Medical Physics. Ivan Tanev Ivanov. Thracian University. 2016
CHAPTER 2. RHEOLOGY AND HYDRODYNAMICS
2.1. Potential energy and forces of interaction between the particles of the substance. Molecular structure of liquids and solids. Polymers and biopolymers. Liquid crystals. Phase transitions.
There are forces of strong interaction between the constituent particles (atoms and molecules) of any substance. These forces are usually expressed by the potential energy of interaction, Ер. Depending on the distance, r, between any pair of particles these forces cause attraction (negative Ер) or repulsion (positive Ер). At greater distances, the forces have electrostatic character - Coulombic forces of attraction or repulsion that decrease with distance, r, as 1/r2. At intermidiate distances the forces cause attraction and, after their discoverer, are called forces of Van der Waals. They originate from the interaction between permanent or induced electric dipoles and decrease with distance, r, as 1/r6. The attraction forces between similar molecules are called cohesion, while those acting between different types of molecules - adhesion. At very small distances only repulsive forces act.
Fig. 2. 1. 1. Potential energy, Ep, of the interaction between two adjacent constituent particles as a function of their displacement, r.
In physics, bodies having high density (liquids and solids) are considered as condensed matter. For condensed matter fig. 2.1.1 shows a typical curve depicting how the Ер of two adjucent constituent particles depends on the distance, r, between them. Aparently, the Ер has a single minimum (Ерo). When the distance between the particles is ro, the potential energy of interaction has minimal value, Ерo. This position of particles is stable one (equilibrium) because each change in the displacement, ro, brings about force that restors back the equilibrium value of this displacement.
In liquid bodies, the value of Epo is closely equal to the average kinetic (thermal) energy of molecules. Therefore, the molecules of a liquid oscilate remaining at their equilibrium position for short time only, after which they jump to other equilibrium position and so on. Thus, a flow could be formed within any liquid. Even more properties originate from this molecular particularity of liquids. Liquids have no own shape and take the shape of the container in which they are placed. Except to flow, liquids readily evaporate, thereby become cooled. Diffusion within the liquids proceeds at a high rate, and gas absorbtion is very strong. The major constituent of human body is water, which is almost a perfect liquid, strongly alleviating transport processes of metabolism and gas exchange.
In solid bodies Epo is much larger than the average thermal energy of the particles. These strong interactive forces compel each particle of solids to stay at one place allowing it only to vibrate around its equilibrium position (node) that is immobile. Thus, solids do not leak, have their own shape and resist its change.
Upon the action of ouside force the solid body changes its dimentions and shape; the distance between its constituent particles also changes accordingly. The mean displacement between each pairs of adjacent particls decreases at contraction and increases at stretching. In both cases, strong forces arise between the particles acting to restore the equilibrium displacement back. These intrinsic forces are called internal tension. Their role is to balance the ouside force and reduce the change in body shape.
In crystalline solids, the nodes form a spatially regular geometric form, called a crystalline lattice. It contains a huge number of identical elementary cells that have the same directions in space. The nodes of the crystalline lattice may containe atoms (atomic lattice), molecules (molecular lattice), ions of different sign (ion lattice), and ions of the same sign (metal lattice). The crystalline bodies have a specific melting temperature above which they pass from a solid to a liquid phase. Upon melting the distances between the constituent particles change, thereby Ep and the internal energy of the body also change and the melting body absorbs heat. This process is referred to as a phase transition of first order. Upon cooling, the reverse process, crystallization, occurs when each particle arrives at a stable position (node) and the crystalline lattice is again build.
The crystal bodies are found in two forms, as single crystals and as polycrystals. Polycrystals are build up of a multitude of adjacent, randomly oriented small crystals (crystallites). Physical properties (mechanical, optical, electrical) of monocrystalline bodies are different in various directions - anisotropism. Some monocrystalline bodies display optical birefringence.
Another example of solids is the amorphous bodies, whereat the equilibrium positions of particles are randomly distributed throughout the volume of the body. Amorphous bodies are isotropic, i.e., their properties are the same in different directions.
Upon heating the amorphous bodies pass from a solid to a viscous liquid state (melt) with a very high viscosity close to that of rubber. However, this process occurs in a wide temperature range. The mean temperature in this range is called glass transition temperature, and the process is referred to as a phase transition of second order. During this type of transition the average distance between particles does not change, and therefore, the internal energy of the body remains constant. Upon cooling of the melt a vitrification process occurs, whereat the constituent particles arrive at randomly selected equilibrium positions without building a crystal lattice. Amorphous bodies and the glass are regarded as supercooled liquids, in which the particles did no succeed to align in a regular crystal lattice due to the very high viscocity of the melt near the point of glass transition.
The oldest and most important example of amorphous matter is glass. It is prepared by cooling the melt of quartz sand. In order to reduce the glass transition temperature, different ingredients are added to the melt. Except its effect on the glass transition temperature, each ingredient imparts some valuable property to the glass. In such a manner various types of glass are produced. Some glasses absorb X-rays (shields for radiation protection), others are able to measure the pH of aqueous media (glass electrodes), or have a certain color and so on.
Natural and synthetic polymers (rubber, Plexiglas, Teflon, cellophane) are other important cases of amorphous bodies. Their molecules are highly elongated or branched and consist of interconnected units of a same or different type. By creating covalent bonds between the polymer molecules a new, very hard polymer material, thermoset, is produced which has a high glass transition temperature. With respect to their thermal properties, the polymers are of two types. Thermoplastics, by contrast to thermoreactive polymers, change their properties reversibly on heating, that is, they recover after cooling back to room temperature. Like crystalline bodies, some polymers have sufficient stiffness and tensile strength, while others are capable of strong reversible deformations. Polymers are easily formed into fibers and webs. They possess high resistace to electric current, moisture and chemical corrosion. Some of them are used in manufacturing artificial prostheses, cellophane membranes for artificial kidney, silicone membrane for artificial lung and for fast cover of open wounds. Some water-soluble polymers (gelatin) are used as a binding medium or a coating in the pills of drugs.
Biopolymers include bones, muscles and other human tissues, which contain biomacromolecules, primarily proteins. They have structures and properties intermediate to those of the crystalline and amorphous bodies. Visco-elasticity is their most prominent mechanical property. The aqueous solutions of biomacromolecules (polysaccharides, proteins) retain their fluidity over a wide temperature range, while they turn into the state of gel on cooling. This is a state similar to solid body whereat without removing the solvent strong cohesive bonds establish between the dissolved particles. The gel state is similar to that of the cytoplasm of cells.
Ceramics is another example of a solid body. It is prepared as a powder mix of various minerals (clay, kaolin, silica sand, etc.) and the mix is baked at high temperature (about 1500°C). The mineral particles partially melt causing their interconnection. Depending on the ratio of the initial minerals the properties of the resulting ceramics (heat and humidity resistance, porosity, resistance to corrosity, etc.) vary. Thus, ceramics with different properties are obtained such as porcelain, faience (majolica), terra cotta and more. In medicine, the ceramics are used to make a variety of prostheses.
Most substances can exist in one of the three basic physical states - solid, liquid and gaseous. Some organic compounds with moderatly elongated polar (more generally, anisotropic) molecules may also exist in the so called liquid crystal state. The latter was discovered in 1888 by the Austrian botanist Raynittser in a newly synthesized compound cholesterilbenzoat. The liquid crystal is an intermediate state between the liquid and solid states and can be obtained only in a narrow temperature range between the melting temperature of the solid crystal and a certain high temperature above which ordinary liquid results. Liquid crystals are distinguished from solid crystals in that the molecules of solids display positional and orientational order,whereas the molecules of liquid crystals display only orientational order.
Liquid crystals can flow, however, they have properties typical of the crystalline bodies - anisotropy, melting point. They are three types - nematic, smectic and cholesteric (Fig. 2.1.2). In smectic liquid crystals, the centers of gravity of the molecules can move only in one plane, while in nematic crystals they can move along the three directions. Nematic liquid crystals represent a fluid of uniformly oriented sticks, while the smektic crystals contain layers made up of sticks tightly packed in a same direction. Biomembranes contain two layers of lipid molecules (lipid bilayer, double layer), which is actually a liquid crystal of smectic type.
Fig. 2.1.2. Arrangement of the molecules of a liquid crystal of nematic type (A), smectic type (B) and cholesteric type (C).
Liquid crystals are thermotropic and lyotropic, the first alter their structure and properties above a certain temperature, the second ones above a certain concentration. The structure and properties of liquid crystals, especially cholesteric ones (e.g. their color) strongly depend on temperature, external electric fields, the presence of vapors of some substances, ect. For example, an external electric field with intensity greater than a certain limit (about 1v) reorients the molecules of liquid crystals parallel or perpendicular to the field lines. This dramatically changes the optical properties of the crystals; light absorption, optical activity, birefringence, color, etc. This change is called a Fredericks transition. This effect is used in portable monitors, televisions and light indicators. This allows the liquid crystals to be used for measuring temperature, as indicator displays, in toxicology.
The esters of cholesterol form cholesteric liquid crystals containing parallel planes. The molecules in each plain are arranged as in the nematic crystals. The neighboring planes are displaced by a certain angle which depends on the temperature and the external electric field. When the temperature approachs a certain value (37°C) the angle will change, resulting in a change in the color of the crystal. This is used for accurate indication of the skin temperature in patients.
The solid phase of the liquid crystal is called gel state. The transition between gel state and liquid crystal state is an example of a first order phase transition. High temperatures also induce a change in the conformation of the macromolecules of biopolymers. In this case, the thermally induced change in the conformation results in abolition of biological activity called denaturation.
2.2. Deformation of solid bodies. Types of deformation and deformation curve. Hooke's law for elastic deformation. Elastic, viscous and viscoelastic deformations of solids
and biological tissues.
Under the action of external forces (mechanical, electrical, etc.) the solid bodies as well as human tissues sustain deformations. As a physical term the deformation (or strain) of a body means any change in its shape and dimensions. During its deformation the solid body generates an intrinsic resistive force which counteracts the ouside deforming force. During their deformation the solid bodies demonstrate their passive mechanical properties - elasticity, hardness, strength and others. In traumatology and orthopedics, in sport medicine and occupational medicine it is necessary to apprehend the passive mechanical properties of biological tissues and substituting prostheses. The deformations and passive mechanical properties of solids and tissues are object of study for the physical discipline rheology (from Greek, rheo – a flow, logos - science).
The main types of deformations in solids are tensile deformation, compression, bending, twisting and shear (angular deformation) - fig. 2.2.1.
Tensile strain: the force acts to pull the body apart;
Compressive strain: the force squeezes the body;
Shear stress: the force causes one part of the body to slide on another part.
Fig. 2.2.1. Main types of deformations.
Let's have a longitudinal tensile deformation of a body and Lo denotes the initial length of the body and L indicates the final length of the deformed body. Then L = L-Lo is called absolute longitudinal deformation and = L / Lo is the relative longitudinal deformation (Fig. 2.2.2).
Upon imposing an outside force to the body, the constituent particles of the body are forcibly displaced from their equilibrium positions. Hence, the distance between these particles changes giving rise to internal force that resists the external force F. With increasing the deformation the internal force increases and, at a given deformation it balances the external force. At this equilibrium state the internal force is denoted as internal stress . This equilibrium state establishes within a very short time. During the longitudinal tensile deformation the arised internal stress is = F/So, where F is the deforming force, and So is the initial cross-section of the body (Fig. 2.2.2).
Each small deformation ( < 1) is always elastic (reversible). This means that after removal of the deforming force, the body recovers back its initial size and shape. Elasticity of a material is its ability to restore back its original shape after tensile deformation when the outside stress or load is removed. The elastic deformations obey the Hook's law: = /E. Here, E is the modulus of elasticity (Young's modulus, elastic modulus). The modulus of elasticity is equal to that stress, , producing deformation L = Lo. The modulus of elasticity is a measure of the resistance the bodies show during the initial step of their deformation. The reciprocal value of E, 1/Е = α is refered to as elasticity coefficient, or elasticity.