Converse App Download
Mitsue Schue
Two seemingly opposite evolutionary patterns of clinal variation in body size and associated life history traits exist in nature. According to Bergmann's rule, body size increases with latitude, a temperature effect. According to the converse Bergmann rule, body size decreases with latitude, a season length effect. A third pattern causally related to the latter is countergradient variation, whereby populations of a given species compensate seasonal limitations at higher latitudes by evolving faster growth and larger body sizes compared to their low latitude conspecifics. We discuss these patterns and argue that they are not mutually exclusive because they are driven by different environmental causes and proximate mechanisms; they therefore can act in conjunction, resulting in any intermediate pattern. Alternatively, Bergmann and converse Bergmann clines can be interpreted as over- and undercompensating countergradient variation, respectively. We illustrate this with data for the wide-spread yellow dung fly, Scathophaga stercoraria (Diptera: Scathophagidae), which in Europe shows a Bergmann cline for size and a converse Bergmann cline (i.e., countergradient variation) for development time. A literature review of the available evidence on arthropod latitudinal clines further shows a patterned continuum of responses. Converse Bergmann clines due to end-of-season time limitations are more common in larger species with longer development times. Our study thus provides a synthesis to the controversy about the importance of Bergmann's rule and the converse Bergmann rule in nature.
Converse flexoelectricity is a mechanical stress induced by an electric polarization gradient. It can appear in any material, irrespective of symmetry, whenever there is an inhomogeneous electric field distribution. This situation invariably happens in piezoresponse force microscopy (PFM), which is a technique whereby a voltage is delivered to the tip of an atomic force microscope in order to stimulate and probe piezoelectricity at the nanoscale. While PFM is the premier technique for studying ferroelectricity and piezoelectricity at the nanoscale, here we show, theoretically and experimentally, that large effective piezoelectric coefficients can be measured in non-piezoelectric dielectrics due to converse flexoelectricity.
Another relevant electromechanical coupling mechanism in PFM is flexoelectricity11,12. Direct flexoelectricity is a property allowed by symmetry in all materials, and it describes the appearance of polarization in response to a strain gradient. Converse and inverse flexoelectricity are the reverse phenomena of strain induced by polarization gradient13 and strain gradient induced by polarization14. The inverse and converse flexoelectric effects have been experimentally demonstrated by, respectively, applying a voltage to a capacitor and measuring its bending14,15 and by applying a voltage across a truncated pyramid so as to generate an inhomogeneous electric field inside it, thus causing the sample to deform16,17,18,19,20.
Flexoelectricity is a two-way coupling between polarization and strain gradient and, conversely, between strain and polarization gradient. Both direct and converse flexoelectricity are characterized by the same fourth rank tensor, and are hence allowed in centrosymmetric materials12,21. The converse-flexoelectric stress is expressed mathematically as:
The exact solution of the problem is complicated due to complex interactions between the tip, the surface and the electromechanical response of the sample. In addition, the shape and size of the contact area is itself a complex function of the amount of force with which the tip is pressing on the surface, the exact tip shape and the elastic properties of both the tip and the sample. Moreover, the applied tip pressure induces strain gradients in the contact region. These also induce polarization by direct flexoelectricity. This polarization self-consistently modifies the electric field in the material by direct flexoelectricity and hence affect the nominal converse flexoelectric response.
In the absence of an analytical solution for this complex problem, we examine the role of converse flexoelectricity on the PFM response using a self-consistent computational model22, based on a linear continuum theory of piezoelectricity23, augmented with flexoelectricity. In this model, direct and converse flexoelectricity are intimately intertwined manifestations of the same coupling. The total electromechanical energy \((\cal H)\), resulting fourth-order continuum equations and a mesh-free discretization method for solving numerically these equations are described in Supplementary Note 1.
An important feature of the response is the size effect, because flexoelectricity is caused by a gradient: the sharper the contact is, the stronger the gradient will be, and thus the larger the flexoelectrically-induced deformation12. Moreover, the contact area between tip and surface grows as a function of the amount of force with which the tip is pressed onto the surface. Accordingly, we can expect that, upon increasing loads, the contact area increases, and thus the AFM tip-induced electric field gradient in the sample decreases. Gradient-induced apparent piezoelectricity will, therefore, be inversely proportional to contact force. This feature emerges as a useful tool to differentiate the converse flexoelectric electromechanical response from the regular piezoelectric response or the electrochemical strain, for which no dependence on the contact area should be observed5,28.
Effective piezoelectric coefficient as a function of contact radius a and force F. The flexoelectric response is obtained for non-piezoelectric SrTiO3 (STO) considering only flexoelectricity (both direct and converse). For comparison, we show the hypothetical response of an archetypal piezoelectric (quartz) assuming an absence of flexoelectricity. The contact radius is obtained under the applied force, in the absence of the tip voltage. The flexoelectric response shows a size-dependent behaviour which can be used as a means to qualitatively distinguish between flexoelectricity and piezoelectricity
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