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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), written by Kostya Trachenko (Professor of Queen Mary University of London) Vadim V. Brazhkin (Director of Moscow Institute of High pressure Physics), translator: Yan Jia, proofread by Zhang Yi

In the 1970s, Nobel laureate physicist Edward Purcell noticed that there is no liquid that is much less viscous than water. In the first paragraph of the essay "Life at low Reynolds number" (Life at low Reynolds number), he wrote: "the viscosity has a wide range, but it all ends in the same place." it puzzles me. "

Purcell said that "it ends somewhere" means that the viscosity of the liquid will not fall below a certain value. In the first footnote of the essay, he pointed out that Victor Weisskopf had explained this phenomenon to him. So far, however, no one has seen a public record of this explanation. Even so, before and after Purcell's article, Weskov himself published a short article entitled "about liquids" (About liquids). It begins with a thought-provoking story about the challenges theoretical physicists face when trying to deduce the state of matter only from quantum mechanics. They can predict the existence of gases and solids, but not liquids.

The point is that liquids are difficult to deal with-which is very clear in the textbook. For example, Lev Landau and Evgeny Lifshitz's Statistical Physics (Statistical Physics) repeatedly emphasized that the thermodynamic properties and temperature dependence of liquids simply cannot be calculated in an analytical form that is suitable for all liquids. The reason lies in the strong intermolecular interaction and the lack of small vibrations that simplify the solid theory. This complexity is reflected in the famous "no small parameter" problem: liquids have neither the weak interaction of gases nor the small atomic shifts of solids. Despite these difficulties, we have developed the thermodynamic theory of liquids based on excitation in liquids, which is currently being carefully tested.

Viscosity minimums at the same time, we can ask theorists whether they fully understand viscosity to answer Purcell's question of why all viscosity coefficients are cut off in the same place. The viscosity coefficient η represents the ability of the fluid to resist shear force and determines important properties such as diffusion and dissipation. In a rarefied gas-like fluid, η is determined by molecules moving within the average free path L and momentum transfer in collisions: specifically, η = ρ vL / 3, where ρ and v are molecular density and average velocity, respectively.

The equation predicts that the viscosity of the gas increases with temperature, because the molecular velocity increases with temperature. This prediction is counterintuitive because the fluid usually gets thinner when heated. Unlike gases, the viscosity of a dense liquid is determined by the vibration of its molecules around the quasi-equilibrium position before jumping to the adjacent position. These jump frequencies increase with the increase of temperature. As a result, the viscosity decreases with the increase of temperature: η = η 0exp (U/kBT), where U is the activation energy.

Viscosity increases at high temperature and decreases at low temperature, which means that it has a minimum. The minimum results from the smooth transition (crossover) between two different viscosity regions: one is the gaseous region, where the kinetic energy of the particles with higher temperature provides a larger momentum transfer, which leads to a larger η; the other is the liquid region, where the pulsation frequency of the particles with lower temperature decreases and the liquid flow velocity slows down, which also leads to greater η.

It is convenient to look at the transition above the critical point, where it is smooth and free from the interference of gas-liquid phase transition. It is considered by means of the kinematic viscosity v = η / ρ, which describes the flow properties of the fluid. The figure on the next page (original page 67) shows the experimental values of several supercritical fluids (supercritical fluids). It is obvious that there are minimums in kinematic viscosity, which can be understood as the transition state between gaseous and liquid behaviors.

The minimum viscosity provides the first clue to the Purcell problem. When the viscosity reaches its minimum, they certainly stop falling. But can each minimum itself be arbitrarily close to zero? (note that we don't discuss superflow in this quick study. Why is it difficult for the minimum value of η to move up or down and to some extent close to the viscosity of water under environmental conditions?

If scientists can calculate the minimum of viscosity, they can answer this question. But as Landau and Liverside discuss in their book, it's complicated. The interaction between molecules is strong and system-dependent. Using only theory and no model input, it is difficult to calculate viscosity parameters even for simple liquids. For molecular liquids, such as water, this is almost impossible.

Experimental kinematic viscosity of inert and molecular liquids. Each fluid exhibits a minimum. The viscosities of helium, hydrogen, oxygen, neon, carbon dioxide and water are drawn at 20 MPA, 50 MPA, 30 MPA, 50 MPA, 30 MPA and 100 MPA, respectively. (source: NIST, https://webbook.nist.gov/chemistry / fluid) Fortunately, the minimum viscosity at the transition position is a special point. In fact, if only approximately considered, the viscosity can be calculated. The minimum v min is only related to the two basic properties of the condensed matter system v min= ω Da2/2 π, where an is the distance between atoms and ω D is the Debye frequency (Debye frequency) of the system. In turn, these two parameters can be related to the radius of the hydrogen atom and the characteristic binding strength set by the Rydberg energy. Then v min becomes

(1) where me is the electronic mass and m is the molecular mass.

The two basic constants "and me" appear in equation (1). Minimal viscosity turned out to be a quantum property! This seems surprising and contradicts the concept of high temperature liquids as a classical system. But (1) reminds us that the nature of the interaction in the condensed state is quantum mechanical, and the particle affects both the Bohr radius and the Rydberg energy.

The basic constant helps to prevent v min from rising or falling sharply. Because v min is inversely proportional to the square root of molecular weight, the viscosity itself is not universal-although it does not change v min much. For different liquids, as shown in the figure, (1) equation (1) predicts that v min should fall in the range of (0.3-1.5) × 10-7m2/s. This range is comfortably close to the experimental value.

Therefore, the answer to Purcell's question is that viscosity stops falling because they have minimums, which are determined by basic constants. Interestingly, the same thing happens to an unrelated liquid property, the thermal diffusivity, which dominates the heat transfer of the liquid. It also shows the minimum given by formula (1). The reason for this is that, like v min, the thermal diffusivity also depends on two parameters, an and ω D.

As shown in formula (2), when m takes the proton mass mp, formula (1) produces a universal amount of v f, that is, the basic kinematic viscosity:

(2) the basic physical constants me and mp are of universal importance. Together with electron charge and the speed of light, they form a dimensionless constant that determines whether the universe is friendly to living things. This is because they affect the formation of stars and the synthesis of heavier elements, including carbon and oxygen, and then form molecular structures that are vital to life.

Basic constants and water basic constants are also friendly to life at a higher level. Biological processes, such as those in cells, are largely dependent on water. For example, if the Planck constant is taken at different values, the viscosity of water will also change-its kinematic viscosity v, which is related to the flow of water, and its kinetic viscosity η, determine its internal friction and diffusion. If the minimum viscosity increases due to a higher value of viscosity, the water will become thicker and the biological process will be different. Life may not exist in its present form, or even at all.

One may hope that cells can still survive in such a universe, thinning the overly sticky water by finding a hotter place. But it doesn't help. The Planck constant sets a minimum of independent temperature at which the viscosity cannot be further reduced. Water and life are indeed in harmony with the quantized harps of the physical world.

We hope Purcell will be pleased with the answer to his question. Unless he heard the answer from Weskov in the 1970s.

references

E. M. Purcell, "Life at low Reynolds number," Am. J. Phys. 45, 3 (1977).

V. F. Weisskopf, "About liquids," Trans. N. Y. Acad. Sci. 38202 (1977).

L. D. Landau, E. M. Lifshitz, Statisticheskaia fizica (Statistical Physics), 2nd ed., Pergamon Press (1969).

Mr. J. E. Proctor, "Modeling of liquid internal energy and heat capacity over a wide pressure- temperature range from first principles," Phys. Fluids 32, 107105 (2020).

Mr. K. Trachenko, V. V. Brazhkin, "Minimal quantum viscosity from fundamental physical constants," Sci. Adv. 6, eaba3747 (2020).

JD. Barrow, The Constants of Nature: From Alpha to Omega- The Numbers That Encode the Deepest Secrets of the Universe, Pantheon Books (2003).

This article is published in FanPu authorized by the American physical Society (AIP). The original text is translated from Kostya Trachenko and Vadim V. Brazhkin, "The quantum mechanics of viscosity", Physics Today 74,66-67 (2021) https://doi.org/ 10.1063 / PT.3.4908

Reproduced from [Kostya Trachenko and Vadim V. Brazhkin, "The quantum mechanics of viscosity", Physics Today 74,66-67 (2021) https://doi.org/10.1063/PT.3.4908], with the permission of the American Institute of Physics.

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