The best texture is a cone

A looser fibre, more piezoelectric spread the grains of a polar fibre about its axis and watched the piezoelectric coefficients change. Each coefficient is a sum of a degree-one part and a degree-three part, and a spread multiplies the first by the average of $P_1(\cos\beta)$ over the grains’ tilts and the second by the average of $P_3(\cos\beta)$. The third-degree part decays faster, so when the two parts have opposite signs a coefficient can grow as the fibre loosens. For a grain whose shear coefficient is six times its longitudinal one, the fibre’s $d_{33}$ rose from 90 pC/N with perfect alignment to 168 at a spread of about thirty degrees before falling.

It closed on the inverse question. A spread about the axis is one family of textures, the one drawing or poling produces, and it is not the best. Which distribution of orientations maximises the fibre’s $d_{33}$ for a given grain? The essay guessed the answer, a cone of grains at the angle where the grain’s own oblique response peaks, and asked how close a realistic texture could come. This essay settles the guess with a geometric argument, and then measures the realistic case.

Two numbers per texture #

A fibre texture is any distribution of grain orientations that looks the same after any rotation about the fibre axis. The grains’ polar axes are tilted from the fibre axis by angles $\beta$ in some distribution, spread uniformly in azimuth, with each grain turned about its own axis at random. What a texture permits showed that averaging a property over such a texture keeps, for each degree $\ell$ of spherical harmonic in the property, only its axially symmetric part, multiplied by $\langle P_\ell(\cos\beta)\rangle$. A polar grain’s piezoelectric tensor has parts of degree one and three only, so every coefficient of every fibre of that grain is

$$d = a\,\langle P_1(\cos\beta)\rangle + b\,\langle P_3(\cos\beta)\rangle ,$$

with $a$ and $b$ fixed by the grain. For piezoelectric purposes a texture is two numbers, and the whole distribution of tilts can be forgotten once they are known.

The two numbers cannot be chosen freely. A texture in which every grain has the same tilt $\beta$, a cone, has the pair $(\cos\beta, P_3(\cos\beta))$, a point on a curve in the plane. Any other texture is a weighted average of cones, so its pair is a weighted average of points on the curve. The pairs any texture can have are therefore exactly the convex hull of the curve, the region its chords sweep out.

The best texture is on the edge #

The picture at the head of this essay is that plane. A coefficient is a linear function of the two numbers, so its level sets are straight lines across the plane, and a linear function takes its largest value over a convex region at the region’s boundary. The boundary is made of arcs of the curve and of straight chords that bridge its bends, and on a chord a linear function lies between its values at the chord’s two ends, which are points of the curve. So the largest value of any single coefficient, over every fibre texture whatever, is taken by a cone. No mixture of tilts can beat the best single tilt, because a mixture’s response is an average of cones’ responses.

For $d_{33}$ that settles the guess of the earlier essay. The cone’s $d_{33}$ is the grain’s own longitudinal response along a direction at $\beta$ from its polar axis, averaged over the azimuth, which is the oblique coefficient

$$d_{33}(\beta) = d_{33}\cos^3\beta + (d_{31} + d_{15})\sin^2\beta\cos\beta .$$

The moment-plane calculation reproduces it at nineteen angles to within $10^{-13}$ pC/N. For the grain with $d_{33} = 90$, $d_{31} = -33$ and $d_{15} = 560$ pC/N, shaped like tetragonal barium titanate, it peaks at 50.7°, at 223 pC/N. The spreads about the axis that the earlier essay computed run along a path inside the hull, and the best of them, 168, sits well short of the edge. They start at the perfect fibre, a corner of the curve, and move inward. They never reach the part of the edge where the best cone is.

Where the best cone sits, in closed form #

The best cone for $d_{33}$ can be found without a search. Write $c = \cos\beta$. On a cone the fibre’s $d_{33}$ is $f(c) = d_{33}\,c^3 + (d_{31} + d_{15})(c - c^3)$, a cubic in $c$, and its turning point is at

$$c^2 = \frac{d_{31} + d_{15}}{3\,(d_{31} + d_{15} - d_{33})} .$$

For the grain with a large shear coefficient that gives $\cos^2\beta = 527/1311$, a cone at 50.65°, the angle the search found. The turning point lies inside the range of cones, strictly between aligned and perpendicular, exactly when $d_{31} + d_{15} > \tfrac32 d_{33}$. That is the inequality the earlier essay derived for a spread about the axis to raise $d_{33}$ rather than lower it, and it is no coincidence. Both are the sign of $f'(1)$, the slope of the cone’s response as the cone first opens away from the axis. A grain whose response rises when tilted a little rewards both a spread and a cone, and a grain whose response falls rewards neither.

The second grain of the earlier essay shows the other case. Its coefficients have the ratios of zinc oxide, $d_{33} = 12$, $d_{31} = -5$ and $d_{15} = -8$ pC/N, and $d_{31} + d_{15} = -13$ is far below $\tfrac32 d_{33} = 18$. Its best texture for $d_{33}$ is perfect alignment, 12 pC/N, and every cone and every spread costs. With a spread of 20° about the best cone, which for this grain is the axis itself, $d_{33}$ is 6.7, barely half. For such a grain the moment plane’s corner, the perfect fibre, is the extreme point in every direction that matters, and the only texture work worth doing is aligning the grains better.

Each coefficient has its own cone #

The argument is about one coefficient at a time, and the answer depends on which.

The same grain’s other coefficients peak on other cones. The transverse coefficient $d_{31}$ is negative with perfect alignment and changes sign, reaching its largest positive value, 104 pC/N, on a cone at 128°, with the polar axes tilted more against the fibre than along it. The shear coefficient is largest with perfect alignment. The hydrostatic coefficient $d_h = d_{33} + 2d_{31}$, which sets a sensor’s response to pressure from all sides, is largest with perfect alignment too, at 24 pC/N. On the cone that maximises $d_{33}$ it falls to 15, because there $d_{31}$ has grown more negative in proportion.

That is the refused claim below: the texture that maximises one coefficient does not maximise another. A designer choosing a texture chooses it for a coefficient, or for a weighted combination of coefficients, which is again a straight line on the moment plane with its own best cone. The plane makes the trade explicit. Each combination is a direction, and its best texture is the point of the curve furthest in that direction.

A realistic cone #

A cone of grains all at exactly 51° is a limit no processing reaches. A real textured ceramic has its grains spread about whatever angle it favours, and the useful question is how much a spread costs.

The cost is modest at first. With a spread of 5° about the best cone the fibre’s $d_{33}$ is 219 pC/N, with 10° it is 209, with 15° it is 197 and with 20° it is 185. That is still above the best spread about the axis, 168 at any width. The best cone angle moves as the spread grows, from 51° to 48°, 44° and 37°. A wide spread about a steep cone puts some grains past the peak of the oblique curve, where the response falls quickly, and moving the cone towards the axis keeps more of them on the rising side. By a spread of 30° the best cone is at 13°, hardly a cone, and its $d_{33}$ of 170 is barely above the cap’s. By 45° the best cone is the axis itself, and the cone family has merged with the caps.

The first few degrees are almost free. A spread of 5° costs 4 pC/N, under two per cent, because the oblique curve is flat at its peak and grains a little either side of 51° respond almost as well as grains exactly on it. The loss grows with the square of the spread at first, as a flat-topped peak’s does, and only past ten degrees does it start to matter. So a cone texture is worth pursuing only if it can be made with a spread of about twenty degrees or less. Beyond that, the simpler texture a poling field produces does as well. The geometry says why. A cone with a wide spread is a mixture of many cones, a point deep inside the hull, and deep inside the hull every texture looks alike.

Where the grains push back #

The whole calculation treats a fibre’s response as the average of its grains’ responses, each grain responding as if it were free. That is the Voigt average, and it is exact only when every grain feels the same strain or the same field. Real grains constrain each other. A grain tilted to 51° deforms differently from its neighbours tilted elsewhere, and the mismatch sets up internal stresses that reduce the response. How much of the 223 survives the constraint depends on the elastic and dielectric anisotropy of the grain, and a self-consistent average, in which each grain sits in a medium with the fibre’s own properties, would be needed to compute it. It has not been done here.

The constraint matters more for a cone than for a cap. In a cap texture the grains are mostly near the axis and near each other in orientation, so the mismatch is small. In a cone every grain is tilted by about 51° in a different azimuth, and neighbours can differ by a hundred degrees in the direction of their polar axes. The Voigt estimate is therefore most optimistic exactly where it predicts the most, and the true advantage of the cone over the cap is smaller than the 223 against 168 computed here. By how much is the question left open.

Which grains reward a cone #

The inequality sorts grains by their own coefficients, and it can be read against the classes that allow them. A grain with a polar axis has a class among the ten polar classes, and only those classes have a spontaneous polarisation that poling can align. Among them the ones with $d_{15}$ large, as in the tetragonal perovskites, reward tilting. The ones with a small shear coefficient reward alignment. Symmetry allows $d_{15}$ in every polar class with a principal axis, and Neumann’s principle counts it as one independent component in $\infty m$ and in $4mm$ alike. It says nothing about its size relative to $d_{33}$. That ratio is what the criterion reads, and it is a property of the material rather than of its class.

The same geometry lies behind the largest piezoelectric responses known. Relaxor-based single crystals whose polar axes lie along the diagonals of a cube, poled along a cube edge about 55° away from them, respond along that edge far more strongly than along their own polar axes. The effect is usually described in terms of engineered domain configurations. In the terms of this essay, those crystals are cones of four polar directions at 55°, and their large shear coefficient comes from the ease with which the polarisation rotates away from the polar axis. A cone texture in a ceramic is the same arrangement made from grains instead of domains.

A property with no odd degrees #

The plane is particular to properties with two odd degrees. A fibre of achiral crystals can rotate light considered the gyration tensor, whose parts have degrees nought and two. Its fibre average depends on one number, $\langle P_2(\cos\beta)\rangle$, and the textures fill an interval rather than a plane, from $-\tfrac12$ for a cone at 90° to $1$ for perfect alignment. A linear property on an interval is largest at one end or the other. So the optical rotation of a fibre is largest either with the grains aligned or with them all perpendicular to the axis, and no intermediate cone can ever be best. The piezoelectric case is richer only because a second odd degree bends the set of attainable textures into a curve with an interior peak.

The group the texture keeps #

A cone texture has the same symmetry as a cap: the fibre’s limiting group, $\infty m$ for a polar fibre, which the seven groups a field can have lists among Curie’s. Symmetry cannot tell the two textures apart, and it permits exactly the same coefficients to both. It is silent about their sizes, and the whole of this essay is about sizes, the part of a material’s behaviour that what symmetry decides about a material leaves to the material. That is the general shape of what Curie’s principle can and cannot say. It decides which effects a combination of causes permits, and the magnitude of a permitted effect is outside it. The moment plane is a way of organising the magnitudes once symmetry has done its work. It is made possible by symmetry, because the fibre’s axial symmetry is what collapses every distribution to two numbers.

What the calculation leaves out #

The grain is a shape, not a material. The coefficients 90, −33 and 560 pC/N have the ratios of tetragonal barium titanate’s single-crystal values and are used here as a shape. No claim is made about a real barium titanate ceramic.

Only fibre textures. A texture that is not symmetric about an axis has more moments than two, and its response is not confined to the plane. Sheet textures, with two preferred directions, would need the full set of degree-one and degree-three harmonics of the orientation distribution, three and seven numbers rather than two.

Poling works against the cone. A polar grain’s polarisation can usually be switched between the few directions its crystal structure allows, and a poling field along the fibre axis chooses, in each grain, the allowed direction closest to the axis. That is what produces a cap. A cone at 51° asks the opposite: every grain’s polarisation held away from the axis. It can be made only by aligning the crystal lattice of each grain so that no allowed polar direction lies nearer the axis than the cone, which is what the relaxor crystals poled along a cube edge achieve. The calculation assumes the texture and says nothing about how to make it.

Only the linear coefficients. The argument that a linear property’s best texture is a single cone needs the property to be linear in the texture, which any averaged tensor property is. A figure of merit that is a ratio, such as the piezoelectric coefficient divided by the permittivity, is not linear in the moments. Its best texture can be a mixture.

What the calculation has to refuse #

The first test ties the moment plane back to something computed without it, the grain’s oblique response. The second ties the cap path to the earlier essay’s sweep, computed by the same formula at three spreads. The third checks the geometric claim on seventy textures, caps and spread cones, every one of which lies inside the hull. Without it the argument that the best texture is a cone would rest on the convexity alone.

Still open: the constrained cone #

The open question is the one the Voigt average hides. A self-consistent calculation would put each grain of the cone texture in an effective medium with the fibre’s own elastic and dielectric constants, solve for the strain and field inside it, and average the resulting response. The cone’s advantage over the cap would shrink, and the question is whether it survives. If it does, the best cone angle would also shift, towards the axis, where neighbouring grains are more alike. The moment plane would no longer be the whole story, since a self-consistent average is not linear in the texture. It would still be the right way to see the Voigt answer the constraint corrects.