Young-Laplace equation drop. sessile drop method

Where s SV , s SL , s LV- coefficients of surface tension of interfacial boundaries solid - gas (steam), solid - liquid, liquid - gas, respectively, q- edge angle. This ratio is now known as Young's formula. This work, however, did not have such an impact on the development of science in this direction, which was published a few months later by Laplace's article (Pierre Simon Laplace). This seems to be due to the fact that Jung avoided the use of mathematical notation, but tried to describe everything verbally, which makes his work seem confusing and unclear. Nevertheless, he is considered today one of the founders of the quantitative theory of capillarity.

The phenomena of cohesion and adhesion, the condensation of vapor into a liquid, the wetting of solids by liquids, and many other simple properties of matter - all indicated the presence of attractive forces many times stronger than gravity, but acting only at very small distances between molecules. As Laplace said, the only condition that follows from observable phenomena and is imposed on these forces is that they are "imperceptible at sensible distances."

The repulsive forces created more trouble. Their presence could not be denied - they must balance the forces of attraction and prevent the complete destruction of matter, but their nature was completely unclear. The question was complicated by the following two erroneous opinions. First, it was often assumed that the acting force of repulsion was heat (as a rule, the opinion of supporters of the caloric theory), since (such was the argument) when a liquid first expands and then boils when heated, so that the molecules are separated over much greater distances than in a solid body. The second erroneous opinion arose from the notion, which leads back to Newton, according to which the observed pressure of a gas is due to static repulsion between molecules, and not due to their collisions with the walls of the vessel, as Daniel Bernoulli argued in vain.

Against this background, it was natural that the first attempts to explain capillarity, or in general the cohesion of liquids, were based on the static aspects of matter. Mechanics was a well understood theoretical branch of science; thermodynamics and kinetic theory were still in the future. In mechanical considerations, the key was the assumption of large but short-range attractive forces. Liquids at rest (whether in a capillary tube or outside it) are obviously in equilibrium, and therefore these attractive forces must be balanced by repulsive forces. Since even less could be said about them than about the forces of attraction, they were often passed over in silence, and, in the words of Rayleigh, "the forces of attraction were left to perform the unthinkable trick of balancing themselves." Laplace 2 was the first to satisfactorily solve this problem, believing that repulsive forces (thermal, as he assumed) can be replaced by internal pressure, which acts everywhere in an incompressible fluid. (This assumption leads at times to ambiguity in 19th-century writings as to what is strictly meant by "pressure in a fluid.") Here is Laplace's calculation of internal pressure. (This conclusion is closer to the conclusions of Maxwell and Rayleigh. The conclusion is given by.)

It must balance the cohesive forces in the fluid, and Laplace identified this with the force per unit area that resists the separation of an infinite fluid body into two far-separable semi-infinite bodies bounded by flat surfaces. The derivation below is closer to those of Maxwell and Rayleigh than to Laplace's original form, but there is no significant difference in the argument.

Let us consider two semi-infinite fluid bodies with strictly flat surfaces, separated by an interlayer (thicknesses l) a pair with a negligible density (Fig. 1), and in each of them we single out a volume element. The first is in the upper body at a height r above the flat surface of the lower body; its volume is dxdydz. The second is in the lower body and has a volume , where the origin of polar coordinates coincides with the position of the first elementary volume. Let f(s) is the force acting between two molecules separated by a distance s, a d- the radius of its action. Since this is always an attractive force, we have

If a  is the density of the number of molecules in both bodies, then the vertical component of the interaction force of two volume elements is equal to

The total attractive force per unit area (positive value) is

(3)

Let u(s) - potential of intermolecular force:

Integrating by parts again, we get

(6)

Internal Laplace pressure K is the force of attraction per unit area between two flat surfaces when they are in contact, i.e. F(0):

(7)

Where is the volume element, which can be written as
. Because the u(r) by assumption is everywhere negative or equal to zero, then K positively. Laplace believed that K large compared to atmospheric pressure, but Jung had to make the first realistic numerical estimate.

The above derivation is based on the implicit assumption that the molecules are uniformly distributed with a density r, i.e. the liquid does not have a discernible structure on a scale of dimensions commensurate with the range of forces d. Without this assumption, it would be impossible to write expressions (2) and (3) in such a simple form, but it would be necessary to find out how the presence of a molecule in the first volume element affects the probability of the presence of a molecule in the second.

The tension per unit length along an arbitrary line on the liquid surface must be equal (in the appropriate system of units) to the work expended to create a unit area of the free surface. This follows from the experiment on liquid film stretching (Fig. 2).

A liquid film is held on the wire frame, attached by the right edge to a freely movable wire. Strength F, required to balance the tension in the double-sided film, is proportional to the length L. Let F = 2sl. Wire offset by distance dx needs work Fsdx=sdA, where dA- increase in area. Thus, the tension per unit length on a single surface, or surface tension s, numerically equal to the surface energy per unit area.

The value of this work can be immediately obtained from expression (6) for F(l). If we take two semi-infinite bodies in contact and separate them by a distance exceeding the radius of action of intermolecular forces, the work per unit area will be defined as

(8)

When separated, two free surfaces are formed, and therefore the work expended can be equated to twice the surface energy per unit area, which is equal to the surface tension:

(9)

Thus, K is the integral of the intermolecular potential, or its zero moment, and H- his first moment. While K inaccessible to direct experiment, H can be found if we can measure the surface tension.

Let ### is the density of cohesive energy at some point in the liquid or gas, i.e. attitude dU/dV where dU- internal energy of small volume ###V liquid or gas containing this point. For the molecular model, we take

(10)

Where r- distance from the considered point. Rayleigh identified the Laplacian K with the difference of this potential 2 ### between a point on a flat surface of the liquid (value 2 ### S) and a dot inside (value 2 ### I). On the surface, integration in (10) is limited by a hemisphere of radius d, and in the inner region is carried out over the entire sphere. Consequently, ### S have half ### I, or

(11)

Consider now a drop of radius R. Calculation f I does not change, but when received f S the integration is now over a more limited volume due to the curvature of the surface. If a ### is the angle between the vector and the fixed radius, then

Then the internal pressure in the drop is

Where H is determined by equation (9). If we took not a spherical drop, but a portion of a liquid with a surface determined by two main radii of curvature R 1 and R 2 , then we would get the internal pressure in the form

(14)

By Euler's theorem, the sum is equal to the sum of the reciprocal radii of curvature of the surface along any two orthogonal tangents.

Because K and H positive and R is positive for a convex surface, then it follows from (13) that the internal pressure in a drop is higher than in a liquid with a flat surface. Conversely, the internal pressure in a fluid bounded by a concave spherical surface is lower than in a fluid with a flat surface, since R in this case it is negative.

These results form the basis of Laplace's theory of capillarity. The equation for the pressure difference (liquid pressure inside a spherical drop of radius R) and (gas pressure outside) is now called the Laplace equation:

(15)

Three ideas are enough - tension at the surface, internal pressure and contact angle, as well as expressions (1) and (15) to solve all problems of ordinary equilibrium capillarity by methods of classical statics. Thus, after the work of Laplace and Young, the foundations of the quantitative theory of capillarity were laid.

Young's results were obtained later by Gauss by the variational method. But all these works (Jung, Laplace and Gauss) had one common flaw, a flaw, so to speak. This shortcoming will be discussed later.

When calculating the pressure inside a curved liquid surface, the Rayleigh potential 2 was introduced ### (10); along the way, it was noted that ### I is the cohesive energy density. For the first time this useful concept was introduced in 1869 by Dupre, who defined it as the work of crushing a piece of matter into its constituent molecules (la travail de désagrégation totale - the work of complete disaggregation).

Inward force acting on a molecule at depth r , is opposite in sign to the outward force that would arise from the molecules in the shaded volume if it were filled uniformly with a density r.

He quotes the conclusion drawn by his colleague F. J. D. Massier as follows. The force acting on the molecule near the surface towards the volume of the liquid is opposite in sign to the force arising from the shaded volume in Fig. 3, since inside the liquid the force of attraction from the spherical volume of radius is equal to zero from symmetry. So the inward force is

This force is positive because f(0 s d) F( d) = 0 due to the oddness of the function f(s). No force acts on a molecule unless it is within a distance d either side of the surface. Therefore, the work done to remove one molecule from a liquid is

Because the u(r) is an even function. This work is equal to minus twice the energy per molecule required to disintegrate the liquid ( doubled, so as not to count the molecules twice: once when they are removed, the other time as part of the medium):

(18)

This is a simple and understandable expression for the internal energy U liquid containing N molecules. Hence it follows that the density of cohesive energy ### is given by expression (10), or

(19)

What coincides with (11) if we remove the index I. Dupre himself obtained the same result in a roundabout way. He counted dU/dV through work against intermolecular forces with a uniform expansion of the liquid cube. This gave him

(20)

Because the K has the form
((7) and (11)), where the constant a is given by

(21)

That integration (20) again leads to (19).

Rayleigh was critical of Dupre's conclusion. He believed that consideration of the work of a homogeneous expansion from a state of balance of cohesive and repulsive intermolecular forces, taking into account only cohesive forces, was unreasonable; before taking such a step, one should have a better knowledge of the nature of the repulsive forces.

We see that in this conclusion, as well as in the conclusions of Young, Laplace and Gauss, the assumption of an abrupt change in the density of the number of molecules of a substance at the phase boundary is essentially used. At the same time, in order for the above reasoning to describe real phenomena in matter, it is necessary to assume that the radius of action of intermolecular forces in matter is much larger than the characteristic distance between particles. But under this assumption, the interface between two phases cannot be sharp - a continuous transitional density profile must arise, in other words, a transition zone 3 .

Attempts have been made to generalize these findings to a continuous transient profile. In particular, Poisson, trying to follow this path, came to the erroneous conclusion that in the presence of a transition profile, surface tension should disappear altogether. Maxwell later showed the fallacy of this conclusion.

However, the very assumption that the radius of action of intermolecular forces in a substance is much larger than the characteristic distance between particles does not correspond to experimental data. In fact, these distances are of the same order. Therefore, a mechanistic consideration in the spirit of Laplace is, in modern terms, the mean field theory. The same is the van der Waals theory, not described here, which gave the famous equation of state for real gases. In all these cases, an accurate calculation requires taking into account the correlations between the particle number densities at different points. This makes the task very difficult.

In the sessile drop method, a liquid of known surface tension is placed on a solid surface using a syringe. The droplet diameter should be between 2 and 5 mm; this ensures that the contact angle is independent of the diameter. In the case of very small droplets, the influence of the surface tension of the liquid itself will be large (spherical drops will form), and in the case of large drops, gravitational forces begin to dominate.

The sessile drop method measures the angle between a solid surface and a liquid at the point of contact of the three phases. The ratio of interfacial and surface tension forces at the point of contact of the three phases can be described by the Young equation, on the basis of which the contact angle can be determined:

A special case is the "captive bubble" method: the contact angle is measured below the surface in the liquid.

Initially, measurements were made using a goniometer (a hand-held contact angle measuring instrument) or a microscope. Modern technologies make it possible to record an image of a drop and obtain all the necessary data using programs.

Static contact angle

With the static method, the droplet size does not change during the entire measurement, but this does not mean that the contact angle always remains constant. On the contrary, the influence of external factors can lead to a change in the contact angle over time. Due to sedimentation, evaporation, and similar chemical or physical interactions, the contact angle will spontaneously change with time.

On the one hand, the static contact angle cannot absolutely estimate the free energy of a solid surface, and on the other hand, it allows one to characterize the time dependence of such processes as ink drying, glue application, absorption and adsorption of liquids on paper.

Changes in properties over time (spreading of a drop) often interfere with research. A spot, a scratch on the sample can also act as a source of error, any inhomogeneous surface will have a negative effect on measurement accuracy, which can be minimized in dynamic methods.

Dynamic contact angle

When measuring the dynamic contact angle, the syringe needle remains in the drop and its volume changes at a constant rate. The dynamic contact angle describes the processes at the solid/liquid interface during an increase in the volume of the drop (incoming angle) or during a decrease in the drop (outflow angle), i.e. during wetting and drying. The boundary does not form instantly; it takes time to achieve dynamic equilibrium. From practice, it is recommended to set the fluid flow to 5 - 15 ml/min, a higher flow rate will only mimic dynamic methods. For highly viscous liquids (e.g. glycerol), the rate of droplet formation will have different limits.

Leaning angle. During the measurement of the impinging angle, the syringe needle remains in the drop throughout the entire experiment. First, a droplet with a diameter of 3-5 mm is formed on the surface (with a needle diameter of 0.5 mm, which is used by KRUSS), and then it spreads over the surface.
At the initial moment, the contact angle does not depend on the droplet size, because strong forces of adhesion with the needle. At a certain droplet size, the contact angle becomes constant, and it is at this moment that measurements must be taken.
This type of measurement has the highest reproducibility. Increasing angles are usually measured to determine the free energy of a surface.

Leaky corner. During the outflow angle measurement, the droplet size decreases as the surface is dried: a large drop (approximately 6 mm in diameter) is placed on the surface and then slowly reduced by suction through the needle.
From the difference between the advancing angle and receding angle, one can conclude that the surface is uneven or chemically inhomogeneous. The receding angle is NOT suitable for SEP calculation.

Methods for assessing the shape of a sessile drop

Young-Laplace method. The most time-consuming, but also the most accurate method for calculating the contact angle. In this method, when constructing the contour of a drop, corrections are taken into account that not only interfacial interactions destroy the shape of the drop, but also the own weight of the liquid. This model assumes that the drop shape is symmetrical, so it cannot be used for dynamic contact angles. For an incoming drop, the contact angle can also only be determined up to 30°.

Length-width method. In this method, the spreading length of a drop and its height are estimated. The contour, which is part of a circle, is inscribed in a rectangle and the contact angle is calculated from the ratio of width and height. This method is more accurate for small droplets, the shapes of which are closer to a sphere. Not suitable for dynamic contact angle as the needle remains in the drop and the height of the drop cannot be accurately determined.

circle method. In this method, the drop is represented as part of a circle, as in the length-width method, however, the contact angle is calculated not using a rectangle, but using a segment of a circle. But unlike the length-width method, the needle remaining in the drop has less effect on the measurement results.

Tangential method 1. The full contour of a sessile drop is fitted to the equation of a conical segment. The derivative of this equation at the point of intersection of the contour and the baseline gives the angle of inclination at the point of contact, i.e. edge angle. This method can be used with dynamic evaluation methods as long as the drop is not severely broken by the needle.

Tangential method 2. The part of the contour of a sessile drop located near the baseline is adapted to a polynomial function of the type y=a + bx + cx 0.5 + d/lnx + e/x 2 . This function was obtained as a result of numerous mathematical simulations. The method is considered accurate but sensitive to contaminants and foreign matter in the liquid. Suitable for determining dynamic contact angles, but it requires precise imaging, especially at the point of phase contact.

The sessile drop method is implemented in DSA contact angle instruments, which are widely used in laboratories to study the properties of surfaces. These devices also allow you to measure the surface and interfacial tension of liquids.

SELECTING THE CONTOUR OF A LIQUID DROP IN THE PROBLEM OF DETERMINING THE SURFACE TENSION

Mizotin M.M. 1 , Krylov A.S. 1 , Protsenko P.V. 2

1 Moscow State University named after M.V. Lomonosov, faculty of VMK

2 Moscow State University named after M.V. Lomonosov, Faculty of Chemistry

Introduction

Surface tension is one of the most important properties of a liquid, and its accurate measurement is essential for the study of various phenomena and the development of technological processes. There are a number of ways to measure surface tension, but among all of them, the sessile or hanging drop method can be distinguished. The main advantages of the method are a very wide range of applications - from light fluid liquids to liquid metals, and the relative simplicity of the experimental setup compared to other methods. Moreover, in connection with the development of digital computing and photographic equipment, it became possible to perform analysis almost instantly.

The essence of the method is as follows: a drop is placed on a horizontal substrate (lying drop method) or suspended on a capillary tube (hanging drop method) and then its photograph in profile is studied. Measuring the geometric parameters of an equilibrium drop, the shape of which is determined by the ratio of the density and surface tension of the liquid, makes it possible to restore the required surface tension. The installation diagram is shown in Fig. one.

Rice. one. 1 – light source (lamp or microscope mirror), 2 – drop on the substrate,

3 - microscope with a digital camera.

Despite the rather well-developed experimental technique, a special expensive setup is still required for capturing a drop. In this paper, we propose an algorithm for an experimental setup of widely available components. The disadvantages of the setup compared to laboratory equipment are compensated by the proposed image processing methods.

sessile drop method

The basic equation of the sessile drop method, the Young-Laplace equation, describes the surface of a drop with rotational symmetry on a horizontal substrate. To solve this problem, an effective technique was proposed, subsequently improved and supplemented.

This technique is based on the numerical differentiation of the Young-Laplace equation. In order to differentiate the Young-Laplace equation, a parametrization of the curve is introduced
, where t is the length of the arc of the curve from the top of the drop (Fig. 2).

Rice. 2. Parameterization of the drop contour.

This parametrization satisfies the condition
, and leads to the system of equations

(1)

with initial conditions
,
,
,
and additional condition
. In the developed software package, the Cauchy problem (1) is solved by the Runge-Kutta method of the fourth order of accuracy.

To restore the parameters of a sessile drop, it is necessary to solve the inverse problem of determining the capillary constant
, droplet apex coordinates
and its radius of curvature as a function of the radius of the horizontal section of the drop on the height above the substrate. This function is measured with an error and, in some cases, measurements of only a part of the drop contour are available. When solving this inverse problem, the error (2) is minimized

between experimental points
and the curve obtained as a result of the numerical solution of problem (2). The difference between the experimental points and the curve is defined as the root of the sum of squared distances from each experimental point to the curve.

In this regard, the following problem of image processing arises: automatic obtaining of the contour of a drop, which is complicated by the presence of dust and debris in the images (which is associated with the use of a conventional camera in "domestic" conditions), as well as variable lighting conditions.

Error function

One of the main parts of the method is the calculation of the error function (2). Calculating the distance between a point and a curve (3)

In this case, it is very difficult, because are unknown to us, and they also need to be found numerically by the one-dimensional search method.

The following algorithm is proposed for efficient calculation of the error function. First, it is necessary to sort all the experimental points so that with increasing point number i the parameter corresponding to it also increased. Then, when searching for a parameter for each next point, you can use the value of the parameter as an initial approximation , and for the first point the initial approximation will be
. See below for more details on drop contouring.

Secondly, the calculation of the error function can be carried out directly in the process of integrating system (1) using the Runge-Kutta method. Indeed, at each iteration, the values are available to us, and the smallest distance from the point can be found by solving equation (4)

Newton's method. That is, when integrating system (1) numerically, it is necessary to monitor the value of function (4) for each next point, and remember the values of the smallest errors, if necessary, reducing the step by to increase the accuracy of the results.

Drop contour selection

As mentioned above, in order to efficiently calculate the error using formula (4), it is necessary to extract the drop contour from the image in such a way that with increasing point number i the parameter corresponding to it also increased. This operation is carried out in 2 stages: direct selection of edges using the Canny detector and selection of connected sequential sets of points from the resulting binary map of the edges.

The following algorithm was developed for edge tracking. First, it is necessary to carry out the edge thinning operation, since the Canny detector does not guarantee that all the received edges will be 1 pixel thick (this situation mainly occurs at the junctions), and such a condition is necessary for further processing. The edge thinning operation can be carried out using one of the known edge thinning methods. In this work, the algorithm was used.

Further processing is based on the analysis of the 3x3 pixel neighborhood around the considered pixel. On fig. The 3 pixel values in the neighborhood are represented by variables , which takes the value 0 or 1.

Rice. 3. 3x3 neighborhood around the pixel in question ,
.

The general scheme of the algorithm for selecting connected sequences of points:

If a
and
, then the intersection of the contours is located in the central pixel.

If a
and , then the end of the contour is in the central pixel.

At the same time, these conditions can be checked quickly and efficiently using lookup tables, since the total number of possible input values is 512 = 2 9 .

Start from one of the found ends of the contours.

Add the current pixel to the list of edge pixels under the current number and mark the current pixel on the edge map with the number of the current edge.

Find among the neighbors of the current pixel the pixel with value 1.

If the found neighbor is not the end of the contour or the intersection and is not marked on the edge map with any number yet, then move the current pixel to the position of the found neighbor and go to step 3. Otherwise, finish filling the current contour and go to the next one (step 2).

Conclusion

Experimental studies of the paraffin oil / decane system at various concentrations using the proposed algorithm have shown the effectiveness of the proposed approach.

The work was supported by the Federal Target Program "Scientific and Scientific-Pedagogical Personnel of Innovative Russia" for 2009–2013.

Literature

Maze C., Burnet G. A Non-linear Regression Method for Calculating the Surface Tension and Contact Angle from the Shape of a Sessile Drop // Surf. sci. 1969. V. 13. P. 451.

Krylov A. S., Vvedensky A. V., Katsnelson A. M., Tugovikov A. E. Software package for determination of surface tension of liquid metals // J. Non-Cryst.solids. 1993. V. 156-158. P. 845.

O. I. del Rio and A. W. Neumann. Axisymmetric Drop Shape Analysis: Computational Methods for the Measurement of Interfacial Properties from the Shape and Dimensions of Pendant and Sessile Drops // Journal of Colloid and Interface Science, Volume 196, Issue 2, 15 December 1997, Pages 136-147.

M. Hoorfar and A. W. Neumann. Recent progress in Axisymmetric Drop Shape Analysis // Advances in Colloid and Interface Science, Volume 121, Issues 1-3, 13 September 2006, Pages 25-49.

Canny, J., A Computational Approach To Edge Detection // IEEE Trans. Pattern Analysis and Machine Intelligence, 8(6):679–698, 1986

Lam L., Lee S.-W., Suen C.Y. Thinning Methodologies - A Comprehensive Survey // IEEE Transactions on Pattern Analysis and Machine Intelligence archive, Volume 14 Issue 9, September 1992.

Z. Guo and R. W. Hall, "Parallel thinning with two-subiteration algorithms", Comm. ACM, vol. 32, no. 3, pp. 359-373, 1989.

DROPLET EDGE DETECTION FOR SURFACE TENSION DETERMINATION

Mizotin M. 1 , Krylov A. 1 , Protsenko P. 2

1 Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics, Laboratory of Mathematical Methods of Image Processing,

2 Lomonosov Moscow State University, Department of Chemistry

Surface tension is one of the key propertied of liquid, thus its measurement is crucial for studying various phenomena such as wetting and development of technological processes. There sessile and pendant drop techniques are one of the most frequently used because of their universality and simplicity of measurement process.

The method is based on studying of the axisymmetric drop profile. The balance of gravity force and surface tension forms the distinct profile shape, thus surface tension can be calculated by the solution of the inverse problem for the Young-Laplace equation.

In this work the method of droplet contour extraction for determination of the surface tension is presented. The key difference of the proposed method is its orientation on inexpensive experimental setup using widely available components such as standard microscope, digital camera and substrate holder. Proposed techniques of image processing allow to avoid most of the problems concerning inferior quality of the drop images acquired by inexpensive setup retaining the measurement accuracy.

The work was supported by the federal target program ”Scientific and scientific-pedagogical personnel of innovative Russia in 2009-2013”.



APPLICATION OF THE MORPHOLOGICAL AMOEBA METHOD FOR ISOLATION
FROMVESSELS ON THE FUNCUS IMAGES

Nasonov A.V. 1 , Chernomorets A.A. 1 , Krylov A.S. 1 , Rodin A.S. 2

Moscow State University named after M.V. Lomonosov,

1 Faculty of Computational Mathematics and Cybernetics, Laboratory of Mathematical Methods of Image Processing /
2 Faculty of Fundamental Medicine, Department of Ophthalmology

In this paper, an algorithm for identifying vessels in fundus images based on the use of the method of morphological amoebas has been developed. The application of the algorithm to the problem of continuation of vessels from a set of points, which are known to be points of vessels, is considered.

1. Introduction

Fundus photographs are used to diagnose retinal diseases. Segmentation and evaluation of the characteristic values of the vessels of the circulatory system of the retina are of great interest in the diagnosis and treatment of many eye diseases.

Isolation of vessels in retinal images is a rather difficult task of image processing due to the high level of noise, uneven illumination, and the presence of objects similar to vessels. Among the methods for detecting vessels on fundus images, the following classes can be distinguished:

A class of methods that use image convolution with a two-dimensional directional filter and then find the response peaks. For segmentation of the vasculature, a two-dimensional linear filter is proposed, the profile of which is a Gaussian. The advantage of this approach is the stable finding of rectilinear sections of vessels and the calculation of their width. However, the method poorly detects thin and tortuous vessels, and false positives are possible for objects that are not vessels, such as exudates.

Methods using ridge detection. In B, primitives are found - short segments lying in the middle of the lines, then, using machine learning methods, primitives are selected that correspond to the vessels, along which the vascular tree is restored.

Methods that use vessel tracking, which includes both the connection of vessels by a pair of points, and the continuation of vessels. The advantages of this approach include the high accuracy of work on thin vessels and the restoration of ruptured vessels. The disadvantage is the complexity of processing branching and intersections of vessels.

Pixel-based classification based on the application of machine learning methods. Here, for each pixel, a feature vector is constructed, on the basis of which it is determined whether the pixel is part of the vessel or not. To train the method, images of the fundus with vessels marked on it by an expert are used. The disadvantages of the method include a large divergence in the opinions of experts.

In this work, the method of morphological amoebae is used to isolate vessels - a morphological method in which a structural element is selected adaptively for each pixel.

2. Morphological amoeba

We use the amoeba morphological method described in , with a modified distance function.

Consider a grayscale image
. Let's represent it as a graph in which each pixel is connected to eight neighboring pixels by edges with some given weights ("cost"). Then for each pixel
you can find the set of all points
, for which the cost of the journey from to
less than t. The resulting set will be a structural element for the pixel.

We use the following pixel spacing function and
:

Factor
sets a low cost of movement in dark areas and a high cost in light areas, thereby preventing the amoeba from spreading to points outside the vessel, and the term penalizes movement between pixels with very different intensity. Parameter sets the significance of the penalty for this transition.

An example of finding amoebas at
shown in fig. one.

Rice. 1. Examples of forms of morphological amoebae. On the left - the original image with marked points where amoeba are calculated, on the right - found structural elements are marked in white.

3. Isolation of vessels using morphological amoebas

To trace the vessels of the circulatory system on the images of the fundus, an algorithm was developed, consisting of the following steps:

4. Results

An example of the operation of the algorithm is shown in Fig. 2.

Rice. 2. The result of vessel isolation using morphological amoebae. On the left - an image of the fundus (green channel), in the center - points that are obviously points of vessels from which amoebas will be built, on the right - the result of the selection of vessels using the proposed method.

Conclusion

The application of the method of morphological amoebae for the identification of vessels in the images of the fundus is considered.

The developed algorithm is planned to be used in an automated system for detecting retinal diseases.

This work was supported by the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” for 2009–2013 and RFBR grant 10-01-00535-a.

Literature

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, M. Goldbaum. Detection of Blood Vessels in Retinal Images Using Two-Dimensional Matched Filters // IEEE Transactions of Medical Imaging, Vol. 8, no. 3, 1989, pp. 263–269.

J.Staal, M.D.Abramoff, M.Niemeijer, M.A.Viergever, B.Ginneken. Ridge-Based Vessel Segmentation in Color Images of the Retina // IEEE Transactions on Medical Imaging, Vol. 23, no. 4, 2004, pp. 504–509.

M.Patasius, V.Marozas, D.Jegelevieius, A.Lukosevieius. Recursive Algorithm for Blood Vessel Detection in Eye Fundus Images: Preliminary Results // IFMBE Proceedings, Vol. 25/11, 2009, pp. 212–215.

J.Soares, J.Leandro, R.Cesar Jr., H.Jelinek, M.Cree. Retinal Vessel Segmentation Using the 2-D Gabor Wavelet and Supervised Classification // IEEE Transactions of Medical Imaging, Vol. 25, no. 9, 2006, pp. 1214–1222.

APPLICATION OF MORPHOLOGICAL AMOEAS METHODFOR BLOOD VESSEL DETECTION IN EYE FUNDUS IMAGES

Nasonov A. 1 , Chernomorets A. 1 , Krylov A. 1 , Rodin A. 2

Lomonosov Moscow State University,
1 Faculty of Computational Mathematics and Cybernetics, Laboratory of Mathematical Methods of Image Processing, /
2 Faculty of Fundamental Medicine, Department of Ophthalmology

An algorithm of blood vessel detection in eye fundus images has been developed. Segmentation and analysis of blood vessels in eye fundus images provides the most important information to diagnose retinal diseases.

Blood vessel detection in eye fundus images is a challenging problem. Images are corrupted by non-uniform illumination and noise. Also some objects can be wrongly detected as blood vessels.

The proposed algorithm is based on the method of morphological amoebas . Morphological amoeba for a given pixel is a set of pixels with the minimal distance to the given pixel less than a threshold t. We use the sum of average intensity value multiplied by Euclidean distance and absolute value of difference between pixel intensity values for the distance. In this case the distance will be small for blood vessels which are usually dark and big for light areas and edges, and the amoeba will be extended along the vessel but not through vessel walls.

The proposed algorithm of blood vessel detection consists of the following steps:

Extract the green channel as the most informative and perform illumination correction using the method . It makes it possible to use unified amoebas parameters for different images.

Find the set of pixels ( p n) in the resulting image which are surely the pixels of the blood vessels

Calculate the amoeba BUT(p i) for every pixel , apply rank filtering to the amoeba mask with 3x3 window: remove the pixels from the mask which have less than 3 neighbor pixels in the mask. The remaining pixels are marked as blood vessel pixels.

If we need to extend the blood vessels, the third step is repeated for all newly added pixels to blood vessels area.

We plan to use the developed algorithm in automatic system of retinal disease detection.

The work was supported by the federal target program ”Scientific and scientific-pedagogical personnel of innovative Russia in 2009-2013” and RFBR grant 10-01-00535-a.

Literature

R. J. Winder, P. J. Morrow, I. N. McRitchie, J. R. Bailie, P. M. Hart. Algorithms for digital image processing in diabetic retinopathy // Computerized Medical Imaging and Graphics, Vol. 33, 2009, 608–622.

M. Welk, M. Breub, O. Vogel. Differential Equations for Morphological Amoebas // Lecture Notes in Computer Science, Vol. 5720/2009, 2009, pp. 104–114.

G. D. Joshi, J. Sivaswamy. Color Retinal Image Enhancement based on Domain Knowledge // Sixth Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP"08), 2008, pp. 591–598.

images Using Tomography method in handwritten ...

The equation is also considered in two-dimensional and one-dimensional space. In two-dimensional space, the Laplace equation is written:

∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 (\displaystyle (\frac (\partial ^(2)u)(\partial x^(2)))+(\frac (\partial ^(2 )u)(\partial y^(2)))=0)

Also in n-dimensional space. In this case, the sum is equal to zero n second derivatives.

Δ = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 + . . . (\displaystyle \Delta =(\frac (\partial ^(2))(\partial x^(2)))+(\frac (\partial ^(2))(\partial y^(2)))+ (\frac (\partial ^(2))(\partial z^(2)))+...)

Note: all of the above applies to Cartesian coordinates in flat space (whatever its dimension). When using other coordinates, the representation of the Laplace operator changes, and, accordingly, the record of the Laplace equation changes (see an example below). These equations are also called the Laplace equation, but to eliminate ambiguity in the terminology, the coordinate system (and, if you want complete clarity, the dimension) is usually added explicitly, for example: "two-dimensional Laplace equation in polar coordinates."

Other forms of Laplace's equation

1 r 2 ∂ ∂ r (r 2 ∂ f ∂ r) + 1 r 2 sin ⁡ θ ∂ ∂ θ (sin ⁡ θ ∂ f ∂ θ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ φ 2 = 0 ( \displaystyle (1 \over r^(2))(\partial \over \partial r)\left(r^(2)(\partial f \over \partial r)\right)+(1 \over r^( 2)\sin \theta )(\partial \over \partial \theta )\left(\sin \theta (\partial f \over \partial \theta )\right)+(1 \over r^(2)\sin ^(2)\theta )(\partial ^(2)f \over \partial \varphi ^(2))=0)

Singular points r = 0 , θ = 0 , θ = π (\displaystyle r=0,\theta =0,\theta =\pi ).

1 r ∂ ∂ r (r ∂ u ∂ r) + 1 r 2 ∂ 2 u ∂ φ 2 = 0 (\displaystyle (\frac (1)(r))(\frac (\partial )(\partial r)) \left(r(\frac (\partial u)(\partial r))\right)+(\frac (1)(r^(2)))(\frac (\partial ^(2)u)(\ partial \varphi ^(2)))=0)

special point.

1 r ∂ ∂ r (r ∂ f ∂ r) + ∂ 2 f ∂ z 2 + 1 r 2 ∂ 2 f ∂ φ 2 = 0 (\displaystyle (1 \over r)(\partial \over \partial r)\ left(r(\partial f \over \partial r)\right)+(\partial ^(2)f \over \partial z^(2))+(1 \over r^(2))(\partial ^ (2)f \over \partial \varphi ^(2))=0)

singular point r = 0 (\displaystyle r=0).

Application of the Laplace equation

The Laplace equation arises in many physical problems of mechanics, heat conduction, electrostatics, hydraulics. The Laplace operator is of great importance in quantum physics, in particular in the Schrödinger equation.

Solutions to the Laplace Equation

Despite the fact that the Laplace equation is one of the simplest in mathematical physics, its solution faces difficulties. The numerical solution is especially difficult because of the irregularity of the functions and the presence of singularities.

Common decision

One-dimensional space

f (x) = C 1 x + C 2 (\displaystyle f(x)=C_(1)x+C_(2))

where C 1 , C 2 (\displaystyle C_(1),C_(2)) are arbitrary constants.

two-dimensional space

The Laplace equation on a two-dimensional space is satisfied by analytic functions. Analytic functions are considered in the theory of functions of a complex variable, and the class of solutions of the Laplace equation can be reduced to a function of a complex variable.

The Laplace equation for two independent variables is formulated as follows

φ x x + φ y y = 0. (\displaystyle \varphi _(xx)+\varphi _(yy)=0.)

Analytic Functions

If a z = x + iy, and

f (z) = u (x , y) + i v (x , y) , (\displaystyle f(z)=u(x,y)+iv(x,y),)

then the Cauchy-Riemann conditions are necessary and sufficient for the function f(z) was analytic:

∂ u ∂ x = ∂ v ∂ y , ∂ u ∂ y = − ∂ v ∂ x . (\displaystyle (\frac (\partial u)(\partial x))=(\frac (\partial v)(\partial y)),~(\frac (\partial u)(\partial y))=- (\frac (\partial v)(\partial x)).)

Both the real and imaginary parts of analytic functions satisfy the Laplace equation. By differentiating the conditions

Leonardo da Vinci is considered the discoverer of capillary phenomena. However, the first accurate observations of capillary phenomena on tubes and glass plates were made by Francis Hawkesby in 1709).

That matter is not infinitely divisible and has an atomic or molecular structure has been a working hypothesis for most scientists since the 18th century. By the end of the 19th century, when a group of positivist physicists pointed out how indirect the proof of the existence of atoms was, there was only a minor reaction to their claim, and as a result, their objections were not refuted until the beginning of this century. If, in retrospect, the doubts seem unfounded to us, we must remember that almost everyone who then believed in the existence of atoms also firmly believed in the material existence of the electromagnetic ether, and in the first half of the 19th century. - often caloric. Nevertheless, the scientists who made the greatest contributions to the theory of gases and liquids used the assumption (usually explicitly) of the discrete structure of matter. The elementary particles of matter were called atoms, or molecules (for example, Laplace), or simply particles (Jung), but we will follow modern concepts and use the word "molecule" for elementary particles that make up a gas, liquid, or solid.

At the beginning of the XIX century. the forces that might exist between molecules were as obscure as the particles themselves. The only force about which there was no doubt was Newtonian gravity. It acts between celestial bodies and, obviously, between one such body (the Earth) and another (for example, an apple) having a laboratory mass; Cavendish had shown shortly before that it also acts between two laboratory masses, and therefore it was assumed that it also acts between molecules. In the early work on liquids one can find the masses of the molecules and the mass densities involved in the equations in which we now have to write the numbers of the molecules and the densities of the numbers of the molecules. In a pure liquid, all molecules have the same mass, so this difference does not matter. But even before 1800 it was clear that the concept of gravitational forces was insufficient to explain capillary phenomena and other properties of liquids. The rise of a liquid in a glass tube does not depend on the thickness of the glass (according to Hawksby, 1709), and thus only the forces from the molecules in the surface layer of the glass act on the molecules in the liquid. Gravitational forces, on the other hand, are only inversely proportional to the square of the distance and, as was known, act freely through the intermediate substance.

The nature of intermolecular forces other than gravity was very obscure, but there was no shortage of speculation. The Jesuit priest Ruggero Giuseppe Boscovich believed that molecules repel each other at very small distances, attract at somewhat greater distances, and then, as the distance increases, alternate between repulsion and attraction with decreasing magnitude. His ideas in the next century influenced both Faraday and Kelvin, but were too complex to be directly useful to those involved in the theory of capillarity. The latter were prudently content with simple hypotheses.

Quink (G.H. Quincke) set up experiments to determine the largest distance at which the action of intermolecular forces is noticeable. He found that for various substances these distances are ~ 1/20000 of a millimeter, i.e. ~ 5 10 -6 cm (data are given according to) .

James Jurin showed that the height to which a liquid rises is determined by the top of the tube that is above the liquid and is independent of the shape of the bottom of the tube. He believed that the rise of the liquid occurs due to the attraction from the inner cylindrical surface of the tube, to which the upper surface of the liquid adjoins. Based on this, he showed that the rise of liquid in tubes of the same substance is inversely proportional to their inner radius.

Clairaut was one of the first to show the need to take into account the attraction between the particles of the liquid itself in order to explain capillary phenomena. However, he did not admit that the distances over which these forces act are imperceptibly small.

In 1751, von Segner introduced the important idea of surface tension by analogy with the mechanical tension of a membrane in the theory of elasticity. Today, the concept of surface tension is commonplace, it is usually the beginning of the study of capillary forces and surface phenomena in educational institutions.

This idea became the key to the further development of the theory. Actually, this was the first step in the study of the phenomenon - a phenomenological concept was introduced that describes the macroscopic behavior of the system. The second step is the derivation of phenomenological concepts and the calculation of the values of quantities based on molecular theory. This step is of great importance, as it is a test of the correctness of one or another molecular theory.

In 1802, John Leslie gave the first correct explanation for the rise of a liquid in a tube, considering the attraction between a solid body and a thin layer of liquid on its surface. He, unlike most previous researchers, did not assume that the force of this attraction is directed upwards (directly to maintain fluid). On the contrary, he showed that attraction is everywhere normal to the surface of a rigid body.

The direct effect of attraction is an increase in pressure in a layer of liquid in contact with a solid, so that the pressure becomes higher than inside the liquid. The result of this is that the layer tends to “spread” over the surface of a solid body, stopped only by gravitational forces. Thus, a glass tube immersed in water is wetted by water wherever it “could crawl”. Rising, the liquid forms a column, the weight of which eventually balances the force that generates the spreading of the liquid.

This theory was not written in mathematical symbols and therefore could not show a quantitative relationship between the attraction of individual particles and the end result. Leslie's theory was later revised using Laplacian mathematical methods by James Ivory in an article on capillary action, under "Fluids, Elevation of", in the supplement to the 4th edition of the Encyclopaedia Britannica, published in 1819.

2. Theories of Jung and Laplace

In 1804, Thomas Young substantiated the theory of capillary phenomena on the principle of surface tension. He also observed the constancy of the wetting angle of a solid surface (contact angle) with a liquid and found a quantitative relationship relating the contact angle to the surface tension coefficients of the corresponding interfacial boundaries. In equilibrium, the contact line should not move on the surface of a solid, which means, said Hawkesby, was a demonstrator at the Royal Society, and his experiments influenced the content of a very lengthy essay on the primary particles of matter and the forces between them, with which Newton completed the edition of his Optics 1717 of the year. cm.

where s SV,sSL,sLV surface tension coefficients of interfacial boundaries solid - gas (steam), solid - liquid, liquid - gas, respectively, q edge angle. This ratio is now known as Young's formula. This work, however, did not have such an impact on the development of science in this direction, which was published a few months later by Laplace's article (Pierre Simon Laplace). This seems to be due to the fact that Jung avoided the use of mathematical notation, but tried to describe everything verbally, which makes his work seem confusing and unclear. Nevertheless, he is considered today one of the founders of the quantitative theory of capillarity.

The phenomena of cohesion and adhesion, the condensation of vapor into a liquid, the wetting of solids by liquids, and many other simple properties of matter all pointed to the presence of attractive forces many times stronger than gravity, but acting only at very small distances between molecules. As Laplace said, the only condition that follows from observable phenomena and is imposed on these forces is that they are "imperceptible at sensible distances."

The repulsive forces created more trouble. Their presence could not be denied - they should balance the forces of attraction and prevent the complete destruction of matter, but their nature was completely unclear. The question was complicated by the following two erroneous opinions. First, it was often assumed that the acting force of repulsion was heat (as a rule, the opinion of supporters of the caloric theory), since (such was the argument) when a liquid first expands and then boils when heated, so that the molecules are separated over much greater distances than in a solid body. The second erroneous opinion arose from the notion, which leads back to Newton, according to which the observed pressure of a gas is due to static repulsion between molecules, and not due to their collisions with the walls of the vessel, as Daniel Bernoulli argued in vain.

Against this background, it was natural that the first attempts to explain capillarity, or in general the cohesion of liquids, were based on the static aspects of matter. Mechanics was a well understood theoretical branch of science; thermodynamics and kinetic theory were still in the future. In mechanical considerations, the key was the assumption of large but short-range attractive forces. Liquids at rest (whether in a capillary tube or outside it) are obviously in equilibrium, and therefore these attractive forces must be balanced by repulsive forces. Since even less could be said about them than about the forces of attraction, they were often passed over in silence, and, in the words of Rayleigh, "the forces of attraction were left to perform the unthinkable trick of balancing themselves." Laplace was the first to solve this problem satisfactorily, believing that repulsive forces (thermal, as he assumed) could be replaced by internal pressure, which acts everywhere in an incompressible fluid. (This assumption leads at times to ambiguity in 19th-century writings as to what is strictly meant by "pressure in a fluid.") Here is Laplace's calculation of internal pressure. (This conclusion is closer to the conclusions of Maxwell and Rayleigh. The conclusion is given by.)

By 1819 he was busy discussing in detail the intermolecular forces of repulsion, which, although still attributed to heat or caloric, had the essential property of decreasing with distance faster than the forces of attraction.

Let us consider two semi-infinite fluid bodies with strictly flat surfaces, separated by an interlayer (thicknesses l) a pair with a negligible density (Fig. 1), and in each of them we single out a volume element. The first is in the upper body at a height r above the flat surface of the lower body; its volume is dxdydz. The second one is located in the lower body and has volume , where the origin of polar coordinates coincides with the position of the first elementary volume. Let f(s) is the force acting between two molecules separated by a distance s, a d- the radius of its action. Since this is always an attractive force, we have

If a r is the density of the number of molecules in both bodies, then the vertical component of the interaction force of two volume elements is equal to

The total attractive force per unit area (positive value) is

Let u(s) is the potential of the intermolecular force:

Integrating by parts again, we get

Internal Laplace pressure K is the force of attraction per unit area between two flat surfaces when they are in contact, i.e. F(0):

where is the volume element, which can be written as . Because the u(r) by assumption is everywhere negative or equal to zero, then K positively. Laplace believed that K large compared to atmospheric pressure, but Jung had to make the first realistic numerical estimate.

(8)

When separated, two free surfaces are formed, and therefore the work expended can be equated to twice the surface energy per unit area, which is equal to the surface tension:

(9)

Thus, K is the integral of the intermolecular potential, or its zero moment, and H is his first moment. While K inaccessible to direct experiment, H can be found if we can measure the surface tension.

Let be the density of cohesive energy at some point of the liquid or gas, i.e. attitude d U / d V where d U— internal energy of small volume V liquid or gas containing this point. For the molecular model, we take

(10)

where r is the distance from the considered point. Rayleigh identified the Laplacian K with the difference of this potential 2 between a point on a flat surface of the liquid (value 2 S) and a dot inside (value 2 I). On the surface, integration in (10) is limited by a hemisphere of radius d, and in the inner region is carried out over the entire sphere. Consequently, S have half I, or

(11)

Consider now a drop of radius R. Calculation f I does not change, but when received f S the integration is now over a more limited volume due to the curvature of the surface. If is the angle between the vector and the fixed radius , then

Then the internal pressure in the drop is

where H is determined by equation (9). If we took not a spherical drop, but a portion of a liquid with a surface determined by two main radii of curvature R1 and R2, then we would get the internal pressure in the form

(14)

By Euler's theorem, the sum is equal to the sum of the reciprocal radii of curvature of the surface along any two orthogonal tangents.

When calculating the pressure inside a curved liquid surface, the Rayleigh potential 2 (10) was introduced; along the way, it was noted that I is the cohesive energy density. For the first time this useful concept was introduced in 1869 by Dupre, who defined it as the work of crushing a piece of matter into its constituent molecules (la travail de désagré gation totale - the work of complete disaggregation).

Inward force acting on a molecule at depth r< d , is opposite in sign to the outward force that would arise from the molecules in the shaded volume if it were filled uniformly with density.

This force is positive because f(0 < s < d) < 0 и F(d) = 0 due to the oddness of the function f(s). No force acts on a molecule unless it is within a distance d either side of the surface. Therefore, the work done to remove one molecule from a liquid is

because the u(r) is an even function. This work is equal to minus twice the energy per molecule required to disintegrate the liquid ( doubled, so as not to count the molecules twice: once when they are removed, the other time as part of the medium):

(18)

This is a simple and understandable expression for the internal energy U liquid containing N molecules. It follows that the cohesive energy density is given by expression (10), or

which coincides with (11) if we remove the index I. Dupre himself obtained the same result in a roundabout way. He counted dU/dV through work against intermolecular forces with a uniform expansion of the liquid cube. This gave him

Because the K has the form ((7) and (11)), where the constant a is given by

(21)

then integration (20) again leads to (19).

Rayleigh criticized Dupree's conclusion. He believed that consideration of the work of a homogeneous expansion from a state of balance of cohesive and repulsive intermolecular forces, taking into account only cohesive forces, was unreasonable; before taking such a step, one should have a better knowledge of the nature of the repulsive forces.

We see that in this conclusion, as well as in the conclusions of Young, Laplace and Gauss, the assumption of an abrupt change in the density of the number of molecules of a substance at the phase boundary is essentially used. At the same time, in order for the above reasoning to describe real phenomena in matter, it is necessary to assume that the radius of action of intermolecular forces in matter is much larger than the characteristic distance between particles. But under this assumption, the interface between two phases cannot be sharp—a continuous transitional density profile must arise, in other words, a transition zone.

3. Gibbs capillarity theory

As often happens, the thermodynamic description turns out to be simpler and more general, without being limited by the shortcomings of specific models.

It was in this way that Gibbs described capillarity in 1878, constructing a purely thermodynamic theory. This theory has become an integral part of Gibbs' thermodynamics. Gibbs' theory of capillarity, without relying directly on any mechanistic models, is devoid of the shortcomings of Laplace's theory; it can rightfully be considered the first detailed developed thermodynamic theory of surface phenomena.

About the theory of capillarity Gibbs can be said that it is very simple and very complex. Simple because Gibbs was able to find a method that allows you to get the most compact and elegant thermodynamic relations, equally applicable to flat and curved surfaces. “One of the main tasks of theoretical research in any field of knowledge,” Gibbs wrote, “is to establish such a point of view from which the object of research is manifested with the greatest simplicity.” Such a point of view in the Gibbs theory of capillarity is the idea of separating surfaces. The use of a visual geometric image of the separating surface and the introduction of redundant quantities made it possible to describe the properties of the surfaces as simply as possible and bypass the issue of the structure and thickness of the surface layer, which was completely unexplored at the time of Gibbs and still remains far from completely resolved. Gibbs excess values (adsorption and others) depend on the position of the dividing surface, and the latter can also be found for reasons of maximum simplicity and convenience.

It is reasonable to choose in each case the separating surface so that it is everywhere perpendicular to the density gradient. If separating surfaces are chosen, then each phase ( l} (l = a, b, g) now corresponds to the volume it occupies V{ l) . Full system scope

Let be the density of the number of molecules of the sort j in the [bulk] phase ( l). Then the total number of molecules of the sort j in the system under consideration is equal to

where is the surface excess of the number of molecules of the sort j(index ( s) means surface - surface). Excesses of other extensive physical quantities are defined in a similar way. Obviously, in the case of, for example, a flat film, is proportional to its area A. Value defined as the surface excess of the number of molecules of the variety j per unit area of the dividing surface is called the adsorption of molecules of the sort j on this surface.

Gibbs used two main positions of the separating surface: one in which the adsorption of one of the components is zero (now this surface is called equimolecular), and a position for which the explicit dependence of the surface energy on the curvature of the surface disappears (this position was called by Gibbs the surface of tension). Gibbs used the equimolecular surface to consider flat liquid surfaces (and surfaces of solids), and the tension surface to consider curved surfaces. For both positions, the number of variables is reduced and maximum mathematical simplicity is achieved.

Now about the complexity of the Gibbs theory. Although very simple mathematically, it is still difficult to understand; this happens for several reasons. First, Gibbs' theory of capillarity cannot be understood in isolation from the entire Gibbsian thermodynamics, which is based on a very general, deductive method. The great generality of a theory always makes it somewhat abstract, which, of course, is reflected in the ease of perception. Secondly, the Gibbs theory of capillarity itself is an extensive, but conditional system that requires the unity of perception without distraction from its individual provisions. An amateurish approach to the study of Gibbs is simply not possible. Finally, an important circumstance is that the entire Gibbs work mentioned is written in a very concise and very difficult language. This work, according to Rayleigh, "is too condensed and difficult, not only for the majority, but, one might say, for all readers." According to Guggenheim, "it is much easier to use the Gibbs formulas than to understand them."

Naturally, the use of the Gibbs formulas without their true understanding led to numerous errors in the interpretation and application of certain provisions of the Gibbs theory of capillarity. Many errors were associated with a misunderstanding of the need to unambiguously determine the position of the separating surface in order to obtain the correct physical result. Errors of this kind were often encountered in the analysis of the dependence of surface tension on surface curvature; even one of the "pillars" of the theory of capillarity, Bakker, did not escape them. An example of errors of another kind is the misinterpretation of chemical potentials when considering surface phenomena and external fields.

Already soon after the publication of the Gibbs theory of capillarity, wishes were expressed for its fuller and more detailed explanation in the scientific literature. In the letter to Gibbs quoted above, Rayleigh suggested that Gibbs himself take over the job. However, this was done much later: Rice prepared a commentary on the entire Gibbs theory, and some of its provisions were commented on in the works of Frumkin, Defey, Rebinder, Guggenheim, Tolman, Buff, Semenchenko and other researchers. Many provisions of the Gibbs theory became clear, and simpler and more effective logical devices were found to justify them.

A typical example is Kondo's spectacular work, which proposed a visual and easy-to-understand method for introducing a tension surface by mentally moving the separating surface. If we write an expression for the energy of an equilibrium two-phase system a - b (a- internal and b- outer phase) with a spherical fracture surface

U = TS - P a V a- P b V b + SA +(22)

and we will mentally change the position of the separating surface, i.e. change its radius r, then, obviously, such physical characteristics as energy U , temperature T, entropy S, pressure R, chemical potential i-th component m i and its mass m i, as well as the total volume of the system V a + V b does not change. As for the volume V a = 4 /3pr 3 and square A = 4pr 2 and surface tension s, then these quantities will depend on the position of the separating surface and therefore for the indicated mental process of change r we get from (22)

- Pa dVa+ Pb dVb + sdA + Ads = 0 (23)

(24)

Equation (24) determines the non-physical (this circumstance is marked with an asterisk) dependence of the surface tension on the position of the separating surface. This dependence is characterized by a single minimum s, which corresponds to the tension surface. Thus, according to Kondo, the surface of tension is such a separating surface for which the surface tension has a minimum value.

Gibbs introduced the tension surface in a different way. He proceeded from the basic equation of the theory of capillarity

(the overline means the excess for an arbitrary dividing surface with principal curvatures FROM 1 and C 2) and considered the physical (rather than purely mental) process of surface curvature for a given position and fixed external conditions.

According to Gibbs, the surface of tension corresponds to such a position of the dividing surface, in which the curvature of the surface layer at constant external parameters does not affect the surface energy and also corresponds to the condition:

¶ s/¶ r =0 (26)

Guggenheim comments on Gibbs' proof thus: "I found Gibbs' consideration difficult, and the more carefully I studied it, the more obscure it seemed to me." This recognition indicates that the understanding of the Gibbs tension surface has been difficult even for specialists in the field of thermodynamics.

As for Kondo's approach, it is clear at first glance. However, it must be ensured that the Gibbs and Kondo tension surfaces are adequate. This can be demonstrated, for example, using the hydrostatic determination of surface tension

Jung mentioned the presence of a density gradient in a layer with a finite thickness, but rejected this effect, considering it insignificant.

P t — local value of the tangential component of the pressure tensor;

r" is the radial coordinate; radii Ra and Rb limit the surface layer.

Differentiation (27) with a mental displacement of the separating surface and the constancy of the physical state (Kondo's approach) leads to equation (24). Differentiation with the curvature of the surface layer and the constancy of the physical state (the Gibbs approach, in this case Ra and Rb variables) gives

(28)

where it is taken into account that P t(Pa) = Pa and P t(Pb) = Pb.

Equations (28) and (24) show that condition (26) is equivalent to condition ( ds/ dr) * = 0 and, therefore, the simpler and more illustrative approach of Kondo is adequate to that of Gibbs.

The introduction of the concept of a separating surface made it possible to mathematically rigorously determine the previously purely intuitive concept of the phase boundary and, therefore, to use exactly defined quantities in the equations. In principle, the thermodynamics of Gibbs surface phenomena describes a very wide range of phenomena, and therefore (apart from awareness, reformulations, more elegant conclusions and proofs) since its inception, very little new has been done in this area. But still, some results, relating mainly to those issues that were not covered by Gibbs, must be mentioned.

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