A rotating incompressible fluid bounded by two concentric spherical rigid surfaces can exhibit purely toroidal free oscillations. The eigenfrequencies are fractions of the angular frequency of rotation. If the bounding surfaces are slightly ellipsoidal, secondary spheroidal fields become existent, and in general, a free mode splits into a doublet with one of which exists only when the inner bounding surface is present. For the real earth, the compressibility of the outer core, the elasticity of the solid earth, and the self-gravitation of the entire earth modify the toroidal core oscillations. The present treatment gives explicitly the effects of these parameters on the eigenfrequencies.
We have constructed a general first-order theory describing those small oscillations of a rotating elliptical earth that are affected by the presence of a liquid outer core. The theory is applicable to free core oscillations and earth tides. Care has been taken to include the effects of the wobble or nutation due to the rotation of the outer core relative to the solid earth. On the basis of the theory the free spheroidal modes of degree 2 and order 1 have been investigated. We have searched for and listed undertones with periods less than 28 hr. No upper limit to the eigenperiods has been detected. It is shown that stable, unstable and neutral polytropic cores are capable of free oscillation. At a period close to the sidereal day the spheroidal mode is accompanied by rigid rotation of the liquid outer core with respect to the solid earth. This is the well-known diurnal wobble of the Earth. It appears probable that the diurnal wobble is one of a class of similar wobbles that involve large toroidal motions in the outer core. Finally, the amplitudes of the 18·6-yr principal nutations has been computed. Excellent agreement is found with observed values.
It is demonstrated that the side-chain engineering of polymer donors and molecular conformation of small-molecule acceptors (SMAs) plays a crucial role in the morphology and photovoltaic performance of polymer solar cells (PSCs). However, the synergetic effect of these two aspects has been rarely reported. Herein, two wide-band gap donor–acceptor (D–A) copolymers (PST-TTC and PSPT-TTC) featuring the same main-chain backbone but different conjugated side chains were developed and combined with three low-band gap symmetric SMAs (ITIC, DTCFO-ICCl, and Y6) with various molecular conformations (S-shape, C-shape, and U-shape) to investigate the synergetic effect of side-chain engineering of polymer donors and conformation manipulation of SMAs on molecular properties, device physics, film morphology, and photovoltaic performance. The results indicate that PST-TTC with an alkylthiothiophene side chain exhibits a broader absorption spectrum, a smaller optical band gap (1.95 vs 1.99 eV), and a deeper-lying highest occupied molecular orbital (HOMO) level (−5.39 vs −5.36 eV) as compared to PSPT-TTC with an alkylthiophenylthiophene side chain. The PSCs were constructed according to the different donor/acceptor combinations based on the two as-synthesized polymers and three selected SMAs. After optimization, PST-TTC paired with S-shaped ITIC provides a power conversion efficiency (PCE) of 10.10% and the PCE can be further improved up to 10.60% when C-shaped DTCFO-ICCl was used to replace ITIC as the acceptor. However, the device performance becomes worse, accompanied with a sharply reduced PCE of 7.36% after the substitution of ITIC for U-shaped Y6. On the contrary, when the paired donor was changed to PSPT-TTC, the PSPT-TTC:Y6-based PSC achieves a remarkably increased PCE of up to 11.46%. These delicate observations suggest that the polymer donor with different conjugated side chains should collaborate with symmetric SMAs possessing various molecular conformations to realize superior morphology and device performance.
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