Most liquids similarly diffuse into each other at a snail's pace when measured by the minute hand of a clock. Yet the rate is high enough to account for much of the vital mixing that goes on in the universe, including the transport fluids across the microscopic distances that are encountered inside the tissues of plants and animals. The rate at which diffusion proceeds is determined by such factors as the absolute temperature of diffusing fluids, their viscosity and the relative size of their constituent molecules.

In 1905 Albert Einstein wrote an equation that relates the size of molecules to the rate at which they diffuse. Few amateurs have used the theory for exploring molecular migrations, because the classical apparatus for investigating the diffusion of liquids, the Tiselius cell, includes such hard-to-make parts as a rectangular glass vessel with parallel and optically flat sides, an interferometer, and an optical train of lenses and filters under the control of precision micrometers. Moreover, the Tiselius cell is tedious to use: the observations may extend over several hours or days, depending on the size of the diffusing molecules.

Last year an elegant solution to the problem of measuring diffusion rates was devised in the form of a simple and inexpensive apparatus by Yasunori Nishijima of the University of Kyoto in Japan and Gerald Oster of the Polytechnic Institute of Brooklyn. Their device consists of a microscope of the kind that sells for about $15, a pair of silvered microscope slides, a source of green light and a microscope camera attachment that can be built at home. The apparatus is as powerful as it is simple. Within five minutes one can determine the rate at which diffusion proceeds, the distance that a molecule migrates during a specified interval and the approximate radius of the molecule. Or, if the radius is known, one can confirm such values as Avogadro's number and the viscosity of the diffusing fluids.

Nishijima and Oster explain that the speed and economy of their technique derive from the fact that it is the average of the square of the distance over which a particle randomly wanders during diffusion that is proportional to the time of its travel. In other words, a particle that spends three minutes diffusing a distance of one millimeter requires nine minutes to wander two millimeters away. Expressed in still another way, the time varies inversely with the square of the distance traveled. Accordingly when one uses a microscope to measure the distance that a particle diffuses, the time required for observation is divided by the square of the microscope's magnifying power. For example, a microscope that magnifies the apparent size of the diffusing fluids by a factor of 50 reduces the time scale of observation by a factor of 2,500. This means that hours spent in observing diffusion processes with a Tiselius cell become seconds with Nishijima and Oster's microdiffusion apparatus. In addition to collapsing the time scale the microdiffusion apparatus requires very small amounts of specimen materials. Moreover, one can measure diffusion processes that proceed over microscopic distances to which the Tiselius cell is insensitive, such as those associated with living cells and the growth of synthetic fibers.

As with the Tiselius cell, the microdiffusion apparatus senses the movement of migrating particles by their effect on the optical properties of the solution into which they diffuse. The speed with which light travels through water is reduced by the addition of most solutes. This can be demonstrated by holding a glass of water up to a light and pouring a thick sugar solution into the water. A pattern of turbulence will be seen resembling the "heat waves" that rise above a hot radiator. The same effect on a smaller scale can be observed by placing a drop of clear water on the slide of a microscope, lighting the stage with a lens focused on a pinhole of light and adding a second drop of heavy sugar solution so that the two make contact.

A far more striking effect is observed when the drops are placed on a microscope slide that has been partially silvered and are covered by a similarly silvered slide that rests on a cover glass at one end. Because the slides are separated at one end the space between them has the form of a thin wedge. If the wedge, when dry, is lighted from the bottom, colored fringes will appear across the upper slide that resemble Newton's rings, the concentric circles of rainbow colors that can be seen if one sets a watch glass on a piece of plate glass. Optically the two effects are identical. Light waves transmitted by the bottom microscope slide are partially reflected by the silvering on the upper slide. Depending on the thickness of the wedge and the wavelength of the light, the light waves tend either to reinforce or to cancel each other. But the thickness of the wedge changes uniformly throughout its length. As a result octaves of color, created by the interference, appear as transverse fringes.

If the wedge contains a medium other than air-a medium that retards the speed of light more than air does-the fringes will crowd closer together as though the wedge had been opened somewhat. A wedge of water, for example, shows more fringes than one of air, and one of sirup shows still more. In short, an interferometric wedge can detect differences in the refractive properties of fluids. Nishijima and Oster took advantage of this effect not only to measure the differing refractive properties of two fluids but also to observe changes in refraction when one fluid diffuses into another. By observing changes in refraction they could investigate the process of diffusion.

Rainbow-colored fringes are not easy to distinguish because the colors of one fringe merge with those of neighboring fringes. If the interferometric wedge is illuminated by light of a single color, however, the fringes appear as alternately dark and light bands of the selected color, particularly if the light proceeds from a narrow source such as a pinhole.

A pair of silvered microscope slides lighted by the green rays of a mercury lamp constitute the essential element of Nishijima and Oster's apparatus. Their lamp is a General Electric Type H-100-A38-4, which operates from a Type 9T64Y3518 ballast. The lamp and ballast are priced at $12.75 and $16.50 respectively. The lamp is mounted at the focus of a lens about two inches in diameter that has a focal length of about two inches. The bundle of parallel rays that proceeds from the lens first passes through a water-filled box with glass sides, which absorbs heat radiation, and then through a filter that absorbs all colors except the desired green. (The filter is a Corning Type 4-64 two inches square; its cost is $4.10.) The filtered light falls on the flat side of the microscope mirror and is directed upward to a condensing lens that brings the rays to focus on a pinhole about .1 millimeter in diameter. This lens should have a diameter of about an inch and a focal length of a little less than an inch. The light is then made parallel again by a small lens, with a diameter of perhaps half an inch and a focal length of about half an inch, above the pinhole. Simple lenses can be used throughout. Color-corrected lenses, although satisfactory, are not needed because the light is monochromatic. The beam from the small lens above the pinhole proceeds through the interferometric wedge and into the objective lens of the microscope.

Brackets for supporting the lenses, the water cell and the filter can be improvised from scrap sheet metal and attached to a firm wooden base, on which the microscope also rests. Ideally the microscope slides should be silvered so that about 85 per cent of the light is reflected and 15 per cent transmitted, but coatings that reflect as little as 50 per cent or even less will work. Microscope slides specially silvered for use in making interferometric wedges can be bought from Henry Prescott of Northfield, Mass., for $3 a pair. The complete optical train is shown in the illustration Figure 1.

The technique of using the apparatus is best demonstrated by a specific example. Assume that the experimenter wants to know the rate at which granulated sugar diffuses into tap water and to measure the radius of the sugar molecule. A 10 per cent (by weight) sirup is made up and a silvered slide is placed on the stage of the microscope with the metalized side facing upward. A single drop of sirup is then placed in the center of the slide and a drop of tap water is added about an eighth of an inch away. A stack of two cover slips is then placed on one end of the slide, and the second slide, with its metalized side down, is placed gently on top to make the wedge. The time to the nearest second and the temperature of the room are recorded at the instant the drops come together.

After the mercury lamp has warmed up and reached normal operating intensity, select for observation a portion of the field where the sharp boundary between the liquids is at right angles to the fringes. Replace the eyepiece with a film holder and bring the image of the fringes to focus in the plane of the film. If a special microscope camera is not at hand, it is not at all difficult to improvise one [see "The Amateur Scientist"; SCIENTIFIC AMERICAN, February, 1961]. Make an exposure after about three minutes, noting the time to the nearest second. The resulting photograph should resemble the illustration in Figure 2. The fringes that correspond to the bulk of the two liquids appear as straight lines at the sides of the photograph. The refractive index across the boundary of the drops varies continuously and causes the fringes to curve at the center of the photograph. Each fringe, whether it is straight or curved, represents a contour line of constant optical distance. A line, AA', drawn parallel to the straight portions of the fringes in the region of constant refraction represents a line of constant geometrical thickness across the wedge. The change of the optical path along this line depends solely on the change of refractive index along the line. Hence the closer the fringes that cross the reference line, the greater the variation of refractive index along the line. The variation can be evaluated by dividing the reference line into any convenient number of equal parts and counting the fringes that cross each part The resulting values are plotted vertically against the divisions of the reference line and joined by a smooth curve, as shown in the graph reproduced below. The value of the magnification is established by substituting a metric scale for the silvered slides on the microscope stage and drawing a line on the photograph that is equal in length to the distance between a pair of magnified millimeter graduations.

A numerical term, the constant of diffusion, is then derived from the graph First, the area enclosed beneath the bell-shaped curve is measured. A planimeter can be used, or the measurement can be approximated closely by placing a sheet of translucent graph paper ruled with rectangular co-ordinates over the graph and counting the squares and fractions of squares that are included within the curve. The maximum height is determined by counting the column of squares that runs up through the top of the curve. In the case of the curve shown below, which corresponds to the fringes illustrated in Figure 2, assume that the height measures 12.7 squares and the area amounts to 61 squares. Assume also that the diffusing drops were in contact 180 seconds when the exposure was made and that the temperature of the room was 25 degrees centigrade.

These four figures representing the height and area of the curve, the time and the temperature are substituted in the top equation on this page, which relates the several factors that determine the diffusion constant. The square of 61 (or 3,721) divided by 4 X 3.14 x 180 X 12.7 X 12.7 (or 364,644) yields the quotient: 1.09 X 10^-2. Next, a scale-correction factor must be applied to transform the arbitrary dimensions of the graph paper into units of the centimeter-gram-second system. Comparative measurement discloses that 4.5 squares of the graph paper equal the length of the magnified scale representing one millimeter on the silvered slides. The metric height of the curve is therefore equal to 12.7 divided by 4.5 or 2.83 millimeters (.283 centimeters). This dimension is squared in the equation, so the quotient, 1.09 X 10^-2, must be multiplied by the square of the scale-correction factor expressed in centimeters: 1.09 X 10^-2 X (1/45)² = 5.4 X 10^-6, the value of the diffusion constant in square centimeters per second for a 10 per cent solution of sucrose diffusing into tap water.

The equation for computing the radius of spherical molecules [Figure 5] is just as simple to evaluate. First, the absolute temperature is found by adding the temperature of the room in degrees centigrade to 273. The result, in the example under discussion, is 298 degrees Kelvin. The absolute temperature must then be multiplied by Boltzmann's constant (the gas constant, 8.3 X 10⁷, divided by Avogadro's number, 6.03 X 10²³): 1.38 X 10^-16. The equation also includes a term for the viscosity of water, a property that varies with temperature. Reference texts list the viscosity of water at 25 degrees C. as 8.9 x 10^-3 poise. Performing the arithmetic, 1.38 X 10^-16 X 298 divided by 6 X 3.14 X 8.9 X 10^-3 x 5.4 x 10^-6 gives the quotient: 4.5 X 10^-8 centimeters, or 4.5 angstrom units-a reasonable value for the average radius of the sugar molecule.

It is also possible to compute the range through which diffusing molecules migrate by applying Einstein's theory of Brownian movement. The theory states that the root mean square displacement equals the square root of twice the product of the diffusion constant multiplied by the elapsed time to the observation. At the end of the first second sugar molecules will have migrated into the water a distance equal to the square root of 2 x 5.4 x 10^-6, or 3.2 X 10^-3 centimeters. After l80 seconds the increased distance will equal the square root of 2 X 5.4 X 10^-6 X 180, or 4.4 X 10^-2 centimeters.

Every now and again in the course of making an experiment the need arises for exposing photographic negatives under lighting conditions that are beyond the range of inexpensive exposure meters. The problem is usually solved by the time-honored technique of cut and try. This solution served R. B. Stambaugh of Chillicothe, Ohio, well enough for several years, but after making an inventory of his scrapped negatives last summer he decided to try for a less costly method of determining exposures. The result is the transistorized exposure meter illustrated in Figure 6.

"Today's high-speed photographic films and large-aperture lenses," he writes, "have made it possible to take pictures at 'available light' levels much too low for the average inexpensive light meter to handle. Sensitive exposure meters of wide range and high accuracy are commercially available-if one is willing to pay as much for them as for the camera-but they are hard to use and invite all sorts of error.

"After mulling over the problem of meter design for some months I eventually hit on an apparatus for measuring light that has proved to be reliable, adaptable to extreme ranges of measurement and inexpensive to build. The device consists essentially of a photosensitive cell, enclosed in a drum that has slot of variable width to admit light, and a meter for indicating amplified cell current. Turning the drum shifts the position of the variable slot and admits more or less light to the cell. The drum is adjusted until the meter indicates a predetermined value of current. At this adjustment the cell always 'sees' the same amount of light and the meter always indicates the same value of current, regardless of the intensity of the external light.

"Adequate sensitivity is provided by a two-transistor amplifier arranged in a balanced bridge circuit, as shown in the accompanying diagram [Figure 7]. The bridge circuit reduces the effects of temperature and change in voltage as the batteries age. The null balance principle of always reading a predetermined value of current eliminates the effect of nonlinearity in the response of the photocell to light and nonuniform amplification by the transistors.

"The entire assembly, including the miniaturized amplifier, drum and photocell, is housed in a box of sheet aluminum that is four inches long, three inches wide and two inches high.

"The slot in the drum is made by cutting a mask from a strip of opaque cardboard that fits smoothly around the juice-can drum. Ten areas are marked off on the strip. The width of each rectangular area is determined from the size of the aperture drum and the markings of the f-number dial. In my instrument, the dial covers nine f numbers in a 180-degree turn, or 20 degrees per f number. Twenty degrees corresponds to .37 inch on my drum, which is 2 1/8 inches in diameter. One cannot easily hand-cut a slot much narrower than about 1/16 inch with the required precision. So I joined two tapered slots end to end and covered the second one with a 16X filter made from exposed film. This stunt enabled me to increase the width of the last five rectangles 16-fold. It is not necessary to cut the slots as a series sharp steps; one needs only to draw a curve that connects the midpoints of the rectangles as shown in the accompanying illustration [Figure 8].

"The filter of desired density was made by taking 16 'pictures' of a blank wall while gradually increasing the exposure. A hole with an area one-sixteenth as large as the sensitive area of my selenium cell was then cut in a piece of opaque paper and used to mask the cell. The masked cell, connected to a microammeter, was exposed to the lamp of my enlarger at a distance such that the pointer of the meter was deflected to half-scale. The paper mask was then replaced by the film. The film was shifted until an area was found that deflected the meter to half-scale. This area was selected as the 16X filter.

"After assembly the meter was calibrated against a good commercial exposure meter by turning the knob until the microammeter read half-scale (25 microamperes) while both instruments were pointed toward a flat-lighted piece of white cardboard. The correct shutter speed, f number and film speed indicated by the commercial meter were transferred to the homemade meter by loosening the setscrew of the dial and, without disturbing the position of the aperture drum, rotating the dial until the f-number scale was opposite the proper shutter speed. When the setscrew in the knob was tightened, the meter was calibrated for every light level within its range of sensitivity.

"The dark zero balance normally will stay in adjustment unless there is an appreciable change in temperature, such as going from indoors to outdoors in winter. Zero balance is checked simply by pressing the 'on' button while holding one hand over the front opening. Balance is established by adjusting the potentiometer with a screwdriver until the meter reads zero when the cell is in darkness.

"It is apparent that the sensitivity of the meter can be increased by substituting a more sensitive photocell for the one used in this design, by increasing the amplification or by changing both the cell and the amplification. I am now testing a circuit that uses a cadmium sulfide cell, for example, which appears to be many times more sensitive than either the selenium or silicon type. The design invites other modifications, particularly with respect to miniaturization."

Bibliography

DIFFUSION UNDER THE MICROSCOPE. Yasunori Nishijima and Gerald Oster in Journal of Chemical Education, Vol. 38, No. 3, pages 114-117; March, 1961.

AN INTRODUCTION TO INTERFEROMETRY. S. Tolansky. Longmans, Green & Co., Inc., 1955.

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