EL ADOLESCENTE BELLO QUE CRECE Y SE RELACIONA

reduced and the length over which the strain transmission occurs is increased. However the depth of the epoxy should not be increased so far that the trans- mission length exceeds the size of the sample plates else only partial trans- mission will occur. C. Reducing the shear modulus of the epoxy also reduces the shear stress in the epoxy and the stress concentration at the sample ends but the shear strain in the epoxy is in- creased so care must be taken to stay within the limits of the epoxy. The third failure point is the epoxy holding the sample to the

sample plates. The thickness of the epoxy layer between the sample and sample plates can be varied to limit the shear stress in the epoxy, see section 2.5.1 and figure 2.22. The maximum shear stress in the epoxy, when there are sample plates above and below to transmit the applied force, is approximatelyτmax=εsamplepEtG/2d, where

Eis the sample’s Young’s modulus,Gis the shear modulus of the

epoxy,d is the epoxy depth. It is therefore best to increase the

depth of the epoxy to limit the maximum shear stress, but one must also bear in mind that the length of sample that needs to be embedded in the epoxy to ensure adequate strain transmission also increases as the epoxy depth is increased. This means the epoxy depth should only be increased while there is a long enough length of spare sample at each end held in the epoxy. Increasing the epoxy depth further reduces the amount of strain transmitted. The shear stress decays roughly exponentially with the distance from the end of the mount with a characteristic length scale of

λ = pEtd/2G, so a distance of 2-3_×λ is desirable to achieve

adequate strain transmission.

As first mentioned in section 2.5.1, for my measurements I used Stycast 2850FT with catalyst 23LV as the epoxy as it is well suited to cryogenic temperatures and has a relatively high shear strength. Stycast is much softer than the sample with a shear

36 Uniaxial Stress Technique

modulus estimated to be∼6 GPa at low temperatures. The sample

plates used in our strain device offer up to 400-500µm of overlap

with the ends of the sample so by fixing the epoxy depth to 25µm

this space is utilised to its full potential. To achieve this separation accurately a foil was fixed to one of the sample plates, further back on the plate than the sample, and polished to 150µm thick so that

when it was combined with the second sample plate sandwiching the sample the correct separation was achieved, see figure 2.14. There still is some uncertainty in making sure the sample sits in the centre between the sample plates above and below the sample but I found in practice that careful application of the epoxy with equal distribution above and below the sample without excess meant the sample naturally found the centre as the epoxy cured. Stycast 2850FT is also a filled epoxy with specified particle diameters up to 45 µm, but in practice, probably due to our preparation and

application method, the larger particles tended to be absent but the smaller particles may have been helping to centre the sample. It was difficult to verify if the sample was centred directly after the mounting stage since there was no line of sight to the side of the sample in our device, but after samples were removed from the rig this was always checked.

The shear stress in the epoxy can also be reduced by using a softer epoxy but at the expense of also increasing the shear strain in the epoxy. A softer epoxy needs to have a proportionally larger yield strain in order to achieve the same sample strains. Stycast 2850FT appears to be ideally suited in this range with a low enough shear modulus to prevent serious stress concentration in the sample, but a large enough yield strength to reach high sample strains. For most epoxies the elastic properties are only known close to room temperature but even comparing these values Stycast is well suited, see table 2.2. Some caution should be taken though when comparing these results directly since the shear tests on which this data are based can be strongly affected by surface preparation, material choice, and epoxy thickness. However once coupled with fact that

Table 2.2:Mechanical properties of select epoxies. Shear modulus and lap shear strength of several epoxies tested at room temperature. Where the shear modulus was unknown, the measured Young’s modulus with an assumed Poisson’s ratio of 0.3 was used to calculate the shear mod- ulus.

Shear modulus Lap shear strength Stycast® _{2850FT [69] ∼4 GPa} ∼40 MPa

(Blasted stainless steel) EPO-TEK®_{H74 [72]}

∼2 GPa _(Unknown)∼11 MPa

EPO-TEK®_{H20E [73]}

∼2 GPa _(Unknown)∼10 MPa

Araldite®_[74]

∼1 GPa _{(Blasted stainless steel)}∼18 MPa

MasterBond®

2.7Conclusions 37

Stycast’s thermal expansion is matched to that of brass it seems an ideal epoxy for our purpose and perhaps had another epoxy been chosen instead during the initial tests of the first uniaxial stress device, a premature failure may have written off the whole idea.

2.7 Conclusions

Traditional uniaxial pressure measurements are technically chal- lenging and extreme care must be taken to ensure high strain homogeneity. The technique described here, where a long narrow bar is strained across a vice, offers significant improvements and can achieve very high strain homogeneity in the central portion of the sample. When experiments are designed to be sensitive to this central portion of the sample this is then a very effective technique.

The strain range now achievable is no longer a weak perturbation but can be a very significant energy scale. To put it in perspective, one would expect that a strain of 1 % can change the Fermi level by approximately 1 % of the band width. For Sr2RuO4and straining along the [100] direction the band width in this direction for theγ

band is∼3 eV [6]. The energy scale of the achievable strain range

is therefore order∼30 meV equivalent to a temperature of∼300 K

or a magnetic field of∼600 T. 0.00 % : ∆εxx b-axis −0.15 % −0.37 % −0.63 % −0.81 % −0.94 % Temperature (K) χ 0(a.u.) 93 94 95 96

Fig. 2.23: YBa₂Cu₃O_6.92susceptibility against temperature. Real part of the susceptibilityχagainst temperature for a sample of YBa₂Cu₃O_6.92compressed along theb-axis.

Figure 2.23 shows preliminary measurements on the high tem- perature superconductor YBa2Cu3O7−δ made with this uniaxial

stress technique. Small concentric coils with diameters 250µm

for the pick-up coil and 1.5 mm excitation coil were placed above the centre of the sample to measure AC magnetic susceptibility. Over a strain range of close to 1 %, an equivalent pressure range of

∼1.6 GPa [76], there is very little broadening of the superconducting

transition, rather just a rigid shift ofTc to higher temperatures.

After carrying out some simple analytic approximations for the strain transmission and further detailed FEA simulations we can provide some guidelines, readily achievable in experiments, for the best procedure for mounting samples. High strain is best achieved by using soft and moderately thick layers of epoxy, bonding a thin sample to rigid sample plates, encasing the sample from above and below. The inhomogeneity from these sample mounts decays over a distance roughly equal to the width of the sample, so length to width aspect ratios greater than∼3:1 should be used to provide a

large enough homogeneous portion in the centre of the sample. Any asymmetry in the sample mounting causes the sample to bend when strained, creating a strain gradient across the sample’s thickness. The strain inhomogeneity can be large if the correct care is not taken but the bending inhomogeneity can be minimised by using long, thin samples. Care must still be taken to stay below the bucking limit however. The soft epoxy leads to some uncertainty

38 Uniaxial Stress Technique

in the exact sample strain achieved so a finite element simulation is required to determine the strain transmission more precisely.

3

The Physics of Sr₂RuO₄

Approaching a Van