Yielding in Crystals

In the last chapter we examined data for the yield strengths exhibited by materials. But what would we expect From our understanding of the structure of solids and the stiffness of the bonds between the atoms, can we estimate what the yield strength should be A simple calculation (given in the next section) overestimates it grossly. This is because real crystals contain defects, dislocations, which move easily. When they move, the crystal deforms the stress needed to move them is the yield strength. Dislocations are the carriers of deformation, much as electrons are the carriers of charge. 

As we showed in Chapter 6 (on the modulus), the slope of the interatomic force-distance curve at the equilibrium separation is proportional to Young s modulus E. Interatomic forces typically drop off to negligible values at a distance of separaHon of the atom centres of 2rg. The maximum in the force-distance curve is typically reached at 1.25ro separation, and if the stress applied to the material is sufficient to exceed this maximum force per bond, fracture is bound to occur. We will denote the stress at which this bond rupture takes place by d, the ideal strength a material cannot be stronger than this. From Fig. 9.1 

More refined estimates of cr are possible, using real interatomic potentials (Chapter 4) they give about E/15 instead of E/8. 

Let us now see whether materials really show this strength. The bar-chart (Fig. 9.2) shows values of Oy/E for materials. The heavy broken line at the top is drawn at the level it/E = 1/15. Glasses, and some ceramics, lie close to this line - they exhibit their ideal strength, and we could not expect them to be stronger than this. Most polymers, too, lie near the line - although they have low yield strengths, these are low because the moduli are low. 

All metals, on the other hand, have yield strengths far below the levels predicted by our calculation - as much as a factor of 10 smaller. Even ceramics, many of them, yield at stresses which are as much as a factor of 10 below their ideal strength. Why is this  

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To a solution of the dicyclohexylamine salt of the dibenzylamine (2.8 g, 5.0 mmol) dissolved in a mixture of methanol (25 mL) and acetic acid (2.4 ml.) was added 5% palladium on carbon (water content 53.3%, 1.2 g, 0.23 mmol Pd). The resulting mixture was stirred for 25 h under hydrogen at atmospheric pressure and ambient temperature. NaOH (2 M) aqueous solution (about 20 mL) was added to the reaction mixture in a water bath to adjust the pH to 5.1 at 30 °C. After the mixture was stirred for 40 min at ambient temperature, it was filtered to remove the catalyst. Filtration and concentration afforded the cleaved amine (949 mg, 97%). Recrystalhzation using EtOAc gave the sodium salt as white crystals (602 mg, 57% as a recovered yield in crystallization). 

A Monte Carlo simulation is fast to perform on a computer, and the presentation of the results is attractive. However, one cannot guarantee that the outcome of a Monte Carlo simulation run twice with the same input variables will yield exactly the same output, making the result less auditable. The more simulation runs performed, the less of a problem this becomes. The simulation as described does not indicate which of the input variables the result is most sensitive to, but one of the routines in Crystal Ball and Risk does allow a sensitivity analysis to be performed as the simulation is run.This is done by calculating the correlation coefficient of each input variable with the outcome (for example between area and UR). The higher the coefficient, the stronger the dependence between the input variable and the outcome. 

Tetrapotassium peroxodiphosphate is produced by electrolysis of a solution containing dipotassium phosphate and potassium fluoride (52). Alkalinity favors the formation of the P20 g anion, whereas the PO anion is produced in larger yields in acidic solution. It is therefore possible to obtain an 80% yield of K4P20g by choosing the proper conditions. The tetrapotassium peroxodiphosphate can be crysta11i2ed from solution by evaporation of water to form a slurry. The crystals can be separated from the slurry and dried. The material is noncorrosive and cannot be catalyticaHy decomposed by iron ions. 

Desyl chloride (a-cbloro-desoxybenzoin, m-chloro-co-phenyl acetophenone) [447-31-4] M 230.7, m 62-64 , 66-67 ,67.5 , 68 . For the purification of small quantities recrystallise from pet ether (b 40-60°), but use MeOH or EtOH for larger quantities. For the latter solvent, dissolve 12.5g of chloride in 45mL of boiling EtOH (95%), filter and the filtrate yields colourless crystals (7.5g) on cooling. A further crop (0.9g) can be obtained by cooling in an ice-salt bath. It turns brown on exposure to sunlight but it is... 

Dried with Linde type 5A molecular sieves or Na2S04 and fractionally distd at reduced pressure. Alternatively, it was refluxed with, and distd from, BaO. Also purified by fractional crystn from the melt and distd from zinc dust. Converted to its phosphate (m 135°) or picrate (m 223°), which were purified by crystn and the free base recovered and distd. [Packer, Vaughn and Wong J Am Chem Soc 80 905 1958.] The procedure for purifying via the picrate comprises the addition of quinoline to picric acid dissolved in the minimum volume of 95% EtOH to yield yellow crystals which are washed with EtOH and air dried before recrystn from acetonitrile. The crystals are dissolved in dimethyl sulfoxide (previously dried over 4A molecular sieves) and passed through a basic alumina column, on which picric acid is adsorbed. The free base in the effluent is extracted with n-pentane and distd under vacuum. Traces of solvent are removed by vapour phase chromatography. [Mooman and Anton J Phys Chem 80 2243 1976.]... 

Vanadium (III) acetonylacetonate [13476-99-8] M 348.3, m 181-184°, 185-190°, pKj 2.92, pK 3.5(for aquo V hydrolysis). Crystd from acetylacetone as brown plates. It can be distilled in small quantities without decomposition. It is soluble in CHCI3 and CgHg and evaporation of a CHCI3 solution yields brown crystals which are washed with cold EtOH and dried in vacuum or at 100° in a CO2 atmosphere. Under moist conditions it readily oxidises [V(AcAc)3 lo V(AcAc)20]. [J Chem Soc 103 78 1913, Inorg Synth 5 105 1957 Anal Chem 30 526 I958 UV J Am Chem Soc 80 5686 1958.]... 

Despite considerable efforts very few membrane proteins have yielded crystals that diffract x-rays to high resolution. In fact, only about a dozen such proteins are currently known, among which are porins (which are outer membrane proteins from bacteria), the enzymes cytochrome c oxidase and prostaglandin synthase, and the light-harvesting complexes and photosynthetic reaction centers involved in photosynthesis. In contrast, many other membrane proteins have yielded small crystals that diffract poorly, or not at all, using conventional x-ray sources. However, using the most advanced synchrotron sources (see Chapter 18) it is now possible to determine x-ray structures from protein crystals as small as 20 pm wide which will permit more membrane protein structures to be elucidated. 

A solution of 6.3 g (0.9 moles) ethoxyacetylene in 50 ml ether is added dropwise during 30 min to a Grignard reagent prepared from 2.18 g (90 mg-atoms) magnesium and 9.81 g (90 mmoles) ethyl bromide. The reaction mixture is stirred for 1 hr at room temperature and then a solution of 3 g (9 mmoles) 3) -acetoxyandrost-5-en-I7-one in 50 ml dry ether is added dropwise. The mixture is refluxed for 1 hr and after cooling to 0° poured into 100 ml of an aqueous ammonium chloride solution. The aqueous solution is extracted with ether, and the organic extract is washed with ammonium chloride solution and water, dried, and evaporated. The residue is chromatographed on 130 g alumina (activity III). Elution with petroleum ether-benzene (1 1) yields, after crystallization from acetone-hexane, 1.27 g (35%) 3j5-acetoxy-17a-ethoxyethynylandrost-5-en-17) -ol mp 138-139° Ho -122°. 

The aqueous phase was rendered acid with concentrated hydrochloric acid (weak Congo red and the separated acid taken up in ether. The isolated crude acid was recrystallized from dibutyl ether, yielding colorless crystals of 4-[(2-methvl-1,2-dicarbobenzoxy-hvdra-zino)-methyl] -benzoic acid, which melted at 112°C. The so-obtained product was sufficiently pure for further reaction. 

A solution of 438 mg of diac in methanol (20 ml) and ammonia solution (SG 0.88 20 ml) was iodinated at 0°C with 1.8 ml 1 N iodine solution. The product was isolated in almost theoretical yield in a manner similar to that described for tetrac. After crystallization from 50% (v/v) methanol, triac was obtained as colorless needles which melted over the range 65°C to 90°C according to the rate of heating. The molten form resolidified at about 110°C and finally melted at 180 C to 181°C without decomposition. The compound, dried at 25°C/3 mm over silica gel, contains methanol of crystallization. 

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