Critical Opalescence

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Louise McDowell
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The phenomenon arises from the large fluctuations in the critical region of the substance. E S R Gopal is at Department of Physics, Indian Institute of Science, Bangalore.

1 Critical Opalescence E S R Gopal Critical opalescence is a striking light scattering phenomenon, which was elegantly explained by Einstein. In the critical region the light scattering is so large that the substance appears milky white in reflected light and brownish dark in transmitted light. The phenomenon arises from the large fluctuations in the critical region of the substance. E S R Gopal is at Department of Physics, Indian Institute of Science, Bangalore. He considers himself to be a student of physics forever. The example of ice, water and steam being the different states of the same material is recorded in the Egyptian and Chinese literature about four thousand years ago. The Critical Region The solid, liquid and gaseous phases of matter have been known since the dawn of history; the example of ice, water and steam being the different states of the same material is recorded in the Egyptian and Chinese literature about four thousand years ago. About ISO years ago, largely due to the careful studies of Andrews in 1860s, it was recognized that the gas can be converted into the liquid phase merely by compression if the temperature is below a critical temperature Tc above which the matter exists as a single fluid phase. For carbon dioxide, studied by Andrews, Tc is nearly 30 C. The so-called permanent gases like oxygen, nitrogen and hydrogen have their critical temperatures well below the room temperature. Indeed the liquefaction of these gases became the method of reaching lower and lower temperatures, culminating in the liquefaction of helium gas at about 4 K by H K Dnnes in This is however a separate story of marching towards the absolute zero of temperature [1, 2]. J H van der Waals gave a clear understanding of the continuity of the liquid and the gas states of matter in To study real gases he corrected the perfect gas equation of state pv=rt (1) by the inclusion of the mutual attraction between the gas molecules and the finite size of the gas molecules. This gives the ~ RESONANCE I April

2 Figure 1. L I p I Q U I o, I ~ FLUID ~T»TC liq + GAS T < Tc -----=>') V equation of state of a gas as [p + (a/vz)](v - b) = RT, (2 ) now called the van der Waals equation of state. The equation gives the p V isotherms as shown in Figure 1. At high temperatures T > > Tc the isotherms are nearly hyperbolic corresponding to the gas state. At low temperatures, T < < T c ' one has the liquid phase at high densities, the vapor phase at low densities and also the two phases coexisting at intermediate densities. The van der Waals equation of state has been very successful in describing the p VT thermodynamic behaviour of gases. The high temperature region T > > Tc corresponds to the Boyle's law behaviour of the gas p V = constant at a given temperature. A general description of this behaviour is conveniently done through the isothermal compressibility x = - (l/v) (0 VI8p)T' (3) which measures the response of the volume to a change in the pressure. In a perfect gas X is easily calculated to be x = (lip) = (VINkT), (4) showing that a gas is very compressible in the low pressure evacuated state as is to be expected. For the dense, say liquid state the compressibility is very small because the atoms or molecules are already packed in a dense state making it difficult to ~ RESONANCE I April 2000

3 compress them further. Note however that as per the van der Waals equation of state, the critical region near Tc'pc' Vc is unusual. Here the isothermal compressibility X = - (l1v)(a VI a p)1" being the reciprocal of the slope of the T = Tc isotherm becomes infinitely large. Clearly something unusual will happen here. Fluctuations Fluctuations are ubiquitous, but are often too small or too quick to be noticed. Normally one thinks that p, V, T and similar quantities are definite and well defined for any system. One however must realize that a quantity like the wind speed varies from place to place and from time to time. Similarly the temperature of a room varies between day and night. The normal processes of measurements are quick enough to measure these variations. Sometimes these variations can be rapid. When water is flowing in a river, there are eddies and other variations -of the speed which are averaged out when we talk of the speed of water in the river. Consider the surface of water in a can. Even when it is a steady state, there are ripples or fluctuations on the surface of the water. The scintillations of a star when seen through the atmosphere are a consequence of the rapid fluctuations in the atmosphere. These types of fluctuations are ubiquitous, but we often do not notice them because they occur in very short time scales or are very small to be noticed easily [3]. A little more thinking shows that a system will deviate or fluctuate from the average value easily if the system also responds easily to external influences or perturbations. Suppose we have a can of water and a can of heavy oil. Even if we strike the can feebly, water shows many more ripples on the surface than the heavy oil which is very viscous and does not respond easily to outside forces. In a similar way if a system is highly compressible, it will show large deviations of density about the steady state value as a result of the inevitable external perturbations on the system. A quantitative relation can be established from the principles of statistical physics. The average value of the deviations from the mean value of say the density p will become zero by the very process of averaging. However the square of the A little more thinking shows that a system will deviate or fluctuate from the average value easily if the system also responds easily to external influences or perturbations. -R-ES-O-N-A-N-C-E--I-A-p-ri-I ~~~

4 fluctuations will all be positive and so the average value of {p - (p) F will not be zero, where () indicates the average value of the quantity. A statistical mechanics calculation gives the average value of the mean square fluctuations as {p-(p) PI (p)2 = (ktxiv). (5) Under normal conditions in say, a gas, this is easily calculated from the earlier expressions to be equal to (l/n). The number of particles N is of the order of and so the mean square fluctuations in density will be unobservably small. However if X becomes very large then the fluctuations will also become very large. Einstein recognized that these large fluctuations are responsible for the very beautiful phenomenon of critical opalescence shown by fluids in their critical region. Light Scattering Rayleigh thus found that the molecular scattering by blue light is many times more than that by red light and so explained the blue colour of the sky as the result of scattering from the air molecules. The scattering of light was first studied in a careful manner by Tyndall in 1859, which is the reason why it is sometimes called Tyndall scattering. A major step to explain the phenomenon was taken in 1871 by Rayleigh. The electric polarisation induced in an atom or a molecule by the electromagnetic field is proportional to oj and the field radiated by such induced oscillating dipoles then becomes proportional to a}. This A,-4 wavelength dependence of the scattered radiation is characteristic of the scattering from molecules, particles or other entities whose dimensions are very small compared to the wavelength of the radiation. Rayleigh thus found that the molecular scattering by blue light is many times more than that by red light and so explained the blue colour of the sky as the result of scattering from the air molecules. Subsequently Tyndall scattering was studied not only from the molecular entities but also from other larger size particles. When the wavelength of light becomes comparable to the size of the scattering particles, like dust particles, the mutual interference from the scattering by the different portions of the dust particle become important and this is called Mie scattering. Here the wavelength dependence becomes complex unlike the simple A,-4 term of the Rayleigh process ~ RESONANCE I April 2000

5 Box 1. Observing Critical Opalescence. A simple experiment can be conducted to observe the striking phenomenon of critical opalescence. Take two commonly available liquids, methyl alcohol, CH 3 0H and carbon-di-sulphide, CS 2 in the weight proportion 20: 80. Pour the liquids into a glass container with a stopper to prevent the evaporation of the liquids and the consequent fumes. At room temperature the heavier liquid CS 2 will remain at the bottom and the lighter liquid CH 3 0H will remain at the top (Figure 2a). Heat the system slowly to say 60 C by immersing the container in a water bath atthe elevated temperature. The two liquids will mutually dissolve into each other and a single phase liquid will be observed above the critical temperature 36 C (Figure 2e). (The critical temperature is sensitive to impurities; 0.5 % of water in the alcohol will increase Tc by nearly 2.5 C. ) The critical opalescence will be observed when the two liquids gradually merge into one ~ ~ _. ~ :. :1. ~ T«Tc (a) T~Tc (b) T = Tc (c) T>Tc (d) T»Tc (e) and the phenomenon will be seen more strikingly when the system is cooled slowly from the one phase region. As Tc is approached a dark region will start appearing at the place where the new meniscus is to form. (Figure 2d). The band will grow to cover almost the entire liquid near Tc (Figure 2c). In reflected light the material will appear milky white. Because of the strong scattering oflight the material will appear to 'glimmer' or 'opalesce'. At the critical temperature the meniscus will start appearing in the middle of the strong dark brown colouration (Figure 2c). When the liquid is cooled below Tc the opalescence will also die down (Figure 2b) gradually from its maximum value. The mutual solubility of one liquid in another is the analogue of the liquid-gas miscibility. The fluctuation dominated aspects of critical phenomena can be studied equally well with binary liquid systems as well as with gas-liquid systems or magnetic systems near their respective critical regions. The observation of the critical opalescence with its strong temperature dependence and the large magnitude of the effect was however very dramatic and could be studied without any special apparatus. It was this striking phenomenon which Einstein clarified as a part of his early studies on statistical physics. He realized that the fluctuations in density in the critical region would lead to the corresponding fluctuations in the refractive index of the medium. These refractive index variations would behave like atomic size scatterers and give rise to the scattering -R-ES-O-N-A-N-C-E--I-A-p-r-il ~

6 The next crucial step was taken by Smoluchowski in 1908 to link the fluctuations in density with the fluctuations of refractive index and hence the scattering of light. This step was completed by Einstein in 1910 with a detailed calculation of the intensity of the scattered light. of light. If the fluctuations are large, the light scattering also becomes large. The 1905 paper of Einstein on Brownian motion, calculating the mean square motions of suspended particles, gave the first idea of how to tackle the problem. Einstein then followed this in 1906 with another paper on the general theory of Brownian motion. In this paper he started with the Boltzmann relation between entropy S and probability W, namely S = kin W + So' (6) A state of entropy x away from the most probable state Xo could be called as a fluctuation state with entropy x. Then one can reframe the above equation as with suitable normalizing factors. This idea was seized by the Polish scientist Marion Smoluchowski to formulate in the same year the more general ideas of treating the entropy of the fluctuating states. The next crucial step was also taken by Smoluchowski in 1908 to link the fluctuations in density with the fluctuations of refractive index and hence the scattering of light. This step was completed by Einstein in 1910 with a detailed calculation of the intensity of the scattered light. (7) Einstein first calculated the mean square amplitude of the fluctuations of density in the case of pure fluids and of composition in the case of fluid mixtures. He then took into account the wavelets scattered from adjacent portions of the media and obtained an expression for the intensity of the scattered light. The differential scattering cross-section (l/v') ds/d W is a convenient quantity to compare with the experiments. It is the flux of light or power scattered into a unit solid angle per unit scattering volume per unit incident intensity oflight and is often called the Rayleigh ratio R. The expression obtained by Einstein for the pure fluid case was ~ RESONANCE I April 2000 (8)

7 Box 2. Einstein and Srnoluchowski. It is interesting that both Albert Einstein and Marion Smoluchowski were working somewhat independently of each other to contribute very significantly to the ideas of fluctuations and related phenomena in statistical physics. Both had obvious admiration for each other. The first beautiful paper of Einstein in 1905 had the title 'Uber die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierted Teilchen' ('On the motion of particles, suspended in stationary liquids and necessitated by the molecular kinetic theory ofheat'), Annalen der Physik, 17 (1905) This was followed a year later by another significant article' Zur Theorie de Brownschen Bewegung' ('Towards the theory of Brownian motion') Annalender Physik 19 (1906) The interestingpaperofsmoluchowski was published in the same year 'Zur k,inetischen Theorie der Brownshen molekular Bewegung und der Suspensionen' ('Towards the kinetic theory of Brownian motions and of suspensions'), Annalen der Physik, 21 (1906) Smoluchowski generously refers to Einstein's work on p.772 ofthis paper with the words "... eine Diskussion der von Einstein befolgten, sehr sinnreichenmethoden einzugehen..." ("... to go into a discussion of the very thoughtful method followed by Einstein..."). The 1908 paper of Smoluchowski linking the fluctuations with the critical opalescence had the title 'Molekular-kinetischen Theorie der Opaleszenz von Gasen im kritischen Zustande, sowie einiger verwandter Erscheinungen' ('Molecular kinetic theory of opalescence of gases in their critical region and of some allied phenomena') Annalen der Physik, 25 ( 1908) The paper of Einstein, giving a complete calculation of the light scattering was in 1910, was titled 'Theorie der Opaleszenz von homogenen Flussigkeiten und Flussigkeitsgemischen in der Nahe des kritishchen Zustandes' ('Theory of the opalescence of homogeneous fluids and fluid mixtures in the neighbourhood ofthe critical state') Annalen der Physik, 33 (1910) ~ It starts on p 1275 with the words "Smoluchowski hat in einer wichtigen theoretischen Arbeit gezeigt..." (" Smoluchowski has shown in an important theoretical work that..." ). Clearly both the persons had great respect for each other's works and were generous in giving credit. In the beginning both were not working at the great centers of research. As is well known Einstein was in the Patents Office, Berne, in 1905 and Smoluchowski in the Lvov University, Lemberg. By 1910 Einstein moved to Zurich and later to Berlin, while Smoluchowski got a position in Cracow by 1913, just four years before his premature death. Einstein himself wrote the obituary note of Smoluchowski in the journal Naturwissenschaften in A good account of their work in establishing the foundations of fluctuation theory is given by Chandrasekhar (1943). where' ' is the dielectric constant. Because of the very significant contributions by Smoluchowski, the equation is sometimes called the Einstein-Smoluchowski equation. Near T the isoc thermal compressibility becomes very large and this makes the light scattering enormously large in the critical region. Einstein also obtained the expression to be used in the case of liquid mixtures where the concentration fluctuations are responsible for the fluctuations in the refractive index of the system. These -R-ES-O--N-A-N-C-E--I-A-p-ri-I ~~

8 Suggested Reading [1] K Mendelssohn, The Quest for Absolute Zero, Weidenfeld & Nicholson Pub. Lid., London, [2] R Srinivasan, Resonance, Vol.l, No.12, p.6, [3] Sriram Ramaswamy, Resonance, VoI.5, No.3, p.16, [4] K R Rao, Resonance, Vo1.2, No.8, p.26, [5] HE Stanley, Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford, [6] B Chu, Laser light Scattering, Academic Press, New York, [7] S Chandrasekhar, Reviews of Modern Physics, 15, 1, expressions were used immediately and successfully to explain and analyze the critical opalescence phenomena, taking into account the multiple scattering of light. This and the experimental verification of the Brownian motion expressions by Perrin soon afterwards became the striking vindications of the molecular statistical theory in the early part of the twentieth century. Later Developments A few remarks on the subsequent developments in the study of critical opalescence should also be made before concluding this short account. Einstein had assumed in his calculations that there is no spatial correlation among the fluctuations. Ornstein and Zernike showed in 1914 that when the fluctuations are growing to be infinitely large the correlations between the different regions of the fluctuations must be taken into account. They introduced the radial distribution function and the structure factor for this purpose and this technique has become the standard way of treating the structural aspects of solids and liquids [4]. In the 1960s it was recognised that the critical fluctuations in a variety of cases like gas-liquid systems, magnetic transitions, binary liquid systems etc., all become very large in their critical regions and that the fluctuation derived properties dominate over the normal equilibrium thermodynamic behaviour characteristic of the system far away from the critical region. Thus the behaviour of these materials in their critical region has universal features, the so-called universality of critical phenomena [5]. The divergence of the isothermal compressibility to infinity (and of the analogous quantities in the other systems) is characterised by critical power law exponents which have universal values. The correlation length among the fluctuations also becomes infinitely large at the critical point, characterised by another universal power law exponent. Lasers became available as very powerful monochromatic and monodirectional sources of light to study the many delicate aspects of light scattering phenomena. A wealth of new and interesting results have come out of these investigations ~ R-E-S-O-N-A-N-C-E-I---A-pr-il

9 about the nature of the universality of critical phenomena and about the subtle features of the light scattering interactions. These are nicely covered by Chu [6]. One cannot help being awestruck by the genius of Einstein whose work not only answered many questions of that period of time but also gave rise to investigations which are being carried out long after his pioneering studies were published. Address for Correspondence E S R Gopal Department of Physics Indian Institute of Science Bangalore , India. GBS on Albert Einstein In 1930, in an entertaining after-dinner toast to Einstein, who was present, George Bernard Shaw made the following remarks: "Religion is always right. Religion solves every problem and thereby abolishes problems from the universe. Religion gives us certainty, stability, peace and the absolute. It protects us against progress which we all dread. Science is the very opposite. Science is always wrong. It never solves a problem without raising ten more problems." Shaw then continued: Copernicus proved that Ptolemy was wrong. Kepler proved that Copernicus was wrong. Galileo proved that Aristotle was wrong. But at that point the sequence broke down, because science then came up for the first time against that incalculable phenomenon, an Englishman. As an Englishman, Newton was able to combine a prodigious mental faculty with the credulities and delusions that would disgrace a rabbit. As an Englishman, he postulated a rectilinear universe because the English always use the word 'square' to denote honesty, truthfulness, in short: rectitude. Newton knew thatthe universe consisted of bodies in motion, and that none of them moved in straight lines, nor ever could. But an Englishman was not daunted by the facts. To explain why all the lines in his rectilinear universe were bent, he invented a force called gravitation and then erected a complex British universe and established it as a religion which was devoutly believed in for 300 years. The book ofthis Newtonian religion was riot that oriental magic thing, the Bible. It was that British and matter-of-fact-thing, a Bradshaw. It gives the stations of all the heavenly bodies, their distances, the rates at which they are travelling, and the hour at which they reach eclipsing points or crash into the earth. Every item is precise, ascertained, absolute and English. "Three hundred years after its establishment a young professor rises calmly in the middle of Europe and says to our astronomers: 'Gentlemen, if you will observe the next eclipse of the sun carefully, you will be able to explain what is wrong with the perihelion of Mercury. ' The civilized Newtonian world replies that, if the dreadful thing is true, if the eclipse makes good the blasphemy, the next thing the young professor will do is to question the existence of gravity. The young professor smiles and says that gravitation is a very useful hypothesis and gives fairly close results in most cases, but that personally he can do without it. He is asked to explain how, if there is no gravitation, the heavenly bodies do not move in straight lines and run clear out of the universe. He replies that no explanation is needed because the universe is not rectilinear and exclusively British: it is curvilinear. The Newtonian universe thereupon drops dead and is supplanted by the Einstein universe. Einstein has not challenged the facts of science but the axioms of science, and science has surrendered to the challenge. " (Blanche Patch (Shaw's secretary), 'Thirty years with G.B.S.', Gollancz, London, 1951.) -RE-S-O-N-A-N-C-E--I-A-p-ri-I ~~

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