In the previous chapter, we introduced the metaphor of the electronic states in a solid being like the seats in the orchestra of a theater, with the empty seats in the balcony representing a band of excited states at higher energy. Metals were described as like an orchestra in which only half the “seats” are occupied with electrons, so that a current could always be induced, because there was no energy barrier separating the highest occupied state and the lowest empty state. For metals, the seats actually correspond to matter-waves that can extend over the entire solid (to explain why this is so would be too much of a digression). This is why metals make such good conductors of electricity and heat—it takes very little energy to move these “free electrons” in response to an external stimulus. In a “covalently bonded” solid like diamond, each seat in the orchestra represents a rigid, directional bond between the atoms.
If there are no direct bonds holding the metal atoms together, what keeps the metallic solid from falling apart? Electricity! Each metal atom initially is electrically neutral, with as many positively charged protons in its nucleus as there are negatively charged electrons in probability clouds surrounding it. Remove an electron from each atom to roam over the solid, and the resulting original atoms are now positively charged. Try to compress the metal, and you are pushing the atoms closer together, and the positively charged atoms (called ions) repel each other, resisting the external pressure. This makes it strong. The lack of directional rigid bonds between atoms in a metal is why they are easy to deform, to pull out into long, thin wires or to flatten into sheets. The forces holding the metal together are not as strong as in most covalently bonded solids—we can easily bend and break a metal paper clip without superpowers!—but they are much stronger than one would expect, given how malleable most metals are.
There is, obviously, a fair amount of chemistry and materials science that determines the strength of a particular metal. For example, consider Adamantium, the strongest metal in the Marvel universe, which is bonded onto the X-Man Wolverine’s skeleton and claws. This miracle metal is strong not because it is dense, but apparently because it is a defect free covalently bonded solid. It takes more energy to remove a covalently bonded atom from all of its neighbors than just localizing a free electron onto an ion in a normal metal. Adamantium must somehow combine the electrical properties of metals with the strong covalent bonds found in diamonds, while avoiding having any network-weakening defects, making the resulting material “unbreakable.”
Defects weaken covalently bonded solids, but can actually make certain metals stronger.82 One way to improve the strength of steel is termed “work-hardening,” in which the metal is intentionally plastically deformed. This generates defects on the atomic scale within the metal. As there are no directional bonds in a metal, it is normally easy to move planes of atoms past each other when bending or drawing the material into a wire. Defects can block this atomic motion, and too many defects essentially form atomic-scale traffic jams that inhibit further deformations of the metal. This increase in strength comes at the expense of ductility. Cold rolled steel has a tensile strength comparable to titanium alloys, but both are flimsy compared with covalently bonded carbon nanotubes.
While we’re on the subject, we should address a common misconception regarding the chemical composition of Captain America’s shield. Wolverine’s claws are composed of pure Adamantium, but Captain America’s shield is a one-of-a kind alloy of steel and Vibranium. The steel is needed to provide rigidity, so that the shield can ricochet off walls and supervillain minions. Vibranium is an extraterrestrial material brought to Earth when a meteorite crashed in the African nation of Wakanda, which is ruled by the superhero the Black Panther. Vibranium has the ability to absorb any and all sound and convert the energy in the sound wave into some other, not-well-specified form, making it the perfect shock absorber, a quality strongly desired in a shield. Sound waves are alternations in pressure or density, and in a solid, sound is transmitted through the vibrations of the atoms. Vibranium possibly converts the atomic vibrations from an absorbed sound wave into an optical transition (which would explain why sometimes a glow is observed when Vibranium is used), thereby conserving energy in the process.
The material that forms Captain America’s shield was the result of a laboratory fluke, when a steel alloy and Vibranium were accidentally fused into an alloy. The conditions under which this metallurgical fusion occurred were not recorded, and this synthesis has never been repeated. In fact, attempts to re-create the metallic alloy used in Cap’s shield led to the synthesis of Adamantium. Captain America’s shield has sometimes been erroneously described as being an alloy of Vibranium and Adamantium, but this is physically impossible. When Captain America was discovered frozen alive in a block of ice by the Avengers (issue # 4, March 1964) he was found with his shield, the same disc he used for most of his superhero career in World War II. This was the same shield he used as a member of the Avengers as he fought along with and led these heroes. The same shield he had in Avengers # 66, when materials scientist extraordinaire Dr. Myron MacLain created Adamantium! Cap’s shield can sometimes appear to defy physics, but it cannot travel backward through time.
That the shield does indeed contain a Vibranium component has been experimentally verified by no less an expert in metallurgy than Stephen Colbert. On his television show, The Colbert Report, Stephen was bequeathed Captain America’s shield when the Sentinel of Liberty’s last will and testament was read.83 Rapping the shield with his knuckles and listening to the resulting “thunk,” Colbert assured his audience that this was “the sound of indestructible Wakandan Vibranium.” He thereby resolved a long-standing mystery in physics: What sound would you hear if you struck a material that absorbed all vibrations!
Back to Wonder Woman, it turns out that her creator himself led a double life. Quite unusual for the 1930s, Marston lived with both his wife and Olive Richard, the same woman whose 1940 Family Circle interview with Marston led to his comic-book career. This ménage à trois lasted from the 1920s till his death in 1947. Marston’s peripatetic academic career may have been a consequence of this unusual family life. The fact that at times the household’s financial support was provided by the working women may have reinforced Marston’s convictions concerning feminism and women’s rights. Photos of Marston from this time period show a man with more than a passing resemblance to the Daily Planet’s Perry White. That he was able to live with two women during the depths of the Depression no doubt testifies to the irresistible allure that accompanies a mastery of both science and comic books!
While there is debate as to whether Olive or wife Elizabeth Holloway Martson served as the inspiration for Wonder Woman, there seems to be a clear paper trail crediting Olive with one of Wonder Woman’s accessories. From Les Daniels’ 2001 excellent history of the character, in another Family Circle interview less than a year after the Amazon Princess’ first comic book appearance, Marston stated that the inspiration for Wonder Woman’s bracelets came from Olive Richard’s wrist jewelry!
DOES SIZE MATTER?
When the heroes of the Marvel universe banded together to fight Loki, the Norse god of mischief in Avengers # 1, Ant Man was front and center. By Avengers # 2, Henry Pym had adopted a new superhero persona, as Giant Man. When the first issue of the Avengers was published, over in Tales to Astonish # 48, Ant Man was facing off against the supervillain the Porcupine. The Porcupine was Alex Gentry, an engineer who uses his technological prowess to develop a quill-coated suit that conceals a host of offensive and defensive weapons—such as tear gas, stun pellets, ammonia (presumably for clean escapes), “liquid fire” (I suspect he means a gas flamethrower), detector mines, liquid cement, and others—with which he embarks on an obligatory crime wave. Now, I must say that, as a physicist, I have worked with many engineers in my academic career, and hardly any of them dress like giant porcupines. Gentry had captured Ant-Man and placed him in a bathtub partially filled with water. Given that he cannot climb up the
slick porcelain walls, Ant Man is forced to endlessly tread water until he becomes exhausted, at which point he would drown.
In the very next issue of Tales to Astonish # 49, perhaps motivated by embarrassment over such pedestrian death traps as a partially filled bathtub, Henry Pym discovered a reverse version of his shrinking potion that enabled him to grow larger than his normal height of six feet. And so Giant-Man was born into the Marvel universe. Eventually Pym became a diminutive crime-fighter again, this time as the flamboyant Yellowjacket, but for the majority of his superhero career, Pym fought for justice as either the oversize Giant Man or Goliath (same hero and same power of super-growth, just different code name and costume).
If one of the functions of comic books is wish-fulfillment for their young readers, then it must be recognized that not many kids fantasize about how cool it would be to be only a few millimeters high or even to dress like a porcupine. Now, to be able to grow to be twelve feet tall, that would be something. And yet, it turns out that being larger than normal carries with it a different, yet no less pressing, set of physical challenges. For one thing, as pointed out in Ultimates #3 (a modern version of the Avengers, featuring a new Giant-Man), your now much-larger (super-dilated) pupils would let much more light into your eyes. You would therefore always need to wear special goggles while enlarged, to avoid overloading your optic nerves.
In addition, there is a fundamental limit to the size that someone could grow; assuming of course that one could grow far beyond his normal height in the first place, which requires just as much of a “miracle exception” as does miniaturization. This limitation is set by the strength of materials (particularly bone) and gravity. Gravity enters the situation because your mass will increase in proportion to your volume if you maintain a constant density. Density is mass divided by volume, so the bigger you are (that is, the larger your volume), the greater your mass if the ratio of the two (density) remains unchanged. You would be a less-than-imposing figure if your super-growth kept your mass fixed.
In fact, just such a situation faced Reed Richards of the Fantastic Four when he encountered the monstrous alien invader Gormuu in Fantastic Four # 271. This issue relates a story that took place before Richards and his three comrades had been exposed to the cosmic rays that would give them their superpowers. Gormuu, a twenty-foot-tall misshapen green creature, was a warrior invader from the planet Kraalo. This story was a nostalgic tip of the hat by writer and artist John Byrne to the Monster Invader from Space stories that dominated Marvel comics in the late 1950s until the Fantastic Four arrived to save both the universe and the financial fortunes of Stan Lee and Jack Kirby. The creatures in these comics, Tales to Astonish, Strange Tales, Journey into Mystery, and Tales of Suspense, were always at least as big as a house and all had names with adjacent double letters, such as Orrgo (the Unconquerable), Bruttu, Googam (son of Goom), and Fin Fang Foom. Gormuu’s competitive advantage regarding world conquest was that his size increased whenever he was struck with any form of “broadcast energy.” Richards, upon examining a ten-foot-long and inches-deep footprint left behind by the already-expanding alien, realizes that the only way to stop this menace is to continue to strike him with more and more broadcast power. Any creature large enough to leave a ten-foot-long footprint behind should also leave an impression several feet deep, if its mass were increasing at the same rate as its volume. Discovering that Gormuu’s growth was at constant mass, not constant density, Richards feeds the alien so much energy that he grows larger than the Earth, and less dense than the background of space, until he becomes an insubstantial and decidedly unthreatening footnote in our nation’s atomic-age history. Bearing in mind the cautionary tale of Gormuu, let us assume that Henry Pym, in his guise as Giant-Man, manages to maintain a constant density as he grows, so that his weight increases in uniform proportion to his volume.
In order to treat the Giant-Man situation mathematically, let’s make a simplifying assumption—n amely, that Henry Pym is a giant box. Of course, he is more naturally described as a large cylinder, but I want to keep the math as simple as possible. Well, if he’s a box, then his volume is the product of his length, height, and width. A box that has a length of ten feet, a width of ten feet, and a height of ten feet will have a volume of 10 feet×10 feet×10 feet, or 1,000 feet cubed. This unit of volume is sometimes called “cubic feet” or feet3, which represents the fact that we multiplied a length (feet) times another length times another length. Now assume that Giant-Man uses his Pym particles to grow twice as large in all directions. So his height is now 20 feet, and his width and length are also 20 feet each. In this case his volume is 20 feet×20 feet×20 feet—that is, 8,000 cubic feet. So, doubling his length in all three directions increases his volume by a factor of eight.
If Giant Man is going to maintain a constant density as he grows, then his mass must increase at the same rate as does his volume, not his length! Doubling his height (as well as his width and breadth) requires that his weight goes up by a factor of eight to keep his density the same. Well, Hank Pym gets heavier the bigger he becomes—so what? The problem is that his weight increases faster than does the ability of his skeleton to carry his weight, so at a certain height, Giant-Man risks breaking his legs simply by standing up. The strength of an object, or its resistance to being deformed by a pull or crushed by a push, depends on how wide it is and not on its length. The technical way to say this is that an object’s “tensile strength” is determined by its cross-sectional area.
Think of a fishing line rated for twenty pounds—that is, for holding up a twenty-pound fish. A heavier fish would snap the line when we tried to lift it onto the boat. If we want to keep a heavier fish suspended, changing the length of the fishing line will be absolutely no help whatsoever. To increase the strength of a fishing line, you don’t increase its length, but rather, its diameter.8484 The wider the fishing line, the greater the area over which the force pulling it is distributed, and the less is the force applied across any given tiny element holding the line together. When a fishing line, or anything for that matter, breaks, the chemical bonds that were holding the material together snap and fly apart. The larger the area that’s available to support a given force, the less stress or strain is applied to any particular molecule, and the less likely it is that a catastrophic failure will occur. As discussed above, when breaks do happen, it is usually because a molecular imperfection or flaw magnifies the applied force locally, leaving the material weaker than if it were uniform and atomically perfect.
The dependence of a material’s strength on its cross-sectional area also limits the distances over which Mr. Fantastic of the Fantastic Four can stretch. As explained in Fantastic Four Annual # 1, he cannot extend the length of his body beyond five hundred yards. A wooden two-by-four board (which has a rectangular cross-section of two inches by four inches) that is three feet long can be supported above the ground by a sawhorse at either end and remain parallel to the floor. A six foot-long board of the same cross-sectional area would sag slightly in the middle, if still supported at the ends. A sixty-foot-long board would droop considerably, while a six mile long board would touch the ground at its center, even ignoring the curvature of the Earth. As Reed points out in Fantastic Four Annual # 1, “the further my body stretches, the weaker my muscles become, so that I cannot exert as much force stretched to a great distance as I can extending for a shorter distance.” Reed Richards, whose understanding of the mass-volume relationship saved the Earth from the terror of Gormuu, is also a walking illustration of the cube-square law of tensile strength.
As our cubical Henry Pym grows into Giant-Man, his volume increases faster than his cross-sectional area. As Giant Man grows larger, his bones naturally increase in size proportionally with the rest of his body. The strength of his femur or vertebrae grows by the square of his expansion rate. But the bigger that Giant-Man becomes, the more weight his bones must support—at constant density his weight increases by the cube of his growth factor. Suppose at
his normal height, Dr. Pym is six feet tall and weighs 185 pounds. His femur at his normal height can support a weight of eighteen thousand pounds while a single vertebra can support eight hundred pounds—nature builds considerable redundancy into some crucial load-bearing structural components. At a height of sixty feet, Giant Man’s expansion factor is ten. His volume therefore grows by a factor of 10×10×10=1,000 while the cross-sectional area of his bones increases only by a factor of 10 × 10 = 100. Henry Pym would now weigh 185,000 pounds, while his vertebrae could support only a weight of 80,000 pounds and his femur could carry 1.8 million pounds. At this height, his skeleton is not able to uniformly support his weight.
In order to keep Giant Man from growing so tall that no villain, even a superpowered one, could pose a credible threat to him, Stan Lee argued in the 1960s that increasing his size placed a biological strain on Henry Pym. His optimal strength was at a height of roughly twelve feet, and if he grew to more than forty or fifty feet, he was as tall as a house but as weak as a kitten. Years later, the metabolic limitation was replaced by a physical one derived from the cube-square law. It is now recognized, as described, for example, in Ultimates # 2, that even if the metabolic issues involving a growth serum are resolved, gravity and physics will still provide stringent limits on your ultimate size.
The fact that volume grows faster than surface area is true even for non-cubical objects. The volume of a sphere is given by the mathematical expression of a constant (4π/3) times the radius of the sphere cubed, or (4π/3) r3, while its surface area is 4π times the radius squared, or 4π r2. (A volume will always have the units of a length cubed—such as cubic feet—while an area has the dimensions of a length squared, such that carpeting is measured in square yards.) Consequently, the rising bubbles in a vat of acid into which Batman and Robin are being slowly lowered provides the Caped Crusader with a textbook illustration of the physical principle of the cube-square law.
The Physics of Superheroes: Spectacular Second Edition Page 36