It is known that the surface of a liquid near the walls of a vessel is curved. The free surface of the liquid, curved near the walls of the vessel, is called the meniscus(Fig. 145).

Let us consider a thin liquid film, the thickness of which can be neglected. In an effort to minimize its free energy, the film creates a pressure difference from different sides. Due to the action of surface tension forces in drops of liquid and inside soap bubbles, additional pressure(the film is compressed until the pressure inside the bubble exceeds atmospheric pressure by the amount of the additional pressure of the film).

Rice. 146.

Let us consider the surface of a liquid resting on some flat contour (Fig. 146, A). If the surface of the liquid is not flat, then its tendency to contract will lead to the appearance of pressure, additional to that experienced by a liquid with a flat surface. In the case of a convex surface, this additional pressure is positive (Fig. 146, b), in the case of a concave surface – negative (Fig. 146, V). In the latter case, the surface layer, trying to contract, stretches the liquid.

The amount of additional pressure, obviously, should increase with increasing surface tension coefficient and surface curvature.

Rice. 147.

Let us calculate the additional pressure for the spherical surface of the liquid. To do this, let us mentally dissect a spherical drop of liquid with a diametrical plane into two hemispheres (Fig. 147). Due to surface tension, both hemispheres are attracted to each other with a force equal to:

.

This force presses both hemispheres against each other along the surface and, therefore, causes additional pressure:

The curvature of a spherical surface is the same everywhere and is determined by the radius of the sphere. Obviously, the smaller , the greater the curvature of the spherical surface.

The excess pressure inside the soap bubble is twice as high, since the film has two surfaces:

Additional pressure causes a change in the liquid level in narrow tubes (capillaries), as a result of which it is sometimes called capillary pressure.

The curvature of an arbitrary surface is usually characterized by the so-called average curvature, which may be different for different points of the surface.

The value gives the curvature of the sphere. In geometry it is proven that the half-sum of the reciprocal radii of curvature for any pair of mutually perpendicular normal sections has the same value:

. (1)

This value is the average curvature of the surface at a given point. In this formula, radii are algebraic quantities. If the center of curvature of a normal section is below a given surface, the corresponding radius of curvature is positive; if the center of curvature lies above the surface, the radius of curvature is negative (Fig. 148).

Rice. 148.

Thus, a non-flat surface can have an average curvature of zero. To do this, it is necessary that the radii of curvature be equal in magnitude and opposite in sign.

For example, for a sphere, the centers of curvature at any point on the surface coincide with the center of the sphere, therefore . For the case of the surface of a circular cylinder of radius we have: , and .

It can be proven that for a surface of any shape the relation is valid:

Substituting expression (1) into formula (2), we obtain the formula for additional pressure under an arbitrary surface, called Laplace's formula(Fig. 148):

. (3)

The radii and in formula (3) are algebraic quantities. If the center of curvature of a normal section is below a given surface, the corresponding radius of curvature is positive; if the center of curvature lies above the surface, the radius of curvature is negative.

Example. If there is a gas bubble in the liquid, then the surface of the bubble, tending to contract, will exert additional pressure on the gas . Let's find the radius of a bubble in water at which the additional pressure is equal to 1 atm. .The coefficient of surface tension of water is equal to . Therefore, for the following value is obtained: .

Properties of the liquid state. Surface layer. Surface tension. Wetting. Laplace's formula. Capillary phenomena.

Liquids are substances that are in a condensed state, which is intermediate between the solid crystalline state and the gaseous state.

The region of existence of liquids is limited on the high-temperature side by its transition to the gaseous state, and on the low-temperature side by its transition to the solid state.

In liquids, the distance between molecules is much smaller than in gases (the density of liquids is ~ 6000 times greater than the density of saturated vapor far from the critical temperature) (Fig. 1).

Fig.1. Water vapor (1) and water (2). Water molecules are enlarged approximately 5 10 7 times

Consequently, the forces of intermolecular interaction in liquids, unlike gases, are the main factor that determines the properties of liquids. Therefore, liquids, like solids, retain their volume and have a free surface. Like solids, liquids are characterized by very low compressibility and resist stretching.

However, the bonding forces between liquid molecules are not so strong as to prevent the layers of liquid from sliding relative to each other. Therefore, liquids, like gases, have fluidity. In the field of gravity, liquids take the shape of the container into which they are poured.

The properties of substances are determined by the movement and interaction of the particles of which they are composed.

In gases, collisions mainly involve two molecules. Consequently, the theory of gases reduces to the solution of the two-body problem, which can be solved exactly. In solids, molecules undergo vibrational motion at the nodes of the crystal lattice in a periodic field created by other molecules. This problem of particle behavior in a periodic field can also be solved exactly.

In liquids, each molecule is surrounded by several others. A problem of this type (the many-body problem), in general, regardless of the nature of the molecules and the nature of their arrangement, has not yet been precisely solved.

Experiments on diffraction of X-rays, neutrons, and electrons helped determine the structure of liquids. Unlike crystals, in which long-range order is observed (regular arrangement of particles in large volumes), in liquids at distances of the order of 3–4 molecular diameters, the order in the arrangement of molecules is disrupted. Consequently, in liquids there is a so-called short-range order in the arrangement of molecules (Fig. 2):

Fig.2. An example of short-range order of liquid molecules and long-range order of molecules of a crystalline substance: 1 – water; 2 – ice

In liquids, molecules undergo small vibrations within limits limited by intermolecular distances. However, from time to time, as a result of fluctuations, a molecule can receive energy from neighboring molecules that is enough to jump to a new equilibrium position. The molecule will remain in the new equilibrium position for some time until, again, as a result of fluctuations, it receives the energy necessary for the jump. The molecule jumps over a distance comparable to the size of the molecule. Vibrations that give way to jumps represent the thermal movement of liquid molecules.

The average time that a molecule is in a state of equilibrium is called relaxation time. As the temperature increases, the energy of the molecules increases, therefore, the probability of fluctuations increases, while the relaxation time decreases:

(1)

Where τ – relaxation time, B– coefficient having the meaning of the vibration period of the molecule, W – activation energy molecules, i.e. energy required to make a molecular jump.

Internal friction in liquids, as in gases, occurs when layers of liquid move due to the transfer of momentum in the direction normal to the direction of movement of layers of liquid. Momentum transfer from layer to layer also occurs during molecular jumps. However, mainly, momentum is transferred due to the interaction (attraction) of molecules of neighboring layers.

In accordance with the mechanism of thermal movement of liquid molecules, the dependence of the viscosity coefficient on temperature has the form:

(2)

Where A– coefficient depending on the jump distance of the molecule, the frequency of its vibrations and temperature, W – activation energy.

Equation (2) – Frenkel-Andrade formula. The temperature dependence of the viscosity coefficient is mainly determined by the exponential factor.

The reciprocal value of viscosity is called fluidity. As the temperature decreases, the viscosity of some liquids increases so much that they practically stop flowing, forming amorphous bodies (glass, plastics, resins, etc.).

Each liquid molecule interacts with neighboring molecules that are within the range of its molecular forces. The results of this interaction are not the same for molecules inside the liquid and on the surface of the liquid. A molecule located inside a liquid interacts with neighboring molecules surrounding it and the resultant force that acts on it is zero (Fig. 3).

Fig.3. Forces acting on liquid molecules

The molecules of the surface layer are under different conditions. The density of vapor above the liquid is much less than the density of the liquid. Therefore, each molecule of the surface layer is acted upon by a resultant force directed normally into the liquid (Fig. 3). The surface layer exerts pressure on the rest of the liquid like an elastic film. Molecules lying in this layer are also attracted to each other (Fig. 4).

Fig.4. Interaction of surface layer molecules

This interaction creates forces directed tangentially to the surface of the liquid and tending to reduce the surface of the liquid.

If an arbitrary line is drawn on the surface of a liquid, then surface tension forces will act along the normal to the line and tangent to the surface. The magnitude of these forces is proportional to the number of molecules located along this line, therefore proportional to the length of the line:

(3)

Where σ – proportionality coefficient, which is called surface tension coefficient:

(4)

The surface tension coefficient is numerically equal to the surface tension force acting per unit length of the contour delimiting the surface of the liquid.

The surface tension coefficient is measured in N/m. Magnitude σ depends on the type of liquid, temperature, and the presence of impurities. Substances that reduce surface tension are called superficially active(alcohol, soap, washing powder, etc.).

To increase the surface area of ​​a liquid, work must be done against surface tension forces. Let's determine the amount of this work. Let there be a frame with a liquid film (for example, soap) and a movable crossbar (Fig. 5).

Fig.5. The movable side of the wire frame is in equilibrium under the action of external force F ext and the resulting surface tension forces F n

Let's stretch the film with a force F ext by dx. Obviously:

Where F n = σL–surface tension force. Then:

Where dS = Ldx– increment of film surface area. From the last equation:

(5)

According to (5), the surface tension coefficient is numerically equal to the work required to increase the surface area by one unit at a constant temperature. From (5) it is clear that σ can be measured in J/m 2.

If a liquid borders another liquid or a solid, then due to the fact that the densities of the substances in contact are comparable, one cannot ignore the interaction of the molecules of the liquid with the molecules of the substances bordering it.

If, upon contact between a liquid and a solid, the interaction between their molecules is stronger than the interaction between the molecules of the liquid itself, then the liquid tends to increase the surface of contact and spreads over the surface of the solid. In this case, the liquid wets the solid. If the interaction between the molecules of the liquid is stronger than the interaction between the molecules of the liquid and the solid, then the liquid reduces the contact surface. In this case, the liquid does not wet solids. For example: water wets glass, but does not wet paraffin; mercury wets metal surfaces, but does not wet glass.

Fig.6. Different shapes of a drop on the surface of a solid for the cases of non-wetting (a) and wetting (b) liquids

Consider a drop of liquid on the surface of a solid (Fig. 7):

Fig.7. Schemes for calculating the equilibrium of a drop on the surface of a solid body for the cases of non-wetting (a) and wetting (b) liquids: 1 - gas, 2 - liquid, 3 - solid

The shape of a drop is determined by the interaction of three media: gas - 1, liquid - 2 and solid - 3. All these media have a common boundary - a circle that encloses the drop. Per element length dl of this contour, surface tension forces will act: F 12 = σ 12 dl– between gas and liquid, F 13 = σ 13 dl- between gas and solid, F 23 = σ 23 dl– between liquid and solid. If dl=1m, then F 12 = σ 12 , F 13 = σ 13 , F 23 = σ 23. Let's consider the case when:

It means that<θ = π (Fig. 7, a). The circle that limits the place of contact of the liquid with the solid body will contract to a point and the drop will take an ellipsoidal or spherical shape. This is a case of complete non-wetting. Complete non-wetting is also observed in the case of: σ 23 > σ 12 + σ 13 .

Another edge case will occur if:

It means that<θ = 0 (Fig. 7b), complete wetting is observed. Complete wetting will also be observed in the case when: σ 13 > σ 12 + σ 23. In this case, there will be no equilibrium, at any angle values θ , and the liquid will spread over the surface of the solid up to the monomolecular layer.

If the drop is in equilibrium, then the resultant of all forces acting on the element of the contour length is zero. The equilibrium condition in this case:

The angle between the tangents to the surface of a solid and to the surface of a liquid, which is measured inside the liquid,called the contact angle.

Its value is determined from (6):

(7)

If σ 13 > σ 23, then cos θ > 0, angle θ sharp - partial wetting occurs if σ 13 < σ 23, then cos θ < 0 – угол θ blunt – partial non-wetting occurs. Thus, the contact angle is a value characterizing the degree of wetting or non-wetting of the liquid

The curvature of the liquid surface results in additional pressure acting on the liquid below this surface. Let us determine the amount of additional pressure under the curved surface of the liquid. Let us select an element of area ∆ on an arbitrary surface of the liquid S(Fig. 8):

Fig.8. To calculate the amount of additional pressure

O′ O– normal to the surface at a point O. Let us determine the surface tension forces acting on the contour elements AB And CD. Surface tension forces F And F′, which act on AB And CD, perpendicular AB And CD and directed tangentially to the surface ∆ S. Let's determine the magnitude of the force F:

Let's break down the power F into two components f 1 and f ′. Force f 1 parallel O′ O and directed into the liquid. This force increases the pressure on the internal areas of the liquid (the second component stretches the surface and does not affect the amount of pressure).

Let us draw a plane perpendicular to ∆ S through points M, O And N. Then R 1 – radius of curvature of the surface in the direction of this plane. Let us draw a plane perpendicular to ∆ S and the first plane. Then R 2 – radius of curvature of the surface in the direction of this plane. In general R 1 ≠ R 2. Let's define the component f 1 . From the picture you can see:

Let's take into account that:

(8)

Strength F′ let us decompose into the same two components and similarly define the component f 2 (not shown in the figure):

(9)

Reasoning similarly, we will determine the components of the forces acting on the elements A.C. And BD, considering that instead R 1 will be R 2:

(10)

Let's find the sum of all four forces acting on the contour ABDC and exerting additional pressure on the internal areas of the liquid:

Let's determine the amount of additional pressure:

Hence:

(11)

Equation (11) is called Laplace's formula. The additional pressure that the curved surface of a liquid exerts on the internal regions of the liquid is called Laplace pressure.

The Laplace pressure is obviously directed towards the center of curvature of the surface. Therefore, in the case of a convex surface, it is directed into the liquid and is added to the normal pressure of the liquid. In the case of a concave surface, the liquid will be under less pressure than the liquid under a flat surface, because Laplace pressure is directed outside the liquid.

If the surface is spherical, then: R 1 = R 2 = R:

If the surface is cylindrical, then: R 1 = R, R 2 = ∞:

If the surface is flat then: R 1 = ∞, R 2 = ∞:

If there are two surfaces, for example, a soap bubble, then the Laplace pressure doubles.

Associated with the phenomena of wetting and non-wetting are the so-called capillary phenomena. If a capillary (a tube of small diameter) is lowered into a liquid, then the surface of the liquid in the capillary takes on a concave shape, close to spherical in the case of wetting and convex in the case of non-wetting. Such surfaces are called menisci.

Capillaries are those tubes in which the radius of the meniscus is approximately equal to the radius of the tube.

Rice. 9. Capillary in wetting (a) and non-wetting (b) liquids

Fig. 10. Rise of liquid in a capillary in case of wetting

In the case of a concave meniscus, the additional pressure is directed towards the center of curvature outside the fluid. Therefore, the pressure under the meniscus is less than the pressure under the flat surface of the liquid in the vessel by the amount of Laplace pressure:

R– meniscus radius, r– radius of the capillary tube.

Consequently, Laplace pressure will cause the liquid in the capillary to rise to such a height h(Fig.9) until the hydrostatic pressure of the liquid column balances the Laplace pressure:

From the last equation:

(12)

Equation (12) is called Jurin's formula. If the liquid does not wet the capillary walls, the meniscus is convex, cos θ < 0, то жидкость в этом случае опускается ниже уровня жидкости в сосуде на такую же глубину h according to formula (12) (Fig. 9).

Consider a convex surface (Fig. 5.18), the curvature of which at the point ABOUT for each of two mutually perpendicular normal sections is different. Let me be the external normal

to the surface at a point ABOUT; MN And R g R 2- main sections. Let us mentally select a surface element AS U and calculate the surface tension forces acting on the segments AB And CD, AC And B.D. believing that AB = CD And AC~BD. For each unit of contour length ABDC surface tension force A surrounding fluid, tending to stretch the surface element AS n in all directions. All forces acting on the side AB, replace with one resultant force A.F. applied to the middle of the segment AB= A/ in perpendicular parallel P, only in them instead Rx will the radius of curvature be £? 2 perpendicular sections R g R. g. Radius R 2 shown in Fig. 5.18 segment P-fi." Hence the resultant AF-* of all normal forces acting on four sides

surface element A5 P, AF~ = DK. +AF, + afs fAF. = V af, yes (rAS n | - -|- -V

The force AF^ presses the surface element A5 P to the layers located below it. Hence the average pressure p cf, due to the curvature of the surface,

To get the pressure on r a at a point, let us direct AS to zero. Moving to the limit of the ratio of AF^ to area asn, on which this force acts, we get AF^dF.

AS n -*o AS n dS n \ R, R 2

But by definition

p. = about 14-+ 4-\ (5 - 8)

p„ = a I ■

Where Rlt R 2- the main radii of curvature at a given point on the surface.

In differential geometry the expression e = -~ ^--\-

J--) is called the average curvature of the surface at the point R.

It has the same meaning for all pairs of normal sections perpendicular to each other.

Expression (5.8) establishing the dependence of the hydrostatic pressure drop r a at the interface between two phases (liquid - liquid, liquid -■ gas or vapor) from interfacial surface tension A and average!! the curvature of the surface 8 at the point under consideration is called Laplace's formula in honor of the French physicist Laplace.

Magnitude r a is added to the capillary pressure p corresponding to a flat surface. If the surface is concave, then a minus sign is placed in formula (5.8). In the general case of an arbitrary surface, the radii of curvature Rx And R 2 may differ from each other both in magnitude and sign. So, for example, at the surface shown in Fig. 5.19, radii of curvature Rx And R 2 in two mutually perpendicular normal sections are different in magnitude and sign. This case may result in positive or negative values r a depending on absolute value Rx And R2. It is generally accepted that if the center of curvature of a normal section is located below the surface, then the corresponding radius of curvature is positive, if above the surface it is negative. Surfaces whose average curvature

at all points is equal to zero e == ~(~--1" - 0, called minimal surfaces. If at one point of such a surface /? 1 >0, then automatically /? 2<С0.

For a sphere, any normal section is a circle of radius R, therefore in formula (5.8) /? x = R2 = R and additional capillary pressure

R. = ~.(5-9)

For a soap bubble due to the existence of its outer and inner surfaces

P*=-~-(5-Yu)

If for a circular cylinder one of the normal sections is considered to be the section running along the generatrix, then Rx= co. The second section perpendicular to it gives a circle of radius

R (R 2 = R). Therefore, in accordance with formula (5.8), the additional capillary pressure under the cylindrical surface

R. = -)|- (5-I)

From expressions (5.9) - (5.11) it is clear that when the shape of the surface changes, only the coefficient in front of the ratio changes a/R. If the surface of the liquid is flat, then R x ~ R 2 = co and therefore p z = 0. In this case, the total pressure

Р = Pi ± р а = Pi ± 0 = p t .

The additional capillary pressure, determined by Laplace's formula, is always directed towards the center of curvature. Therefore, for a convex surface it is directed inside the liquid, for a concave surface it is directed outward. In the first case, it is added to the capillary pressure p h in the second, it is subtracted from it. Mathematically, this is taken into account by the fact that for a convex surface the radius of curvature is considered positive, for a concave surface it is considered negative.

The qualitative dependence of the additional capillary pressure on the curvature of the surface can be observed in the following experiment (Fig. 5.20). ends And I'm B glass tee is immersed in a solution of soapy water. As a result, both ends of the tee are covered with soap film. Taking the tee out of the solution, through the process WITH blow two soap bubbles. As a rule, due to various reasons, bubbles have different sizes. If you close hole C, the larger bubble will gradually inflate, and the smaller one will contract. This convinces us that the capillary pressure caused by the curvature of the surface increases with decreasing radius of curvature.

To get an idea of ​​the value of the additional cap:pillar pressure, let’s calculate it for a drop with a diameter of 1 micron (clouds often consist of approximately such drops):

2a 2.72.75-Yu- 3 „ mgt

r --=-==-= 0.1455 MPa.

5.8. Wetting

Surface tension is possessed not only by the free surface of a liquid, but also by the interface between two liquids, a liquid and a solid, and also by the free surface of a solid. In all cases, surface energy is defined as the difference between the energy of molecules at the interface and the energy in the bulk of the corresponding phase. In this case, the value of surface energy at the interface depends on the properties of both phases. So, for example, at the water-air boundary a = 72.75-10 ~ 3 N/m (at 20 °C and normal atmospheric pressure), at the water-ether boundary a= 12-10 3 N/m, and at the water-mercury boundary a = 427-10~ 3 N/m.

Molecules (atoms, ions) located on the surface of a solid body experience attraction from one side. Therefore, solids, like liquids, have surface tension.

Experience shows that a drop of liquid located on the surface of a solid substrate takes on one shape or another depending on the nature of the solid, the liquid and the environment in which they are located. To reduce the potential energy in the gravitational field, a liquid always tends to take a form in which its center of mass occupies the lowest position. This tendency leads to the spreading of liquid over the surface of a solid. On the other hand, surface tension forces tend to give the liquid a shape that corresponds to a minimum of surface energy. The competition between these forces leads to the creation of one form or another.

Spontaneous increase in the area of ​​the solid-liquid or liquid phase boundary A- liquid IN under the influence of molecular cohesive forces is called spreading.

Let us find out the reasons leading to the spreading of a drop over the surface. Per molecule WITH(Fig. 5.21, A), located at the point of contact of a drop of liquid with a solid substrate, with one

On both sides there are attractive forces of liquid molecules, the resultant of which is Fj_ directed along the bisector of the contact angle on the other - molecules of a solid body, the resultant of which F 2 perpendicular to its surface. Resultant R of these two forces is tilted to the left of the vertical, as shown in the figure. In this case, the tendency of the liquid to position its surface perpendicular to R will lead to its spreading (wetting).

The process of liquid spreading stops when the angle Ф (it is called regional) between the tangent to the surface of the liquid at the point WITH and the surface of a solid body reaches a certain limiting value rt k, characteristic of each liquid-solid pair. If the contact angle is acute

(0 ^ ■& ^ -), then the liquid wets the surface of the solid

body and the smaller it is, the better. At $k= 0, complete wetting occurs, in which the liquid spreads over the surface until a monomolecular film is formed. Wetting is usually observed at the interface of three phases, one of which is a solid (phase 3), and the other two - immiscible liquids or liquid and gas (phases / and 2) (see Fig. 5.21, c).

If strength Fx more than F. 2, i.e., from the liquid side the attractive force on the selected molecule is greater than from the solid side, then the contact angle $ will be large and the picture looks as shown in Fig. 5.21, b. In this case, the angle Ф is obtuse (i/2< § ^ я) и жидкость частично (при неравенстве) или полностью (при равенстве) не смачивает твердую подложку. По отношению к стеклу такой несмачивающей жидкостью яв­ляется, например, ртуть, гдесозд = - 1. Однако та же самая ртуть хорошо смачивает другую твердую подложку, например цинк.

These considerations can be expressed quantitatively in

based on the following ideas. Let us denote by o"i_ 2, °1-з, 0-2 -3 respectively, surface tension at the boundary of liquid - gas, solid - gas and liquid -■ solid surface. The directions of action of these forces in the section will be depicted by arrows (Fig. 5.22). The following surface tension forces act on a drop of liquid located on a solid substrate: at the boundary / - 3 -ffi-з, tending to stretch the drop, and on the border 2 - 3 -Og-z. tending to pull it towards the center. Surface tension 04-2 at the boundary 1-2 directed tangentially to the surface of the drop at a point WITH. If the contact angle Ф is acute, then the projection of the force cri_ 2 onto the plane of the solid substrate (ov 2 cos Ф) will coincide in the direction with о 2.-з (Fig. 5.22; A). In this case, the actions of both forces

will add up. If the angle ft is obtuse, as shown in Fig. 5.21, b, then cos ft is negative and the projection cri._ 2 cosft will coincide in direction with O1-.3. When a drop is in equilibrium on a solid substrate, the following equality must be observed:

= 02-3 + SG1-2 soeF. (5.12)

This equation was derived in 1805 Mr. Jung and named after him. Attitude

B =---^- = cos ft

called wetting criterion.

Thus, the contact angle ft depends only on the surface tensions at the boundaries of the corresponding media, determined by their nature, and does not depend on the shape of the vessel and the magnitude of gravity. When equality (5.12) is not complied with, the following cases may occur. If 01-3 greater than the right side of the equation (5.12), then the drop will spread and the angle ft-■ will decrease. It may happen that cos ft increases so much that the right side of the equality (5,12) becomes equal to o"b_ 3, then equilibrium of the drop will occur in an extended state. If ov_ 3 is so large that even at cos ft = 1 left side of equality (5.12) more right (01 _z > 0 2 -з + o"i_ 2)> then the drop will stretch into a liquid film. If the right side of the equality (5.12) more than o"i 3, then the drop contracts to the center, the angle ft increases, and cos ft decreases accordingly until equilibrium occurs. When cos ft becomes negative, the drop will take the shape shown in Fig. 5.22, b. If it turns out that 0 2 - 3 so great that even at cos ft = -1 (ft = i) right side of equality (5.12) there will be more o"i-z (01 -z <02 h- 01-2)1 then in the absence of gravity the drop will contract into a ball. This case can be observed in small drops of mercury on the surface of glass.

The wetting criterion can be expressed in terms of the work of adhesion and cohesion. Adhesion A a is the occurrence of a connection between the surface layers of two dissimilar (solid or liquid) bodies (phases) brought into contact. A special case of adhesion, when the contacting bodies are identical, is called cohesion(denoted A c). Adhesion is characterized by the specific work spent on separating bodies. This work is calculated per unit area of ​​contact between the surfaces and depends on how they are separated: by shear along the interface or by separation in a direction perpendicular to the surface. For two different bodies (phases) A And IN it can be expressed by the equation

A a= hundred +and in-One-in,

Where A A, and in, and A - in- coefficients of surface tension of phases A and B at the boundary with air and between them.

In the case of cohesion, for each of the phases A and B we have:

АШ = 2а A, A<*> = 2a c.

For the drop we are considering

L S| =2a]_ 2 ; A a= ffi^ 3 -f ai_ 2 - sb-z-

Hence the wetting criterion can be expressed by the equality

IN - With

Thus, as the difference increases 2A a-L with wetting improves.

Note that the coefficients cti-z andОо„ 3 are usually identified with the surface tension of a solid at the boundaries with gas and liquid, while in a state of thermodynamic equilibrium the surface of a solid is usually covered with an equilibrium adsorption layer of the substance forming the drop. Therefore, when accurately solving the problem for equilibrium contact angles, the values ​​of cri_ 3 and (Tg-z., generally speaking, should be attributed not to the solid body itself, but to the adsorption layer covering it, the thermodynamic properties of which are determined by the force field of the solid substrate.

Wetting phenomena are especially pronounced in zero gravity. The study of liquid in a state of space weightlessness was first carried out by the Soviet pilot-cosmonaut P.R. Popovich on the Vostok-4 spacecraft. In the ship's cabin there was a spherical glass flask half filled with water. Since water completely wets clean glass (O = 0), under weightless conditions it spread over the entire surface and closed the air inside the flask. Thus, the interface between glass and air disappeared, which turned out to be energetically beneficial. However, the contact angle i) between the surface of the liquid and the walls of the flask and in a state of weightlessness remained the same as it was on Earth.

The phenomena of wetting and non-wetting are widely used in technology and everyday life. For example, to make a fabric water-repellent, it is treated with a hydrophobizing (impairing water wetting) substance (soap, oleic acid, etc.). These substances form a thin film around the fibers, increasing the surface tension at the water-fabric interface, but only slightly changing it at the fabric-air interface. In this case, the contact angle O increases upon contact with water. In this case, if the pores are small, water does not penetrate into them, but is retained by the convex surface film and collects in drops that easily roll off the material.

The sanding liquid does not flow out through very small openings. For example, if the threads from which the sieve is woven are covered with paraffin, then you can carry water in it, if, of course, the layer of liquid is small. Thanks to this property, waterfowl insects running quickly through the water do not wet their paws. Good wetting is necessary when painting, gluing, soldering, dispersing solids in a liquid medium, etc.

A rubber ball or soap bubble can remain in equilibrium only if the air pressure inside them is a certain amount greater than the pressure of the outside air. Let's calculate the excess of internal pressure over external pressure.

Let the soap bubble have a radius and let the excess pressure inside it over the external pressure be equal to To increase the volume of the bubble by a vanishingly small amount, you need to expend work that goes into increasing the free energy of the surface of the bubble and is equal to where a is the surface tension of the soap film, the size of one of the surfaces of the bubble ( For simplicity, we neglect the difference between the radii of the inner and outer surfaces). So we have the equation

on the other side,

Substituting expressions for into the above equation, we get:

According to the law of reaction, the pressure produced by the bubble on the air inside it has the same value.

If instead of a bubble that has two surface films, we consider a drop that has only one surface, then we will come to the conclusion that the surface film exerts a pressure on the inside of the drop equal to

where is the radius of the drop.

In general, due to the curvature of the surface layer of the liquid, excess pressure is created: positive under the convex surface and negative under the concave surface. Thus, in the presence of curvature, the surface layer of the liquid becomes a source of force directed from the convex side of the layer to the concave side.

Rice. 226. To an explanation of Laplace's formula.

Laplace gave a formula for excess pressure suitable for the case when the surface of the liquid has any shape allowed by the physical nature of the liquid state. This Laplace formula has the following form:

where have the following meaning. At some point on the surface of the liquid (Fig. 226), you need to imagine a normal and through this normal draw two mutually perpendicular planes that intersect the surface of the liquid along the curves and The radii of curvature of these curves at the point are denoted by

It is easy to see that from Laplace’s formula for a flat surface of a liquid we obtain a for a spherical surface, as we derived earlier.

If the surface were “saddle-shaped”, then the curves would lie on opposite sides of the tangent plane in

point then the radii would have different signs. In geometry it is proven that for the so-called minimal surfaces, i.e. those having the smallest possible area for a given contour, the sum is equal to zero everywhere. Soap films that tighten a wire circuit have precisely this property.

Foam is a collection of bubbles that have common walls. The curvature of such a wall (defined by the expression + is proportional to the pressure difference on both sides of the wall.

If the end of a clean glass rod is immersed in clean water and the rod is removed, we will see a drop of water hanging at the end. It is obvious that water molecules are more attracted to glass molecules than to each other.

Similarly, a drop of mercury can be lifted with a copper stick. In such cases, the solid is said to be wetted by the liquid.

It will be different if we dip a clean glass rod into pure mercury or if we lower a glass rod covered with fat into water: here the rod, taken out of the liquid, does not carry away a single drop of this latter. In these cases it is said that the liquid does not wet the solid.

Rice. 227. The arrows show the directions of forces with which the surface layer acts on the column of liquid underneath it.

If you immerse a narrow, clean glass tube in water, the water in the tube will rise to a certain height in defiance of gravity (Fig. 227, a). Narrow tubes are called capillaries, or capillaries, and hence the phenomenon itself is called capillarity. Liquids that wet the walls of a capillary tube undergo capillary rise. Liquids that do not wet the walls of the capillary (for example, mercury in a glass tube) undergo, as shown in Fig. 227, b, lowering. Capillary rises and falls are greater, the narrower the capillaries.

Capillary rises and falls are caused by excess pressure, which arises due to the curvature of the liquid surface. In fact, in a tube that is wetted by a liquid, the liquid forms a concave meniscus. According to what has been said

in the previous paragraph, the surface of such a meniscus will develop a force directed from bottom to top, and this force will support a column of liquid in the tube despite the action of gravity. On the contrary, in a tube that is not wetted by liquid, a convex meniscus will result; it will give a downward force and, therefore, lower the liquid level,

Let us derive the relationship between the surface tension of the liquid, its density, the radius of the tube, and the height of the column rising in the tube. Let the liquid “completely wet” the walls of the tube (like water a glass tube), so that at the point where it meets the tube, the surface of the liquid is tangent to the surface of the tube. This contact takes place along a contour whose length is. Due to surface tension, the contour will develop a force and this force applied to the column will balance the force of its gravity, equal to where is the acceleration of gravity.

Thus,

that is, the height of the capillary rise is proportional to surface tension and inversely proportional to the radius of the tube and the density of the liquid.

The same formula (11) for capillary rise can be obtained as a consequence of Laplace’s formula (10) or (in the case of a symmetrical surface under consideration) formula (9). One can reason like this: in a liquid under a concave surface, the pressure is reduced by an amount; therefore, in equilibrium, when the pressure at the level of the free surface of a liquid poured into a vessel is equal to the pressure of the liquid in the capillary at the same level, the liquid column in the capillary must have such a height that it the pressure balanced the pressure deficit created by the concavity of the meniscal surface. Therefore, this is where formula (11) comes from.

Reasoning similarly, we are convinced that when the liquid “does not wet at all” the walls of the capillary, at equilibrium it will be in the capillary at a level lowered by a height determined by the same formula (11).

Measuring capillary rise is one of the simple ways to determine the value of a.

In Fig. 228 shows the capillary rise of liquid between two plates forming a dihedral angle. It is not difficult to imagine that the rising liquid will be limited at the top

hyperbole; the asymptotes of this hyperbola will be the edges of the dihedral angle and the line lying at the level of the liquid in the vessel.

Let us consider the equilibrium conditions of a liquid in contact with a solid wall (Fig. 229). Let us denote the excess free energy of each square centimeter of the surface of a solid body 3 bordering a vacuum or gas 2 by When a layer of any liquid, wetting the surface of a solid body, spreads over it, the solid-gas interface is replaced by the solid-liquid interface, and the free energy of this new surface will be different. Obviously, the decrease in the free energy of each square centimeter of the surface of a solid body is equal to the work of forces under the influence of which 1 cm of the perimeter of the liquid film moves a distance of 1 cm in the direction perpendicular to the perimeter of the film. Therefore, the difference can be considered as a force applied to 1 cm of the perimeter of the liquid film, acting tangentially to the surface of the solid and causing the liquid to move along the surface of the solid. However, the spreading of a liquid over the surface of a solid body is accompanied by an increase in the surface between liquid 1 and vacuum or gas 2, which is prevented by the surface tension of the liquid. In the general case, when a solid body is not completely wetted by a liquid, the force (as shown in Fig. 229, a) is directed under a certain angle to the surface of a solid body; this angle is called the contact angle. We see, therefore, that a liquid bordering a solid body will be in equilibrium when

From this we find that the contact angle at which, at equilibrium, the free surface of the liquid meets the surface

Rice. 228. Capillary rise of liquid between plates forming a dihedral angle.

Rice. 229. Liquid wets a solid wall (a); does not wet the hard wall

solid body, is determined by the formula

From the meaning of the derivation of formula (12), it is clear that this formula remains valid for the case when the liquid does not wet the solid (Fig. 229, b); then the contact angle will be obtuse; the absence of wetting means that (i.e., the free energy of a solid body at its interface with a vacuum or gas is less than at the interface of the same body with a liquid; in other words, in this case, when a liquid moves along the surface of a solid body, there will be no work be carried out, but, on the contrary, work will need to be expended in order to carry out such a movement of fluid).

With complete wetting, the contact angle and with complete absence of wetting, the contact angle depends on the nature of the contacting substances and on the temperature. If you tilt the wall of a vessel, the contact angle does not change.

Formula (12) explains the shape of a drop lying on a horizontal plane. On a solid support, which is wetted by a liquid, the drop takes the shape shown in Fig. 230; if the support is not wetted, then the drop shape shown in Fig. is obtained. 231, where the contact angle is obtuse.

Rice. 230. Drop of wetting liquid.

Rice. 231. A drop of non-wetting liquid.

Absolutely clean glass is completely wetted by water, ethyl alcohol, methyl alcohol, chloroform, and benzene. For mercury on clean glass, the contact angle is 52° (for a freshly formed drop 41°), for turpentine 17°, for ether 16°.

When the liquid completely wets the stand, no drops appear, but the liquid spreads over the entire surface. This happens, for example, with a drop of water on an absolutely clean glass plate. But usually the glass plate is somewhat dirty, which prevents the drop from spreading and creates a measurable contact angle.

Rice. 232. Oil drop on water

The considerations on the basis of which the formula was derived can also be applied to the case when instead of a solid body we have a second liquid, for example, when an oil drop floats on the surface of water (Fig. 232). But in this case the directions of the forces are no longer opposite; When a liquid comes into contact with a solid, the normal component of the surface

tension is balanced by the resistance of the solid wall, but this does not occur when liquids come into contact; therefore, in this case, the equilibrium condition should be written differently, namely as the equality of the total force and the geometric sum (taken with the opposite sign) of the forces

If, for example, olive oil floats on water, then din/cm, din/cm and dan/cm. Thus, here the surface tension at the interface of air and water is greater than the sum of both surface tensions that the oil has in relation to both air and water; we will therefore have unlimited spreading of the drop. The thickness of the oil layer will reach the size of one molecule (about cm), and then the layer will begin to disintegrate. But if the water is contaminated, its surface tension becomes lower, and then a large drop of oil may remain on the surface after a very thin layer of oil has spread through the water.

A liquid that penetrates, due to the action of molecular forces, into a thin gap between two surfaces of solids has a wedging effect on these surfaces. The wedging effect of thin layers of liquid was experimentally proven by the skillful experiments of Prof. B.V. Deryagin, who also developed the theory of this phenomenon and explained the Rehbinder effect on the basis of the wedging action of the liquid (§ 46).

In contact with another medium, it is in special conditions compared to the rest of the liquid mass. The forces acting on each molecule of the surface layer of the liquid bordering the vapor are directed towards the volume of the liquid, that is, into the liquid. As a result, work is required to move a molecule from the depth of the liquid to the surface. If at a constant temperature the surface area is increased by an infinitesimal amount dS, then the work required for this will be equal to. The work to increase the surface area is done against the forces of surface tension, which tend to reduce the surface. Therefore, the work of the surface tension forces themselves to increase the surface area of ​​the liquid will be equal to:

Here the proportionality coefficient σ is called surface tension coefficient and is determined by the amount of work done by surface tension forces based on the change in surface area per unit. In SI, the surface tension coefficient is measured in J/m 2.

Molecules of the surface layer of a liquid have excess potential energy compared to deep molecules, which is directly proportional to the surface area of ​​the liquid:

The increase in potential energy of the surface layer is associated only with the increase in surface area: . Surface tension forces are conservative forces, therefore the equality holds: . Surface tension forces tend to reduce the potential energy of the liquid surface. Typically, the energy that can be converted into work is called free energy U S . Therefore, we can write it down. Using the concept of free energy, we can write formula (6.36) as follows: . Using the last equality we can determine surface tension coefficient as a physical quantity numerically equal to the free energy of a unit surface area of ​​a liquid.

The effect of surface tension forces can be observed using a simple experiment on a thin film of liquid (for example, soap solution) that envelops a rectangular wire frame, one side of which can be mixed (Fig. 6.11). Let us assume that the movable side, length l, is acted upon by an external force F B , moving the movable side of the frame uniformly over a very small distance dh. The elementary work of this force will be equal to , since the force and displacement are co-directed. Since the film has two surfaces and, surface tension forces F are directed along each of them, the vector sum of which is equal to the external force. The modulus of the external force is equal to twice the modulus of one of the surface tension forces: . The minimum work done by an external force is equal in magnitude to the sum of the work done by surface tension forces: . The amount of work done by the surface tension force will be determined as follows:

, Where . From here. That is surface tension coefficient can be defined as a value equal to the force of surface tension acting tangentially to the surface of the liquid per unit length of the dividing line. Surface tension forces tend to reduce the surface area of ​​a liquid. This is noticeable for small volumes of liquid, when it takes the form of droplets-balls. As is known, it is the spherical surface that has the minimum area for a given volume. A liquid taken in large quantities, under the influence of gravity, spreads over the surface on which it is located. As is known, the force of gravity depends on the mass of the body, therefore its value also decreases as the mass decreases and at a certain mass becomes comparable or even much less than the value of the surface tension force. In this case, the force of gravity can be neglected. If a liquid is in a state of weightlessness, then even with a large volume its surface tends to be spherical. This is confirmed by the famous Plateau experience. If you select two liquids with the same density, then the effect of gravity on one of them (taken in a smaller quantity) will be compensated by the Archimedean force and it will take the shape of a ball. Under this condition, it will float inside another liquid.

Let's consider what happens to a drop of liquid 1, bordering on one side with steam 3, on the other side with liquid 2 (Fig. 6.12). Let us choose a very small element of the interface between all three substances dl. Then the surface tension forces at the interfaces between the media will be directed tangentially to the contour of the interfaces and are equal to:

We neglect the effect of gravity. Liquid drop 1 is in equilibrium if the following conditions are met:

(6.38)

Substituting (6.37) into (6.38), reducing both sides of equalities (6.38) by dl, squaring both sides of equalities (6.38) and adding them, we obtain:

where is the angle between the tangents to the dividing lines of the media, called edge angle.

Analysis of equation (6.39) shows that when we obtain and liquid 1 completely wets the surface of liquid 2, spreading over it in a thin layer ( complete wetting phenomenon ).

A similar phenomenon can be observed when a thin layer of liquid 1 spreads over the surface of a solid body 2. Sometimes, on the contrary, the liquid does not spread over the surface of a solid body. If , That and liquid 1 does not completely wet solid body 2 ( phenomenon of complete non-wetting ). In this case, there is only one point of contact between liquid 1 and solid 2. Complete wetting or non-wetting are limiting cases. You can really watch partial wetting , when the contact angle is acute () and partial non-wetting , when the contact angle is obtuse ( ).

In Figure 6.13 A cases of partial wetting are shown, and in Fig. 6.13 b examples of partial non-wetting are given. The considered cases show that the presence of surface tension forces of adjacent liquids or liquids on the surface of a solid body leads to curvature of the surfaces of liquids.

Let's consider the forces acting on a curved surface. The curvature of a liquid surface results in forces acting on the liquid below that surface. If the surface is spherical, then surface tension forces are applied to any element of the circumference (see Fig. 6.14), directed tangentially to the surface and tending to shorten it. The resultant of these forces is directed towards the center of the sphere.

Per unit surface area, this resultant force exerts additional pressure, which is experienced by the fluid under the curved surface. This additional pressure is called Laplace pressure . It is always directed towards the center of curvature of the surface. Figure 6.15 shows examples of concave and convex spherical surfaces and shows the Laplace pressures, respectively.

Let us determine the value of Laplace pressure for a spherical, cylindrical and any surface.

Spherical surface. Drop of liquid. As the radius of the sphere decreases (Fig. 6.16), the surface energy decreases, and the work is done by the forces acting in the drop. Consequently, the volume of liquid under a spherical surface is always somewhat compressed, that is, it experiences Laplace pressure, directed radially to the center of curvature. If, under the influence of this pressure, the ball reduces its volume by dV, then the amount of compression work will be determined by the formula:

The decrease in surface energy occurred by an amount determined by the formula: (6.41)

The decrease in surface energy occurred due to the work of compression, therefore, dA=dU S. Equating the right-hand sides of equalities (6.40) and (6.41), and also taking into account that and , we obtain the Laplace pressure: (6.42)

The volume of liquid under a cylindrical surface, as well as under a spherical one, is always somewhat compressed, that is, it experiences Laplace pressure directed radially to the center of curvature. If, under the influence of this pressure, the cylinder reduces its volume by dV, then the magnitude of the compression work will be determined by formula (6.40), only the magnitude of the Laplace pressure and the increment in volume will be different. The decrease in surface energy occurred by the amount determined by formula (6.41). The decrease in surface energy occurred due to the work of compression, therefore, dA=dU S. Equating the right-hand sides of equalities (6.40) and (6.41), and also taking into account that for a cylindrical surface and , we obtain the Laplace pressure:

Using formula (6.45), we can go to formulas (6.42) and (6.44). So for a spherical surface, therefore, formula (6.45) will be simplified to formula (6.42); for cylindrical surface r 1 = r, a , then formula (6.45) will be simplified to formula (6.44). To distinguish a convex surface from a concave one, it is customary to assume that the Laplace pressure is positive for a convex surface, and accordingly, the radius of curvature of the convex surface will also be positive. For a concave surface, the radius of curvature and the Laplace pressure are considered negative.