with a similar length of 15&#;16&#;mm and a cross-sectional diameter of 17&#;mm (by crosslinking in the mould). Two samples were LCE 10 and LCE 40, and one sample was PDMS, used mainly for comparison and benchmarking. Each sample was mounted on a dynamic shaker that sends elastic waves longitudinally through the bottom surface of the sample, as illustrated in Fig. 4. A dynamic laser interferometer was used to pick up the acceleration at the top surface (the output signal from the structure), and an accelerometer was used to measure the acceleration at the bottom surface (the input signal to the structure).
Fig. 4: Vibration attenuation test.The amplitude of the signal (measured in [dB] as a ratio to the incident amplitude) transmitted through the elastomer cylinder is plotted against the vibration frequency. The primary standing wave resonance near 1&#;kHz is found in all materials, with underdamped PDMS showing several secondary resonance peaks. The overdamped LCE systems have the Q-factor equal to 0.4 (i.e. the damping ratio ζ&#;=&#;120%), compared with PDMS: Q&#;=&#;2.4 (ζ&#;=&#;21%). Supplementary Fig. S4 shows graphical representations of resonance modes.
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Figure 4 shows the damped-oscillator wave resonance of the system, again comparing LCE materials with the standard PDMS elastomer. The results show resonances of the three samples at around &#;Hz (see Supplementary Fig. S7 for detailed modal analyses). The resonance of the PDMS has a peak value of 15&#;dB. In reference to the PDMS resonance peak, the LCE samples showed a much lower peak response which was around 4.5&#;dB. These peak values of the LCE samples are more than two times less than that of the PDMS sample for similar size, form and vibration mode. This suggests the existence of higher damping values in the LCE samples in comparison to the PDMS.
In comparison to the results in the impact case shown earlier, a wide-area transducer generating longitudinal compression waves shows no effect of the LCE dynamic softness. In the impact case, the waves of compressional deformation propagate and attenuate similarly to all elastomers. We measured the speed of longitudinal elastic wave travelling along the cylinder length (ca. 15&#;mm), as c&#;=&#;49&#;m/s for PDMS, 43&#;m/s for LCE10, and 62&#;m/s for LCE40. Given that the density of all elastomers is similar, ca. &#;kg/m3, these values confirm that it is the shear (rubber) modulus that is responsible for the acoustic wave: the typical values of rubber modulus are G&#;=&#;1&#;2&#;MPa (for the modulus at a given frequency one should consult the Master Curve in Fig. 2a), which closely matches the measured wave speed \(c\approx \sqrt{G/\rho }\) (the precise value of wave speed is affected by many additional factors: from the sample shape to the elastic impedance). It is also consistent that LCE10 and PDMS had approximately the same crosslinking density (hence a similar rubber modulus), and therefore a comparable wave speed, while LCE40 had a higher crosslinking density, and accordingly, a slightly higher wave speed. For comparison, the bulk modulus of all elastomers is approximately the same, K&#;=&#;1&#;2&#;GPa, predicting a typical speed of longitudinal compression wave in the order of &#;m/s: such a fast wave zips across the 15&#;mm distance in ca. 15&#;μs, and therefore would affect the measurement above 60&#;kHz, well outside our testing range.
The attenuation of the acoustic waves in the elastomers was measured via the damping of the primary resonance peak, which was found around 1&#;kHz in all materials, consistently with the separately measured wave speed and the distance to travel. The quality factor Q of the resonance, defined as the frequency-to-bandwidth ratio of the resonator is calculated using the &#;3&#;dB method24, was determined as Q&#;=&#;2.4 for PDMS (the damping ratio ζ&#;=&#;21%, indicating the underdamped material). This is consistent with several secondary resonances seen in the PDMS transmission data in Fig. 4 and also with the secondary vibrations in the impact test seen in Fig. 3a. In contrast, the LCE materials are heavily overdamped: Q&#;=&#;0.42 (ζ&#;=&#;120% for LCE40, and Q&#;=&#;0.45 (ζ&#;=&#;110%) for LCE10. The difference between LCE10 and LCE40 is within the uncertainty of the test, influenced by precise details of the surface cut, the cylinder width and length, and even the placement of the transducer. It is consistent with the theoretical understanding that the anomalous vibration dissipation, on the microscopic level, is caused by the local rotations of the nematic director, which occur on the length scale well below the network mesh size given by crosslinking density (an extensive discussion of this theory can be found in refs. 4,11,12).
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Now that the winter solstice is around the corner, we&#;re all aware of the need to heat our living, shopping and working spaces. The lake states have a huge and increasing wood inventory, tremendous untapped potential for producing renewable heat.
Almost everyone is aware of old-fashioned fireplaces, wood stoves and outdoor wood boilers. These devices offer benefits to the residential user. Sometimes, there are air quality issues associated with the devices, often due to the use of wet fuels and excessive damping of the air flow. However, there are a range of advanced, highly efficient and very dependable technologies for residential, commercial and institutional applications, especially for schools, hospitals, municipal facilities and similar structures.
Advanced wood chip systems work well when heating spaces of more than about 50,000 square feet, although this breakpoint is not set in stone. Capital costs of advanced wood chip systems are higher than those for fossil fuel systems, but operating costs are lower. Current low prices for natural gas have increased the payback period for wood chip systems, if there is access to natural gas.
One of the nation&#;s premiere wood chip heating system manufacturers is based in the Upper Peninsula of Michigan. Messersmith Manufacturing has built, delivered and installed over a hundred institutional-sized systems around the country.
An important factor when considering a wood chip system is identifying local wood chip suppliers. Not all areas of the lake states have ready access to contractors with wood chippers. Determining the quality of available chips is a critical factor in designing a wood chip system. These systems are often the least expensive heating alternative available, but a thorough feasibility study is necessary to be certain.
Smaller spaces may be better heated with wood pellets or cordwood. Wood pellet systems have their greatest potential for residences and small businesses that are off the natural gas grid. In Michigan, roughly a third of all households heat with propane, fuel oil or electricity. Advanced wood pellet stoves, furnaces and boilers offer a less expensive alternative and are equally as reliable and convenient as fossil fuel systems.
One of the current hurdles in expanding wood pellet heat is the lack of a bulk delivery service, which is common in the more advanced renewable energy economies. Similar to fossil fuel deliveries of propane or fuel oil, trucks supply wood pellets once or twice each heating season. All the homeowner needs to do is move the thermostat.
For rural locations, there are a wide range of modern cordwood boilers and indoor stoves that heat homes, small businesses and resorts. The critical question is how much wood is a user willing to process? And, is the user willing to keep these home fires stoked on a daily basis?
When the wood supply is gathered from the forest by the user and the cost of that labor is chalked-up to recreation, these systems are quite inexpensive. If cordwood is purchased, the costs are typically below that for propane, fuel oil and electricity but not necessarily that of natural gas.
On a community scale, groups of buildings can be heated and cooled from a central boiler plant connected to a piping network. These &#;district energy&#; systems are highly efficient and deliver the least costly heating and cooling. Downtown St. Paul, Minnesota, is heated and cooled with a district energy system fueled on municipal solid wood waste. In southeastern Michigan, Bordine&#;s Grand Blanc plant nursery facility uses a district energy system to heat over 11 acres of greenhouses.
There is a wide variety of designs available to provide low-cost, dependable and clean renewable thermal energy from wood-based fuels.
Two of the more common concerns about wood-based thermal energy involve our forest inventories and atmospheric carbon.
The lake states&#; wood volumes have been increasing for a century. Michigan has one of the highest rates of annual growth in the nation, more than twice that of mortality and removals. So, while forests do, indeed, have limits to wood supply, there is no current shortage of wood. There is room to expand the wood heat sector.
The issue about wood supply is more about availability than inventory, an important distinction. Our forests offer a great opportunity for contributing to our economy and sustainable communities, while at the same time increasing environmental benefits and healthier forests.
Carbon additions to the atmosphere are sometimes cited as concerns in the face of climate change. However, you can burn trees for fuel for a thousand years, or a million, with no increase in carbon within the carbon cycle. The carbon from burning wood came from the atmosphere just a short while ago. In fact, managed forest landscapes actually sequester more carbon than unmanaged forest landscapes.
On the other hand, no matter how little fossil fuel you burn, atmospheric carbon dioxide (CO2) will be increased. It is true that, compared with wood, burning fossil fuel may produce less CO2 for the amount of energy received, but that is irrelevant because wood is already part of the carbon cycle and fossil fuel carbon is not.
Are you interested in learning more about wood-based thermal energy? If so, contact the Statewide Wood Energy Team at Michigan Wood Energy. If you&#;re going to warm yourself with a fire, it&#;s always better to burn above-ground, plant-derived carbon than that long-buried underground.
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