Monthly Archives: September 2013

Electrons and Holes

The only empty electronic states in the silicon crystal are in the CB (Figure 5.1c). An electron placed in the CB is free to move around the crystal and also respond to an applied electric field because there are plenty of neighboring empty energy levels. An electron in the CB can easily gain energy from the field and move to higher energy lev­els because these states are empty. Generally we can treat an electron in the CB as if it were free within the crystal with certain modifications to its mass, as explained later in Section 5.1.3.

Since the only empty states are in the CB, the excitation of an electron from the VB requires a minimum energy of Eg. Figure 5.3a shows what happens when a pho­ton of energy hv > Eg is incident on an electron in the VB. This electron absorbs the incident photon and gains sufficient energy to surmount the energy gap Eg and reach the CB. Consequently, a free electron and a “hole,” corresponding to a missing elec­tron in the VB, are created. In some semiconductors such as Si and Ge, the photon ab­sorption process also involves lattice vibrations (vibrations of the Si atoms), which we have not shown in Figure 5.3b.

Electrons and Holes

Electron energy

подпись: electron energy

Figure 5.3

A photon with an energy greater than Eg can excite an electron from the VB to the CB.

When a photon breaks a Si-Si bond, a free electron and a hole in the Si—Si bond are created.

подпись: figure 5.3
a photon with an energy greater than eg can excite an electron from the vb to the cb.
when a photon breaks a si-si bond, a free electron and a hole in the si—si bond are created.

(b)

подпись: (b)Although in this specific example a photon of energy hv > Eg creates an electron – hole pair, this is not necessary. In fact, in the absence of radiation, there is an electron – hole generation process going on in the sample as a result of thermal generation. Due to thermal energy, the atoms in the crystal are constantly vibrating, which corresponds to the bonds between the Si atoms being periodically deformed. In a certain region, the atoms, at some instant, may be moving in such a way that a bond becomes over­stretched, as pictorially depicted in Figure 5.4. This will result in the overstretched bond rupturing and hence releasing an electron into the CB (the electron effectively

Figure 5.4 Thermal vibrations of atoms can break bonds and thereby create electron-hole pairs.

Electrons and HolesBecomes “free”). The empty electronic state of the missing electron in the bond is what we call a hole in the valence band. The free electron, which is in the CB, can wander around the crystal and contribute to the electrical conduction when an electric field is applied. The region remaining around the hole in the VB is positively charged because a charge of — e has been removed from an otherwise neutral region of the crystal. This hole, denoted as /i+, can also wander around the crystal as if it were free. This is be­cause an electron in a neighboring bond can “jump,” that is, tunnel, into the hole to fill the vacant electronic state at this site and thereby create a hole at its original position. This is effectively equivalent to the hole being displaced in the opposite direction, as illustrated in Figure 5.5a. This single step can reoccur, causing the hole to be further displaced. As a result, the hole moves around the crystal as if it were a free positively charged entity, as pictured in Figure 5.5a to d. Its motion is quite independent from that of the original electron. When an electric field is applied, the hole will drift in the di­rection of the field and hence contribute to electrical conduction. It is now apparent that there are essentially two types of charge carriers in semiconductors: electrons and holes. A hole is effectively an empty electronic state in the VB that behaves as if it were a positively charged “particle” free to respond to an applied electric field.

When a wandering electron in the CB meets a hole in the VB, the electron has found an empty state of lower energy and therefore occupies the hole. The electron falls from the CB to the VB to fill the hole, as depicted in Figure 5.5e and f. This is called recombination and results in the annihilation of an electron in the CB and a hole in the VB. The excess energy of the electron falling from CB to VB in certain semiconductors such as GaAs and InP is emitted as a photon. In Si and Ge the excess energy is lost as lattice vibrations (heat).

It must be emphasized that the illustrations in Figure 5.5 are pedagogical pictorial visualizations of hole motion based on classical notions and cannot be taken too seriously, as discussed in more advanced texts (see also Section 5.11). We should remember that the electron has a wavefunction in the crystal that is extended and not localized, as the pictures in Figure 5.5 imply. Further, the hole is a concept that corre­sponds to an empty valence band wavefunction that normally has an electron. Again, we cannot localize the hole to a particular site, as the pictures in Figure 5.5 imply.

Tiiilffili

(d)

H*

Warn

ih

IimwiS

CB

A

VB

H+

Electrons and Holes
Electrons and Holes
Electrons and Holes

N

SlHiMii

 

If)

 

Electrons and Holes Electrons and Holes Electrons and Holes

Figure 5.5 A pictorial illustration of a hole in the valence band wandering around the crystal due to the tunneling of electrons from neighboring bonds.

Conduction in Semiconductors

When an electric field is applied across a semiconductor as shown in Figure 5.6, the energy bands bend. The total electron energy E is KE + PE, but now there is an addi­tional electrostatic PE contribution that is not constant in an applied electric field. A uniform electric field ‘Ex implies a linearly decreasing potential V (x), by virtue of (dV/dx) = —’Ex, that is, V = —Ax + B. This means that the PE, —eV(x), of the electron is now eAx — eB, which increases linearly across the sample. All the energy levels and hence the energy bands must therefore tilt up in the x direction, as shown in Figure 5.6, in the presence of an applied field.

Under the action of “Ex, the electron in the CB moves to the left and immediately starts gaining energy from the field. When the electron collides with a thermal vibra­tion of a Si atom, it loses some of this energy and thus “falls” down in energy in the CB. After the collision, the electron starts to accelerate again, until the next collision, and so on. We recognize this process as the drift of the electron in an applied field, as illustrated in Figure 5.6. The drift velocity v* of the electron is ixeT, x where ixe is the drift mobility of the electron. In a similar fashion, the holes in the VB also drift in an applied field, but here the drift is along the field. Notice that when a hole gains energy, it moves “down” in the VB because the potential energy of the hole is of opposite sign to that of the electron.

Electrostatic PE(x)

Electrons and Holes

Electrons and Holes

(b)

подпись: (b)

Figure 5.6 When an electric field is applied, electrons in the CB and holes in the VB can drift and contribute to the conductivity.

A simplified illustration of drift in Јx.

Applied field bends the energy bands since the electrostatic PE of the electron is

-eV(x) and V(x) decreases in the direction of ЈX/ whereas PE increases.

подпись: figure 5.6 when an electric field is applied, electrons in the cb and holes in the vb can drift and contribute to the conductivity.
a simplified illustration of drift in јx.
applied field bends the energy bands since the electrostatic pe of the electron is
-ev(x) and v(x) decreases in the direction of јx/ whereas pe increases.
»

W

I

W

ЈP

D

C

<D

‘o

(a)

подпись: (a)X

Since both electrons and holes contribute to electrical conduction, we may write the current density 7, from its definition, as

J = envde + epvdh

Where n is the electron concentration in the CB, p is the hole concentration in the VB, and Vde and Vdh are the drift velocities of electrons and holes in response to an applied electric field *EX> Thus,

[5.1]

Where iie and fXh are the electron and hole drift mobilities. In Chapter 2 we derived the drift mobility fie of the electrons in a conductor as

Vde — fte’E’x

Lie =

Exf

Where re is the mean free time between scattering events and me is the electronic mass. The ideas on electron motion in metals can also be applied to the electron motion in the CB of a semiconductor to rederive Equation 5.3. We must, however, use an effective mass m* for the electron in the crystal rather than the mass me in free space. A “free” electron in a crystal is not entirely free because as it moves it interacts with the potential energy (PE) of the ions in the solid and therefore experiences various internal forces. The effective mass m * accounts for these internal forces in such a way that we can relate the acceleration a of the electron in the CB to an external force Fext (e. g., —e*Ex) by Fcxt = m*a just as we do for the electron in vacuum by Fex, = mea. In applying the

M,

And

[5.2]

Vdh = HhZ;

[5.3]

Electrons and Holes

Electron and hole drift velocities

 

Electrons and Holes

Fex t = m* a type of description to the motion of the electron, we are assuming, of course, that the effective mass of the electron can be calculated or measured experimentally. It is important to remark that the true behavior is governed by the solution of the Schrцdinger equation in a periodic lattice (crystal) from which it can be shown that we can indeed describe the inertial resistance of the electron to acceleration in terms of an effective mass m*. The effective mass depends on the interaction of the electron with its environment within the crystal.

We can now speculate on whether the hole can also have a mass. As long as we view mass as resistance to acceleration, that is, inertia, there is no reason why the hole should not have a mass. Accelerating the hole means accelerating electrons tunneling from bond to bond in the opposite direction. Therefore it is apparent that the hole will have a nonzero finite inertial mass because otherwise the smallest external force will impart an infinite acceleration to it. If we represent the effective mass of the hole in the VB by mЈ, then the hole drift mobility will be

[5.4]

подпись: [5.4]Exh

№h — * ml

Where rh is the mean free time between scattering events for holes.

Conductivity of a

Semiconductor

подпись: conductivity of a
semiconductor

[5.5]

подпись: [5.5]Taking Equation 5.1 for the current density further, we can write the conductivity of a semiconductor as

A = en/ie + epfih

Where n andp are the electron and hole concentrations in the CB and VB, respectively. This is a general equation valid for all semiconductors.

INTRINSIC SEMICONDUCTORS

Silicon Crystal and Energy Band Diagram

The electronic configuration of an isolated Si atom is [Ne]3s2p2. However, in the vicinity of other atoms, the 3s and 3p energy levels are so close that the interactions result in the four orbitals ^ (3s), ^ (3/?*), V’ (3 py), and ^ (3 pz) mixing together to form four new hybrid orbitals (called V’tyb) that are symmetrically directed as far away from each other as possible (toward the comers of a tetrahedron). In two dimensions, we can simply view the orbitals pictorially as in Figure 5.1a. The four hybrid orbitals, W» each have one electron so that they are half-occupied. Therefore, a V’hyb orbital of one Si atom can overlap a Vfyb orbital of a neighboring Si atom to form a covalent bond with two spin-paired electrons. In this manner one Si atom bonds with four other Si atoms by overlapping the half-occupied xlrhyb orbitals, as illustrated in Figure 5.1b.

INTRINSIC SEMICONDUCTORS

V’hyb orbitals

Valence

Electron

Conduction band (CB) Empty of electrons at 0 K.

Band gap = E l

F ~ 1

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1

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(b)

Si ion core (+4e) (a)

 

Electron energy

A

 

INTRINSIC SEMICONDUCTORS
INTRINSIC SEMICONDUCTORS

Valence band (VB)

Full of electrons at 0 K.

 

(c)

 

INTRINSIC SEMICONDUCTORS INTRINSIC SEMICONDUCTORS

Figure 5.1

A simplified two-dimensional illustration of a Si atom with four hybrid orbitals T/^yb – Each orbital has one electron.

A simplified two-dimensional view of a region of the Si crystal showing covalent bonds.

The energy band diagram at absolute zero of temperature.

Figure 5.2 A two-dimensional pictorial view of the Si crystal showing covalent bonds as two lines where each line is a valence electron.

INTRINSIC SEMICONDUCTORSEach Si-Si bond corresponds to a bonding orbital, yjrB, obtained by overlapping two neighboring V^hyb orbitals. Each bonding orbital (jfB) has two spin-paired electrons and is therefore full. Neighboring Si atoms can also form covalent bonds with other Si atoms, thus forming a three-dimensional network of Si atoms. The resulting structure is the Si crystal in which each Si atom bonds with four Si atoms in a tetrahedral arrangement. The crystal structure is that of a diamond, which was described in Chapter 1. We can imagine the Si crystal in two dimensions as depicted in Figure 5.1b. The electrons in the covalent bonds are the valence electrons.

The energy band diagram of the silicon crystal is shown in Figure 5.1c.[13] The vertical axis is the electron energy in the crystal. The valence band (VB) contains those electronic states that correspond to the overlap of bonding orbitals (irB). Since all the bonding orbitals (V^) are full with valence electrons in the crystal, the VB is also full with these valence electrons at a temperature of absolute zero. The conduction band (CB) contains electronic states that are at higher energies, those corresponding to the overlap of antibonding orbitals. The CB is separated from the VB by an energy gap Egy called the bandgap. The energy level Ev marks the top of the VB and Ec marks the bottom of the CB. The energy distance from Ec to the vacuum level, the width of the CB, is called the electron affinity x* The gen­eral energy band diagram in Figure 5.1c applies to all crystalline semiconductors with appropriate changes in the energies.

The electrons shown in the VB in Figure 5.1c are those in the covalent bonds be­tween the Si atoms in Figure 5.1b. An electron in the VB, however, is not localized to an atomic site but extends throughout the whole solid. Although the electrons appear localized in Figure 5.lb, at the bonding orbitals between the Si atoms this is not, in fact, true. In the crystal, the electrons can tunnel from one bond to another and exchange places. If we were to work out the wavefunction of a valence electron in the Si crystal, we would find that it extends throughout the whole solid. This means that the electrons in the covalent bonds are indistinguishable. We cannot label an electron from the start and say that the electron is in the covalent bond between these two atoms.

We can crudely represent the silicon crystal in two dimensions as shown in Figure 5.2. Each covalent bond between Si atoms is represented by two lines corre­sponding to two spin-paired electrons. Each line represents a valence electron.

Semiconductors

In this chapter we develop a basic understanding of the properties of intrinsic and extrinsic semiconductors. Although most of our discussions and examples will be based on Si, the ideas are applicable to Ge and to the compound semiconductors such as GaAs, InP, and others. By intrinsic Si we mean an ideal perfect crystal of Si that has no impurities or crystal defects such as dislocations and grain boundaries. The crystal thus consists of Si atoms perfectly bonded to each other in the diamond structure. At temperatures above absolute zero, we know that the Si atoms in the crystal lattice will be vibrating with a distribution of energies. Even though the average energy of the vi­brations is at most 3kT and incapable of breaking the Si-Si bond, a few of the lattice vibrations in certain crystal regions may nonetheless be sufficiently energetic to “rup­ture” a Si-Si bond. When a Si-Si bond is broken, a “free” electron is created that can wander around the crystal and also contribute to electrical conduction in the presence of an applied field. The broken bond has a missing electron that causes this region to be positively charged. The vacancy left behind by the missing electron in the bonding orbital is called a hole. An electron in a neighboring bond can readily tunnel into this broken bond and fill it, thereby effectively causing the hole to be displaced to the orig­inal position of the tunneling electron. By electron tunneling from a neighboring bond, holes are therefore also free to wander around the crystal and also contribute to elec­trical conduction in the presence of an applied field. In an intrinsic semiconductor, the number of thermally generated electrons is equal to the number of holes (broken bonds). In an extrinsic semiconductor, impurities are added to the semiconductor that can contribute either excess electrons or excess holes. For example, when an impurity such as arsenic is added to Si, each As atom acts as a donor and contributes a free elec­tron to the crystal. Since these electrons do not come from broken bonds, the numbers of electrons and holes are not equal in an extrinsic semiconductor, and the As-doped Si in this example will have excess electrons. It will be an n-type Si since electrical con­duction will be mainly due to the motion of electrons. It is also possible to obtain a p-type Si crystal in which hole concentration is in excess of the electron concentration due to, for example, boron doping.

DEFINING TERMS

Average energy Eav of an electron in a metal is deter­mined by the Fermi-Dirac statistics and the density of states. It increases with the Fermi energy and also with the temperature.

Boltzmann statistics describes the behavior of a collection of particles (e. g., gas atoms) in terms of their energy distribution. It specifies the number of particles N(E) with given energy, through N(E) oc exp (-E/kT), where k is the Boltzmann constant. The description is nonquantum mechanical in that there is no restriction on the number of particles that can have the same state (the same wavefunction) with an energy E. Also, it applies when there are only a few particles compared to the number of possible states, so the likelihood of two particles having the same state becomes negligible. This is generally the case for thermally excited electrons in the conduction band of a semiconductor, where there are many more states than electrons. The kinetic energy distribution of gas molecules in a tank obeys the Boltzmann statistics.

Cathode is a negative electrode. It emits electrons or attracts positive charges, that is, cations.

Debye frequency is the maximum frequency of lat­tice vibrations that can exist in a particular crystal. It is the cut-off frequency for lattice vibrations.

Debye temperature is a characteristic temperature of a particular crystal above which nearly all the atoms are vibrating in accordance with the kinetic molecular theory, that is, each atom has an average energy (potential + kinetic) of 3kT due to atomic vi­brations, and the heat capacity is determined by the Dulong-Petit rule.

Density of states g(E) is the number of electron states [e. g., wavefunctions, ^(n, I, mt, ms)] per unit energy per unit volume. Thus, g(E)dE is the number of states in the energy range E to (E + dE) per unit volume.

Density of vibrational states is the number of lattice vibrational modes per unit angular frequency range.

Dispersion relation relates the angular frequency co and the wavevector K of a wave. In a crystal lattice, the coupling of atomic oscillations leads to a particular relationship between co and K which determines the allowed lattice waves and their group velocities. The dispersion relation is specific to the crystal structure, that is, it depends on the lattice, basis, and bonding.

Effective electron mass m*e represents the inertial re­sistance of an electron inside a crystal against an accel­eration imposed by an external force, such as the ap­plied electric field. If Fext = eEx is the external applied force due to the applied field <EX, then the effective mass m* determines the acceleration a of the electron by eEx = m*a. This takes into account the effect of the internal fields on the motion of the elec­tron. In vacuum where there are no internal fields, m*e is the mass in vacuum me.

Fermi-Dirac statistics determines the probability of an electron occupying a state at an energy level E. This takes into account that a collection of electrons must obey the Pauli exclusion principle. The Fermi-Dirac function quantifies this probability via f(E) = 1/{1 + exp[(Ј — EF)/kT]}, where EF is the Fermi energy.

Fermi energy is the maximum energy of the electrons in a metal at 0 K.

Field emission is the tunneling of an electron from the surface of a metal into vacuum, due to the application of a strong electric field (typically Ј > 109Vm_1).

Group velocity is the velocity at which traveling waves carry energy. If co is the angular frequency and K is the wavevector of a wave, then the group velocity vg = dco/dK.

Harmonic oscillator is an oscillating system, for ex­ample, two masses joined by a spring, that can be de­scribed by simple harmonic motion. In quantum me­chanics, the energy of a harmonic oscillator is quantized and can only increase or decrease by a dis­crete amount fico. The minimum energy of a harmonic oscillator is not zero but hco (see zero-point energy).

Lattice wave is a wave in a crystal due to coupled os­cillations of the atoms. Lattice waves may be traveling or stationary waves.

Linear combination of atomic orbitals (LCAO) is a

Method for obtaining the electron wavefunction in the molecule from a linear combination of individual atomic wavefunctions. For example, when two H atoms A and B come together, the electron wavefunctions, based on LCAO, are

Fa = fs(A) + fs(B) fb = fls(A) ~ fs(B)

Where j/s(A) and fs(B) are atomic wavefunctions centered around the H atoms A and B, respectively. The l/a and f/b represent molecular orbital wavefunctions for the electron; they reflect the behavior of the elec­tron, or its probability distribution, in the molecule.

Mode or state of lattice vibration is a distinct, inde­pendent way in which a crystal lattice can vibrate with its own particular frequency co and wavevector K. There are only a finite number of vibrational modes in a crystal.

Molecular orbital wavefunction, or simply molecu­lar orbital, is a wavefunction for an electron within a system of two or more nuclei (e. g., molecule). A mo­lecular orbital determines the probability distribution of the electron within the molecule, just as the atomic orbital determines the electron’s probability distribu­tion within the atom. A molecular orbital can take two electrons with opposite spins.

Orbital is a region of space in an atom or molecule where an electron with a given energy may be found. An orbit, which is a well-defined path for^a-electron, cannot be used to describe the whereabouts of the elec­tron in an atom or molecule because the electron has a probability distribution. Orbitals are generally repre­sented by a surface within which the total probability is high, for example, 90 percent.

Orbital wavefunction, or simply orbital, describes the spatial dependence of the electron. The orbital is f (r, 0, 0), which depends on n, t, and mt, and the spin dependence ms is excluded.

Phonon is a quantum of lattice vibrational energy of magnitude tico, where co is the vibrational angular fre­quency. A phonon has a momentum h K where K is the wavevector of the lattice wave.

Seebeck effect is the development of a built-in poten­tial difference across a material as a result of a temper­ature gradient. If dV is the built-in potential across a temperature difference dT, then the Seebeck coeffi­cient S is defined as S = dV/dT. The coefficient gauges the magnitude of the Seebeck effect. Only the net Seebeck voltage difference between different met­als can be measured. The principle of the thermocouple is based on the Seebeck effect.

State is a possible wavefunction for the electron that defines its spatial (orbital) and spin properties, for example, /s(n, t, mt, ms) is a state of the elec­tron. From the Schrцdinger equation, each state cor­responds to a certain electron energy E. We thus speak of a state with energy E, state of energy E, or even an energy state. Generally there may be more than one state jr with the same energy E.

Thermionic emission is the emission of electrons from the surface of a heated metal.

Work function is the minimum energy needed to free an electron from the metal at a temperature of absolute zero. It is the energy separation of the Fermi level from the vacuum level.

Zero-point energy is the minimum energy of a har­monic oscillator fico. Even at 0 K, an oscillator in quantum mechanics will have a finite amount of en­ergy which is its zero-point energy. Heisenberg’s un­certainty principle does not allow a harmonic oscillator to have zero energy because that would mean no un­certainty in the momentum and consequently an infi­nite uncertainty in space (Apx Ax > fi).

Questions and problems

Phase of an atomic orbital

What is the functional form of a Is wavefunction i/^(r)? Sketch schematically the atomic wave­function Vrb(‘*) as a function of distance from the nucleus.

What is the total wavefunction (r, f) ?

What is meant by two wavefunctions (A) and (B) that are out of phase?

Sketch schematically the two wavefunctions (A) and Vs (B) at one instant.

Molecular orbitals and atomic orbitals Consider a linear chain of four identical atoms representing a hypothetical molecule. Suppose that each atomic wavefunction is a Is wavefunction. This system of identical atoms has a center of symmetry C with respect to the center of the molecule (midway between the second and the third atom), and all molecular wavefunctions must be either symmetric or antisym­metric about C.

Using the LCAO principle, sketch the possible molecular orbitals.

Sketch the probability distributions | yjr |2.

If more nodes in the wavefunction lead to greater energies, order the energies of the molecular orbitals. Note: The electron wavefunctions, and the related probability distributions, in a simple potential energy well that are shown in Figure 3.15 can be used as a rough guide toward finding the appropriate njotectr* lar wavefunctions in the four-atom symmetric molecule. For example, if we were to smooth the electron potential energy in the four-atom molecule into a constant potential energy, that is, generate a potential energy well, we should be able to modify or distort, without flipping, the molecular orbitals to somewhat resemble yjr to ^4 sketched in Figure 3.15. Consider also that the number of nodes increases from none for V^i to three for in Figure 3.15.

Diamond and tin Germanium, silicon, and diamond have the same crystal structure, that of diamond. Bonding in each case involves sp3 hybridization. The bonding energy decreases as we go from C to Si to Ge, as noted in Table 4.7.

What would you expect for the bandgap of diamond? How does it compare with the experimental value of 5.5 eV?

Tin has a tetragonal crystal structure, which makes it different than its group members, diamond, silicon, and germanium.

Is it a metal or a semiconductor?

What experiments do you think would expose its semiconductor properties?

Table 4.7

Property

Diamond

Silicon

Germanium

Tin

Melting temperature, °C

3800

1417

937

232

Covalent radius, nm

0.077

0.117

0.122

0.146

Bond energy, eV

3.60

1.84

1.7

1.2

First ionization energy, eV

11.26

8.15

7.88

7.33

Bandgap, eV

?

1.12

0.67

?

Compound III-V Semiconductors Indium as an element is a metal. It has a valency of III. Sb as an element is a metal and has a valency of V. InSb is a semiconductor, with each atom bonding to four neighbors, just like in silicon. Explain how this is possible and why InSb is a semiconductor and not a metal alloy. (Consider the electronic structure and sp3 hybridization for each atom.)

Compound II-VI semiconductors CdTe is a semiconductor, with each atom bonding to four neigh­bors, just like in silicon. In terms of covalent bonding and the positions of Cd and Те in the Periodic Table, explain how this is possible. Would you expect the bonding in CdTe to have more ionic character than that in III-V semiconductors?

*4.6 Density of states for a two-dimensional electron gas Consider a two-dimensional electron gas in

Which the electrons are restricted to move freely within a square area a2 in the xy plane. Following the procedure in Section 4.5, show that the density of states g(E) is constant (independent of energy).

Fermi energy of Cu The Fermi energy of electrons in copper at room temperature is 7.0 eV. The elec­tron drift mobility in copper, from Hall effect measurements, is 33 cm2 V-1 s~].

What is the speed vp of conduction electrons with energies around Ep in copper? By how many times is this larger than the average thermal speed ^thermal of electrons, if they behaved like an ideal gas (Maxwell-Boltzmann statistics)? Why is vF much larger than ^thermal?

What is the De Broglie wavelength of these electrons? Will the electrons get diffracted by the lat­tice planes in copper, given that interplanar separation in Cu = 2.09 A? (Solution guide: Diffrac­tion of waves occurs when 2d sin# = k, which is the Bragg condition. Find the relationship be­tween к and d that results in sin# > 1 and hence no diffraction.)

Calculate the mean free path of electrons at Ep and comment.

Free electron model, Fermi energy, and density of states Na and Au both are valency I metals; that is, each atom donates one electron to the sea of conduction electrons. Calculate the Fermi energy (in eV) of each at 300 К and 0 K. Calculate the mean speed of all the conduction electrons and also the speed of electrons at Ep for each metal. Calculate the density of states as states per eV cm-3 at the Fermi energy and also at the center of the band, to be taken at (Ep + Ф)/2. (See Table 4.1 for Ф.)

Fermi energy and electron concentration Consider the metals in Table 4.8 from Groups I, II, and III in the Periodic Table. Calculate the Fermi energies at absolute zero, and compare the values with the ex­perimental values. What is your conclusion?

Table 4.8

Ef (eV) Erie V)

Metal Group Mat Density (g cm-3) [Calculated] [Experiment]

TOC o "1-5" h z Cu I 63.55 8.96 — 6.5

Zn II 65.38 7.14 — 11.0

A1 III 27 2.70 — 11.8

Temperature dependence of the Fermi energy

Given that the Fermi energy for Cu is 7.0 eV at absolute zero, calculate the Ef at 300 K. What is

The percentage change in Ј> and what is your conclusion?

Given the Fermi energy for Cu at absolute zero, calculate the average energy and mean speed per conduction electron at absolute zero and 300 K, and comment.

X-ray emission spectrum from sodium Structure of the Na atom is [Ne]3s1. Figure 4.59a shows the formation of the 3,s and 3p energy bands in Na as a function of intemuclear separation. Figure 4.59b shows the X-ray emission spectrum (called the L-band) from crystalline sodium in the soft X-ray range as explained in Example 4.6.

From Figure 4.59a, estimate the nearest neighbor equilibrium separation between Na atoms in the

Crystal if some electrons in the 3s band spill over into the states in the 3p band.

Explain the origin of the X-ray emission band in Figure 4.59b and the reason for calling it the

L-band.

What is the Fermi energy of the electrons in Na from Figure 4.59b?

Taking the valency of Na to be I, what is the expected Fermi energy and how does it compare with

(b)

подпись: 
(b)
That in part (c)?

DEFINING TERMS

>

<D

ЛP

<D

W

0.5 1

Intemuclear distance (nm)

Figure 4.59

Energy band formation in sodium.

/.-emission band of X-rays from sodium.

I SOURCE: (b) Data extracted from W. M. Cadt and D. H. Tomboulian, Phys. Rev., 59, 1941, p. 381.

Conductivity of metals in the free electron model Consider the general expression for the conduc­tivity of metals in terms of the density of states g(Ep) at Ep given by

A = e2v2Fxg{EF)

Show that within the free electron theory, this reduces to a — e2nz/me, the Drude expression.

In1′

Mean free path and conductivity in the free electron model

подпись: mean free path and conductivity in the free electron modelMean free path of conduction electrons in a metal Show that within the free electron theory, the mean free path t and conductivity a are related by

87 x 1<

^/3*2/3*

Calculate i for Cu and Au, given each metal’s resistivity of 17 nf2 m and 22 nЈ2 m, respectively, and that each has a valency of I. We are used to seeing a <x n. Can you explain why a <x n2/3?

*4.14 Low-temperature heat capacity of metals The heat capacity of conduction electrons in a metal is proportional to the temperature. The overall heat capacity of a metal is determined by the lattice heat ca­pacity, except at the lowest temperatures. If 8E{ is the increase in the total energy of the conduction elec­trons (per unit volume) and ST is the increase in the temperature of the metal as a result of heat addition, Et has been calculated as follows:

Oo

DEFINING TERMS

Where Et (0)is the total energy per unit volume at 0 K, n is the concentration of conduction electrons, and Efo is the Fermi energy at 0 K. Show that the heat capacity per unit volume due to conduction electrons in the free electron model of metals is

DEFINING TERMS

[4.84]

подпись: [4.84]Heat capacity of

Conduction

Electrons

Where y = {n2/2){nk2/Efo)’ Calculate Ce for Cu, and then using the Debye equation for the lattice heat capacity, find Cv for Cu at 10 K. Compare the two values and comment. What is the comparison at room temperature? (Note: Cvoiume = Cmoiar(p/Mat), where p is the density in g cm-3, CvoiUme is in J K-1 cm“3, and Mat is the atomic mass in g mol-1.)

Secondary emission and photomultiplier tubes When an energetic (high velocity) projectile elec­tron collides with a material with a low work function, it can cause electron emission from the surface. This phenomenon is called secondary emission. It is fruitfully utilized in photomultiplier tubes as il­lustrated in Figure 4.60. The tube is evacuated and has a photocathode for receiving photons as a signal. An incoming photon causes photoemission of an electron from the photocathode material. The electron is then accelerated by a positive voltage applied to an electrode called a dynode which has a work func­tion that easily allows secondary emission. When the accelerated electron strikes dynode D, it can release several electrons. All these electrons, the original and the secondary electrons, are then acceler­ated by the more positive voltage applied to dynode D2. On impact with Ј>2, further electrons are re­leased by secondary emission. The secondary emission process continues at each dynode stage until the final electrode, called the anode, is reached whereupon all the electrons are collected which results in a signal. Typical applications for photomultiplier tubes are in X-ray and nuclear medical instruments

D

D

 

Dynodes

 

Photomultiplier tubes.

I SOURCE: Courtesy of Hamamatsu.

 

3

 

Vacuum — tube

 

—•

Signal

 

Anode

 

DEFINING TERMS DEFINING TERMS

(X-ray CT scanner, positron CT scanner, gamma camera, etc.), radiation measuring instruments (e. g., radon counter), X-ray diffractometers, and radiation measurement in high-energy physics research.

A particular photomultiplier tube has the following properties. The photocathode is made of a semiconductor-type material with Eg ~ 1 eV, an electron affinity x of 0.4 eV, and a quantum efficiency of 20 percent at 400 nm. Quantum efficiency is defined as the number of photoemitted electrons per absorbed photon. The diameter of the photocathode is 18 mm. There are 10 dynode electrodes and an ap­plied voltage of 1250 V between the photocathode and anode. Assume that this voltage is equally dis­tributed among all the electrodes.

What is the longest threshold wavelength for the phototube?

What is the maximum kinetic energy of the emitted electron if the photocathode is illuminated with a 400 nm radiation?

What is the emission current from the photocathode at 400 nm illumination?

What is the KE of the electron as it strikes the first dynode electrode?

It has been found that the tube has a gain of 106 electrons per incident photon. What is the average

Number of secondary electrons released at each dynode?

Thermoelectric effects and Ep Consider a thermocouple pair that consists of gold and aluminum. One junction is at 100 °C and the other is at 0 °C. A voltmeter (with a very large input resistance) is in­serted into the aluminum wire. Use the properties of Au and A1 in Table 4.3 to estimate the emf regis­tered by the voltmeter and identify the positive end.

The thermocouple equation Although inputting the measured emf for V in the thermocouple equa­tion V — aAT + b(AT)2 leads to a quadratic equation, which in principle can be solved for A7 in general AT is related to the measured emf via

AT = aV + a2V2 + <23 V3 H

With the coefficients a, 02* etc., determined for each pair of TCs. By carrying out a Taylor’s expansion of the TC equation, find the first two coefficients a and «2- Using an emf table for the K-type thermo­couple or Figure 4.33, evaluate a and <22­

Thermionic emission A vacuum tube is required to have a cathode operating at 800 °C and providing an emission (saturation) current of 10 A. What should be the surface area of the cathode for the two ma­terials in Table 4.9? What should be the operating temperature for the Th on W cathode, if it is to have the same surface area as the oxide-coated cathode?

Table 4.9

Be (Am 2 K-2)

<D(eV)

Th on W

3 x 104

2.6

Oxide coating

100

1

Field-assisted emission in MOS devices Metal-oxide-semiconductor (MOS) transistors in micro­electronics have a metal gate on an Si02 insulating layer on the surface of a doped Si crystal. Consider this as a parallel plate capacitor. Suppose the gate is an A1 electrode of area 50 fim x 50 fim and has a voltage of 10 V with respect to the Si crystal. Consider two thicknesses for the Si02, (a) 100 A and (b) 40 A, where (1 A = 10“10 m). The work function of A1 is 4.2 eV, but this refers to electron emission into vacuum, whereas in this case, the electron is emitted into the oxide. The potential energy barrier be­tween A1 and Si02 is about 3.1 eV, and the field-emission current density is given by Equation 4.46a and

Calculate the field-emission current for the two cases. For simplicity, take me to be the electron mass in free space. What is your conclusion?

4.20

подпись: 4.20CNTs and field emission The electric field at the tip of a sharp emitter is much greater than the “applied field,” *E0. The applied field is simply defined as Vg Id where d is the distance from the cathode tip to the gate or the grid; it represents the average nearly uniform field that would exist if the tip were replaced by a flat surface so that the cathode and the gate would almost constitute a parallel plate capacitor. The tip ex­periences an effective field Ј that is much greater than which is expressed by a field enhancement fac­tor /3 that depends on the geometry of the cathode-gate emitter, and the shape of the emitter; Ј = ^fE(). Further, we can take 4>^2 O % <Ј>3//2 in Equation 4.46. The final expression for the field-emission current density then becomes

6.44 x 107<1>3/2

W„ )

, 1.5×10« ,2 /10.4 (

J = – —- exp(^jexp(-

Fowler – Nordheim field emission current

 

DEFINING TERMS

[4.85]

 

4.21

подпись: 4.21Where <Ј> is in eV. For a particular CNT emitter, <Ј> = 4.9 eV. Estimate the applied field required to achieve a field-emission current density of 100 mA cm2 in the absence of field enhancement (ft = 1) and with a field enhancement of p = 800 (typical value for a CNT emitter).

Nordheim-Fowler field emission in an FED Table 4.10 shows the results of I-V measurements on a Motorola FED microemitter. By a suitable plot show that the I-V follows the Nordheim-Fowler emis­sion characteristics. Can you estimate <I>?

Table 4.10 Tests on a Motorola FED micro field emitter

VG 40.0 42 44 46 48 50 52 53.8 56.2 58.2 60.4

/emission 0.40 2.14 9.40 20.4 34.1 61 93.8 142.5 202 279 367

*4.25 Overlapping bands Consider Cu and Ni with their density of states as schematically sketched in Fig­ure 4.61. Both have overlapping 3d and 4s bands, but the 3d band is very narrow compared to the 4s

Band. In the case of Cu the band is full, whereas in Ni, it is only partially filled.

In Cu, do the electrons in the 3d band contribute to electrical conduction? Explain.

In Ni, do electrons in both bands contribute to conduction? Explain.

Do electrons have the same effective mass in the two bands? Explain.

Can an electron in the 4s band with energy around Ep become scattered into the 3d band as a re­

Sult of a scattering process? Consider both metals.

> E

подпись: > e DEFINING TERMSScattering of electrons from the 4s band to the 3d band and vice versa can be viewed as an additional scattering process. How would you expect the resistivity of Ni to compare with that of Cu, even though Ni has two valence electrons and nearly the same density as Cu? In which case would you ex­pect a stronger temperature dependence for the resistivity?

DEFINING TERMS

> E

Figure 4.61 Density of states and electron filling in Cu and Ni.

*4.26 Overlapping bands at Ep and higher resistivity Figure 4.61 shows the density of states for Cu (or

Ag) and Ni (or Pd). The d band in Cu is filled, and only electrons at Ef in the s band make a contribu­tion to the conductivity. In Ni, on the other hand, there are electrons at Ep both in the s and d bands. The d band is narrow compared with the s band, and the electron’s effective mass in this d band is large; for simplicity, we will assume m* is “infinite” in this band. Consequently, the d-band electrons cannot be accelerated by the field (infinite m*), have a negligible drift mobility, and make no contribution to the conductivity. Electrons in the s band can become scattered by phonons into the d band, and hence be­come relatively immobile until they are scattered back into the s band when they can drift again. Con­sider Ni and one particular conduction electron at Ep starting in the s band. Sketch schematically the magnitude of the velocity gained vx — ux from the field Ex as a function of time for 10 scattering events; vx and ux are the instantaneous and initial velocities, and | vx — ux | increases linearly with time, as the electron accelerates in the s band and then drops to zero upon scattering. If rss is the mean time for s to 5-band scattering, rsd is for 5-band to d-band scattering, rds is for d-band to 5-band scattering, assume the following sequence of 10 events in your sketch: r^, rss, ?sd> 4s. *ss, *sd,

What would a similar sketch look like for Cu? Suppose that we wish to apply Equation 4.27. What does g(Ep) and r represent? What is the most important factor that makes Ni more resistive than Cu? Con­sider Matthiessen’s rule. (Note: There are also electron spin related effects on the resistivity of Ni, but for simplicity these have been neglected.)

4.27 Grtineisen’s law A1 and Cu both have metallic bonding and the same crystal structure. Assuming that the Griineisen’s parameter у for A1 is the same as that for Cu, у = 0.23, estimate the linear expansion coefficient X of Al, given that its bulk modulus К = 75 GPa, cs = 900 J K~1 kg”1, and p = 2.7 g cm-3. Compare your estimate with the experimental value of 23.5 x 10-6 K-1.

DEFINING TERMS

First point-contact transistor invented at Bell Labs. I SOURCE: Courtesy of Bell Labs.

DEFINING TERMS

The three inventors of the transistor: William Shockley (seated), John Bardeen (left), and Walter Brattain (right) in 1948; the three inventors shared the Nobel prize in 1956.

I SOURCE: Courtesy of Bell Labs.

Joint Lubrication

21.1

21.2

21.3

подпись: 21.1
21.2
21.3
Introduction Tribology

Friction • Wear and Surface Damage

Lubrication

21.4

21.5

21.6

21.7

21.8

Michael J. Furey

Mechanical Engineering Depar tment

And Center for Biomedical

Engineering, Virginia Polytechnic

Institute and State University 21.9

подпись: 21.4
21.5
21.6
21.7
21.8
michael j. furey
mechanical engineering depar tment
and center for biomedical
engineering, virginia polytechnic
institute and state university 21.9
Hydrodynamic Lubrication Theories • Transition from Hydrodynamic to Boundary Lubrication Synovial Joints

Theories on the Lubrication of Natural and Normal Synovial Joints

In Vitro Cartilage Wear Studies Biotribology and Arthritis: Are There Connections? Recapitulation and Final Comments Terms and Definitions • Experimental Contact Systems •

Fluids and Materials Used as Lubricants in In Vivo Studies •

The Preoccupation with Rheology and Friction • The Probable Existence of Various Lubrication Regimes • Recent Developments Conclusions

“The Fabric of the Joints in the Human Body is a subject so much the more entertaining, as it must strike every one that considers it attentively with an Idea of fine Mechanical Composition. Wherever the Motion of one Bone upon another is requisite, there we find an excellent Apparatus for rendering that Motion safe and free: We see, for Instance, the Extremity of one Bone molded into an orbicular Cavity, to receive the Head of another, in order to afford it an extensive Play. Both are covered with a smooth elastic Crust, to prevent mutual Abrasion; connected with strong Ligaments, to prevent Dislocation; and inclosed in a Bag that contains a proper Fluid Deposited there, for lubricating the Two contiguous Surfaces. So much in general"

The above is the opening paragraph of the classic paragraph of the classic paper by the surgeon, Sir William Hunter, “Of the Structure and Diseases of Articulating Cartilages” which he read to a meeting of the Royal Society, June 2, 1743 [1]. Since then, a great deal of research has been carried out on the subject of synovial joint lubrication. However, the mechanisms involved are still unknown.

Introduction

The purpose of this article is twofold: (1) to introduce the reader to the subject of tribology—the study of friction, wear, and lubrication; and (2) to extend this to the topic of biotribology, which includes the lubrication of natural synovial joints. It is not meant to be an exhaustive review of joint lubrication theories; space does not permit this. Instead, major concepts or principles will be discussed not only in

The light of what is known about synovial joint lubrication but perhaps more importantly what is not known. Several references are given for those who wish to learn more about the topic. It is clear that synovial joints are by far the most complex and sophisticated tribological systems that exist. We shall see that although numerous theories have been put forth to attempt to explain joint lubrication, the mech­anisms involved are still far from being understood. And when one begins to examine possible connections between tribology and degenerative joint disease or osteoarthritis, the picture is even more complex and controversial. Finally, this article does not treat the (1) tribological behavior of artificial joints or partial joint replacements, (2) the possible use of elastic or poroplastic materials as artificial cartilage, and (3) new developments in cartilage repair using transplanted chondrocytes. These are separate topics, which would require detailed discussion and additional space.

Tribology

The word tribology, derived from the Greek “to rub,” covers all frictional processes between solid bodies moving relative to one another that are in contact [2]. Thus tribology may be defined as the study of friction, wear, and lubrication.

Tribological processes are involved whenever one solid slides or rolls against another, as in bearings, cams, gears, piston rings and cylinders, machining and metalworking, grinding, rock drilling, sliding electrical contacts, frictional welding, brakes, the striking of a match, music from a cello, articulation of human synovial joints (e. g., hip joints), machinery, and in numerous less obvious processes (e. g., walking, holding, stopping, writing, and the use of fasteners such as nails, screws, and bolts).

Tribology is a multidisciplinary subject involving at least the areas of materials science, solid and surface mechanics, surface science and chemistry, rheology, engineering, mathematics, and even biology and bio­chemistry. Although tribology is still an emerging science, interest in the phenomena of friction, wear, and lubrication is an ancient one. Unlike thermodynamics, there are no generally accepted laws in tribology. But there are some important basic principles needed to understand any study of lubrication and wear and even more so in a study of biotribology or biological lubrication phenomena. These basic principles follow.

Friction

Much of the early work in tribology was in the area of friction—possibly because frictional effects are more readily demonstrated and measured. Generally, early theories of friction dealt with dry or unlu­bricated systems. The problem was often treated strictly from a mechanical viewpoint, with little or no regard for the environment, surface films, or chemistry.

In the first place, friction may be defined as the tangential resistance that is offered to the sliding of one solid body over another. Friction is the result of many factors and cannot be treated as something as singular as density or even viscosity. Postulated sources of friction have included (1) the lifting of one asperity over another (increase in potential energy), (2) the interlocking of asperities followed by shear, (3) interlocking followed by plastic deformation or plowing, (4) adhesion followed by shear, (5) elastic hysteresis and waves of deformation, (6) adhesion or interlocking followed by tensile failure, (7) intermolecular attraction, (8) electrostatic effects, and (9) viscous drag. The coefficient of friction, indicated in the literature by ц or f is defined as the ratio F/W where F = friction force and W = the normal load. It is emphasized that friction is a force and not a property of a solid material or lubricant.

Wear and Surface Damage

One definition of wear in a tribological sense is that it is the progressive loss of substance from the operating surface of a body as a result of relative motion at the surface. In comparison with friction, very little theoretical work has been done on the extremely important area of wear and surface damage. This is not too surprising in view of the complexity of wear and how little is known of the mechanisms by which it can occur. Variations in wear can be, and often are, enormous compared with variations in friction. For example, practically all the coefficients of sliding friction for diverse dry or lubricated systems fall within a relatively narrow range of 0.1 to 1. In some cases (e. g., certain regimes of hydrodynamic or “boundary” lubrication), the coefficient of friction may be <0.1 and as low as 0.001. In other cases (e. g., very clean unlubricated metals in vacuum), friction coefficients may exceed one. Reduction of friction by a factor of two through changes in design, materials, or lubricant would be a reasonable, although not always attainable, goal. On the other hand, it is not uncommon for wear rates to vary by a factor of 100, 1,000, or even more.

For systems consisting of common materials (e. g., metals, polymers, ceramics), there are at least four main mechanisms by which wear and surface damage can occur between solids in relative motion:

Abrasive wear, (2) adhesive wear, (3) fatigue wear, and (4) chemical or corrosive wear. A fifth, fretting wear and fretting corrosion, combines elements of more than one mechanism. For complex biological materials such as articular cartilage, most likely other mechanisms are involved.

Again, wear is the removal of material. The idea that friction causes wear and therefore, low friction means low wear, is a common mistake. Brief descriptions of five types of wear; abrasive, adhesive, fatigue, chemical or corrosive, and fretting—may be found in [2] as well as in other references in this article. Next, it may be useful to consider some of the major concepts of lubrication.

Lubrication

Lubrication is a process of reducing friction and/or wear (or other forms of surface damage) between relatively moving surfaces by the application of a solid, liquid, or gaseous substance (i. e., a lubricant). Since friction and wear do not necessarily correlate with each other, the use of the word and in place of and/or in the above definition is a common mistake to be avoided. The primary function of a lubricant is to reduce friction or wear or both between moving surfaces in contact with each other.

Examples of lubricants are wide and varied. They include automotive engine oils, wheel bearing greases, transmission fluids, electrical contact lubricants, rolling oils, cutting fluids, preservative oils, gear oils, jet fuels, instrument oils, turbine oils, textile lubricants, machine oils, jet engine lubricants, air, water, molten glass, liquid metals, oxide films, talcum powder, graphite, molybdenum disulfide, waxes, soaps, polymers, and the synovial fluid in human joints.

A few general principles of lubrication may be mentioned here.

The lubricant must be present at the place where it can function.

Almost any substance under carefully selected or special conditions can be shown to reduce friction or wear in a particular test, but that does not mean these substances are lubricants.

Friction and wear do not necessarily go together. This is an extremely important principle which applies to nonlubricated (dry) as well as lubricated systems. It is particularly true under conditions of “boundary lubrication,” to be discussed later. An additive may reduce friction and increase wear, reduce wear and increase friction, reduce both or increase both. Although the reasons are not fully understood, this is an experimental observation. Thus, friction and wear should be thought of as separate phenomena—an important point when we discuss theories of synovial joint lubrication.

The effective or active lubricating film in a particular system may or may not consist of the original or bulk lubricant phase.

In a broad sense, it may be considered that the main function of a lubricant is to keep the surfaces apart so that interaction (e. g., adhesion, plowing, and shear) between the solids cannot occur; thus friction and wear can be reduced or controlled.

The following regimes or types of lubrication may be considered in the order of increasing severity or decreasing lubricant film thickness (Fig. 21.1):

Hydrodynamic lubrication

Elastohydrodynamic lubrication

Transition from hydrodynamic and elastohydrodynamic lubrication to boundary lubrication

Boundary lubrication.

Joint Lubrication

FIGURE 21.1 Regimes of lubrication

A fifth regime, sometimes referred to as dry or unlubricated, may also be considered as an extreme or limit. In addition, there is another form of lubrication that does not require relative movement of the bodies either parallel or perpendicular to the surface, i. e., as in externally pressurized hydrostatic or aerostatic bearings.

Hydrodynamic Lubrication Theories

In hydrodynamic lubrication, the load is supported by the pressure developed due to relative motion and the geometry of the system. In the regime of hydrodynamic or fluid film lubrication, there is no contact between the solids. The film thickness is governed by the bulk physical properties of the lubricants, the most important being viscosity; friction arises purely from shearing of viscous lubricant.

Contributions to our knowledge of hydrodynamic lubrication, with special focus on journal bearings, have been made by numerous investigators including Reynolds. The classic Reynolds treatment consid­ered the equilibrium of a fluid element and the pressure and shear forces on this element. In this treatment, eight assumptions were made (e. g., surface curvature is large compared to lubricant film thickness, fluid is Newtonian, flow is laminar, viscosity is constant through film thickness). Velocity distributions due to relative motion and pressure buildup were developed and added together. The solution of the basic Reynolds equation for a particular bearing configuration results in a pressure distribution throughout the film as a function of viscosity, film shape, and velocity.

The total load W and frictional (viscous) drag F can be calculated from this information. For rotating disks with parallel axes, the “simple” Reynolds equation yields:

H Onn □

-*■ □ 4.9 n (21.1)

R □ W H

Where ho is the minimum lubricant film thickness, n is the absolute viscosity, U is the average velocity (U + U2)/2, Wis the applied normal load per unit width of disk, and R is the reduced radius of curvature (1/R = 1R + 1/R2).

The dimensionless term (n U/W) is sometimes referred to as the hydrodynamic factor. It can be seen that doubling either the viscosity or velocity doubles the film thickness, and that doubling the applied load halves the film thickness. This regime of lubrication is sometimes referred to as the rigid isoviscous
or classical Martin condition, since the solid bodies are assumed to be perfectly rigid (non-deformable), and the fluid is assumed to have a constant viscosity.

At high loads with systems such as gears, ball bearings, and other high-contact-stress geometries, two

Additional factors have been considered in further developments of the hydrodynamic theory of lubri

Cation. One of these is that the surfaces deform elastically; this leads to a localized change in geometry

More favorable to lubrication. The second is that the lubricant becomes more viscous under the high pressure existing in the contact zone, according to relationships such as:

Joint Lubrication

(21.2)

Where n is the viscosity at pressure p, no is the viscosity at atmospheric pressure po, and a is the pressure – viscosity coefficient (e. g., in Pa-1). In this concept, the lubricant pressures existing in the contact zone approximate those of dry contact Hertzian stress. This is the regime of elastohydrodynamic lubrication, sometimes abbreviated as EHL or EHD. It may also be described as the elastic-viscous type or mode of lubrication, since elastic deformation exists and the fluid viscosity is considerably greater due to the pressure effect.

The comparable Dowson-Higginson expression for minimum film thickness between cylinders or disks in contact with parallel axes is:

Joint Lubrication

0.7

подпись: 0.7

0.54

подпись: 0.54

0.03

подпись: 0.03(21.3)

The term E’ represents the reduced modulus of elasticity:

‘2

подпись: 
'2
(21.4)

Where E is the modulus, v is Poisson’s ratio, and the subscripts 1 and 2 refer to the two solids in contact. All the other terms are the same as previously stated. In addition to the hydrodynamic factor (nU/W), a pressure-viscosity factor (aW/R), and an elastic deformation factor (W/RE’) can be considered. Thus, properties of both the lubricant and the solids as materials are included. In examining the elastohydro – dynamic film thickness equations, it can be seen that the velocity U is an important factor (ho rc U 07) but the load W is rather unimportant (ho rc W-013).

Experimental confirmation of the elastohydrodynamic lubrication theory has been obtained in certain selected systems using electrical capacitance, x-ray transmission, and optical interference techniques to determine film thickness and shape under dynamic conditions. Research is continuing in this area, including studies on micro-EHL or asperity lubrication mechanisms, since surfaces are never perfectly smooth. These studies may lead to a better understanding of not only lubricant film formation in high – contact-stress systems but lubricant film failure as well.

Two other possible types of hydrodynamic lubrication, rigid-viscous and elastic-isoviscous, complete the matrix of four, considering the two factors of elastic deformation and pressure-viscosity effects. In addition, squeeze film lubrication can occur when surfaces approach one another. For more information on hydrodynamic and elastohydrodynamic lubrication, see Cameron [3] and Dowson and Higginson [4].

Transition from Hydrodynamic to Boundary Lubrication

Although prevention of contact is probably the most important function of a lubricant, there is still much to be learned about the transition from hydrodynamic and elastohydrodynamic lubrication to
boundary lubrication. This is the region in which lubrication goes from the desirable hydrodynamic condition of no contact to the less acceptable “boundary” condition, where increased contact usually leads to higher friction and wear. This regime is sometimes referred to as a condition of mixed lubrication.

Several examples of experimental approaches to thin-film lubrication have been reported [3]. It is important in examining these techniques to make the distinction between methods that are used to determine lubricant film thickness under hydrodynamic or elastohydrodynamic conditions (e. g., optical interference, electrical capacitance, or x-ray transmission), and methods that are used to determine the occurrence or frequency of contact. As we will see later, most experimental studies of synovial joint lubrication have focused on friction measurements, using the information to determine the lubrication regime involved; this approach can be misleading.

Boundary Lubrication

Although there is no generally accepted definition of boundary lubrication, it is often described as a condition of lubrication in which the friction and wear between two surfaces in relative motion are determined by the surface properties of the solids and the chemical nature of the lubricant rather than its viscosity. An example of the difficulty in defining boundary lubrication can be seen if the term bulk viscosity is used in place of viscosity in the preceding sentence—another frequent form. This opens the door to the inclusion of elastohydrodynamic effects which depend in part on the influence of pressure on viscosity. Increased friction under these circumstances could be attributed to increased viscous drag rather than solid-solid contact. According to another common definition, boundary lubrication occurs or exists when the surfaces of the bearing solids are separated by films of molecular thickness. That may be true, but it ignores the possibility that “boundary” layer surface films may indeed be very thick (i. e.,

20, or 100 molecular layers). The difficulty is that boundary lubrication is complex.

Although a considerable amount of research has been done on this topic, an understanding of the basic mechanisms and processes involved is by no means complete. Therefore, definitions of boundary lubrication tend to be nonoperational. This is an extremely important regime of lubrication because it involves more extensive solid-solid contact and interaction as well as generally greater friction, wear, and surface damage. In many practical systems, the occurrence of the boundary lubrication regime is unavoid­able or at least quite common. The condition can be brought about by high loads, low relative sliding speeds (including zero for stop-and-go, motion reversal, or reciprocating elements) and low lubricant viscosity—factors that are important in the transition from hydrodynamic to boundary lubrication.

The most important factor in boundary lubrication is the chemistry of the tribological system—the contacting solids and total environment including lubricants. More particularly, the surface chemistry and interactions occurring with and on the solid surfaces are important. This includes factors such as physisorption, chemisorption, intermolecular forces, surface chemical reactions, and the nature, struc­ture, and properties of thin films on solid surfaces. It also includes many other effects brought on by the process of moving one solid over another, such as: (1) changes in topography and the area of contact, (2) high surface temperatures, (3) the generation of fresh reactive metal surfaces by the removal of oxide and other layers, (4) catalysis, (5) the generation of electrical charges, and (6) the emission of charged particles such as electrons.

In examining the action of boundary lubricant compounds in reducing friction or wear or both between solids in sliding contact, it may be helpful to consider at least the following five modes of film formation on or protection of surfaces: (1) physisorption, (2) chemisorption, (3) chemical reactions with the solid surface, (4) chemical reactions on the solid surface, and (5) mere interposition of a solid or other material. These modes of surface protection are discussed in more detail in [2].

The beneficial and harmful effects of minor changes in chemistry of the environment (e. g., the lubricant) are often enormous in comparison with hydrodynamic and elastohydrodynamic effects. Thus, the surface and chemical properties of the solid materials used in tribological applications become especially important. One might expect that this would also be the case in biological (e. g., human joint) lubrication where biochemistry is very likely an important factor.

Joint Lubrication

FIGURE 21.2 In any tribological system, friction, wear, and surface damage depend on four interrelated factors

General Comments on Tribological Processes

It is important to recognize that friction and wear depend upon four major factors, i. e., materials, design, operating conditions, and total environment (Fig. 21.2). This four-block figure may be useful as a guide in thinking about synovial joint lubrication either from a theoretical or experimental viewpoint—the topic discussed in the next section.

Readers are cautioned against the use of various terms in tribology which are either vaguely defined or not defined at all. These would include such terms as “lubricating ability,” “lubricity,” and even “boundary lubrication.” For example, do “boundary lubricating properties” refer to effects on friction or effects on wear and damage? It makes a difference. It is emphasized once again that friction and wear are different phenomena. Low friction does not necessarily mean low wear. We will see several examples of this common error in the discussion of joint lubrication research.

Synovial Joints

Examples of natural synovial or movable joints include the human hip, knee, elbow, ankle, finger, and shoulder. A simplified representation of a synovial joint is shown in Fig. 21.3. The bones are covered by a thin layer of articular cartilage bathed in synovial fluid confined by synovial membrane. Synovial joints are truly remarkable systems—providing the basis of movement by allowing bones to articulate on one another with minimal friction and wear. Unfortunately, various joint diseases occur even among the young—causing pain, loss of freedom of movement, or instability.

Synovial joints are complex, sophisticated systems not yet fully understood. The loads are surprisingly high and the relative motion is complex. Articular cartilage has the deceptive appearance of simplicity and uniformity. But it is an extremely complex material with unusual properties. Basically, it consists of water (approximately 75%) enmeshed in a network of collagen fibers and proteoglycans with high molecular weight. In a way, cartilage could be considered as one of Nature’s composite materials. Articular cartilage also has no blood supply, no nerves, and very few cells (chondrocytes).

The other major component of an articular joint is synovial fluid, named by Paracelsus after “synovia” (egg-white). It is essentially a dialysate of blood plasma with added hyaluronic acid. Synovial fluid contains complex proteins, polysaccharides, and other compounds. Its chief constituent is water (approximately 85%). Synovial fluid functions as a joint lubricant, nutrient for cartilage, and carrier for waste products.

Joint Lubrication

FIGURE 21.3 Representation of a synovial joint

For more information on the biochemistry, structure, and properties of articular cartilage, Freeman [5], Sokoloff [6], Stockwell [7] and articles referenced in these works are suggested.

Theories on the Lubrication of Natural and Normal Synovial Joints

As stated, the word tribology means the study of friction, wear, and lubrication. Therefore, biotribology may be thought of as the study of biological lubrication processes, e. g., as in synovial joints. A surprisingly large number of concepts and theories of synovial joint lubrication have been proposed [8-10] (as shown in Table 21.1). And even if similar ideas are grouped together, there are still well over a dozen funda­mentally different theories. These have included a wide range of lubrication concepts, e. g., hydrodynamic, hydrostatic, elasto-hydrodynamic, squeeze-film, “boundary”, mixed-regime, “weeping”, osmotic, synovial mucin gel, “boosted”, lipid, electrostatic, porous layers, and special forms of boundary lubrication (e. g., “lubricating glycoproteins”, structuring of boundary water “surface-active” phospholipids). This chapter will not review these numerous theories, but excellent reviews on the lubrication of synovial joints have been written by McCutchen [11], Swanson [12], and Higginsworth and Unsworth [13]. The book edited by Dumbleton is also recommended [14]. In addition, theses by Droogendijk [15] and Burkhardt [16] contain extensive and detailed reviews of theories of joint lubrication.

McCutchen was the first to propose an entirely new concept of lubrication, “weeping lubrication”, applied to synovial joint action [17,18]. He considered unique and special properties of cartilage and how this could affect flow and lubrication. The work of Mow et al. continued along a more complex and sophisticated approach in which a biomechanical model is proposed for the study of the dynamic interaction between synovial fluid and articular cartilage [19,20]. These ideas are combined in the more recent work of Ateshian [21] which uses a framework of the biphasic theory of articular cartilage to model interstitial fluid pressurization. Several additional studies have also been made of effects of porosity and compliance, including the behavior of elastic layers, in producing hydrodynamic and squeeze-film lubrication. A good review in this area was given by Unsworth who discussed both human and artificial joints [22].

The following general observations are offered on the theories of synovial joint lubrication that have been proposed:

Most of the theories are strictly mechanical or rheological—involving such factors as deformation, pressure, and fluid flow.

There is a preoccupation with friction, which of course is very low for articular cartilage systems.

None of the theories consider wear—which is neither the same as friction nor related to it.

The detailed structure, biochemistry, complexity, and living nature of the total articular cartilage – synovial fluid system are generally ignored.

These are only general impressions. And although mechanical/rheological concepts seem dominant (with a focus on friction), wear and biochemistry are not completely ignored. For example, Simon [23] abraded articular cartilage from human patellae and canine femoral heads with a stainless steel rotary file, measuring the depth of penetration with time and the amount of wear debris generated. Cartilage wear was also studied experimentally by Bloebaum and Wilson [24], Radin and Paul [25], and Lipshitz, Etheredge, and Glimcher [26-28]. The latter researchers carried out several in vitro studies of wear of articular cartilage using bovine cartilage plugs or specimens in sliding contact against stainless steel plates. They developed a means of measuring cartilage wear by determining the hydroxyproline content of both the lubricant and solid wear debris. Using this system and technique, effects of variables such as time, applied load, and chemical modification of articular cartilage on wear and profile changes were deter­mined. This work is of particular importance in that they addressed the question of cartilage wear and damage rather than friction, recognizing that wear and friction are different phenomena.

Special note is also made of two researchers, Swann and Sokoloff, who considered biochemistry as an important factor in synovial joint lubrication. Swann et al. very carefully isolated fractions of bovine synovial fluid using sequential sedimentation techniques and gel permeation chromatography. They found a high molecular weight glycoprotein to be the major constituent in the articular lubrication fraction from bovine synovial fluid and called this LGP-I (from lubricating glycoprotein). This was based on friction measurements using cartilage in sliding contact against a glass disc. An excellent summary of this work with additional references is presented in a chapter by Swann in The Joints and Synovial Fluid: I [6].

Sokoloff et al. examined the “boundary lubricating ability” of several synovial fluids using a latex-glass test system and cartilage specimens obtained at necropsy from knees [29]. Measurements were made of friction. The research was extended to other in vitro friction tests using cartilage obtained from the nasal septum of cows and widely differing artificial surfaces [30]. As a result of this work, a new model of boundary lubrication by synovial fluid was proposed—the structuring of boundary water. The postulate involves adsorption of one part of a glycoprotein on a surface followed by the formation of hydration shells around the polar portions of the adsorbed glycoprotein; the net result is a thin layer of viscous “structured” water at the surface. This work is of particular interest in that it involves not only a specific and more detailed mechanism of boundary lubrication in synovial joints but also takes into account the possible importance of water in this system.

In more recent research by Jay, an interaction between hyaluronic acid and a “purified synovial lubricating factor” (PSLF) was observed, suggesting a possible synergistic action in the boundary lubri­cation of synovial joints [31]. The definition of “lubricating ability” was based on friction measurements made with a latex-covered stainless steel stud in oscillating contact against polished glass.

The above summary of major synovial joint lubrication theories is taken from [10,31] as well as the thesis by Burkhardt [33].

Two more recent studies are of interest since cartilage wear was considered although not as a part of a theory of joint lubrication. Stachowiak et al. [34] investigated the friction and wear characteristics of adult rat femur cartilage against a stainless steel plate using an environmental scanning microscope (ESM) to examine damaged cartilage. One finding was evidence of a load limit to lubrication of cartilage, beyond which high friction and damage occurred. Another study, by Hayes et al. [35] on the influence of crystals on cartilage wear, is particularly interesting not only in the findings reported (e. g., certain crystals can increase cartilage wear), but also in the full description of the biochemical techniques used.

Mechanism

Authors

Date

1.

Hydrodynamic

MacConnail

1932

2.

Boundary

Jones

1934

3.

Hydrodynamic

Jones

1936

4.

Boundary

Charnley

1959

5.

Weeping

McCutchen

1959

6.

Floating

Barnett and Cobbold

1962

7.

Elastohydrodynamic

Tanner

1966

Dowson

1967

8.

Thixotropic/elastic fluid

Dintenfass

1963

9.

Osmotic (boundary)

McCutchen

1966

10.

Squeeze-film

Fein

1966

Higginson et al.

1974

11.

Synovial gel

Maroudas

1967

12.

Thin-film

Faber et al.

1967

13.

Combinations of hydrostatic,

Linn

1968

Boundary, & EHL

14.

Boosted

Walker et al.

1968

15.

Lipid

Little et al.

1969

16.

Weeping + boundary

McCutchen and Wilkins

1969

McCutchen

1969

17.

Boundary

Caygill and West

1969

18.

Fat (or mucin)

Freeman et al.

1970

19.

Electrostatic

Roberts

1971

20.

Boundary + fluid squeeze-film

Radin and Paul

1972

21.

Mixed

Unsworth et al.

1974

22.

Imbibe/exudate composite model

Ling

1974

23.

Complex biomechanical model

Mow et al.

1974

Mansour and Mow

1977

24.

Two porous layer model

Dinnar

1974

25.

Boundary

Reimann et al.

1975

26.

Squeeze-film + fluid film + boundary

Unsworth, Dowson et al.

1975

27.

Compliant bearing model

Rybicki

1977

28.

Lubricating glycoproteins

Swann et al.

1977

29.

Structuring of boundary water

Sokoloff et al.

1979

30.

Surface flow

Kenyon

1980

31.

Lubricin

Swann et al.

1985

32.

Micro-EHL

Dowson and Jin

1986

33.

Lubricating factor

Jay

1992

34.

Lipidic component

LaBerge et al.

1993

35.

Constitutive modeling of cartilage

Lai et al.

1993

36.

Asperity model

Yao et al.

1993

37.

Bingham fluid

Tandon et al.

1994

38.

Filtration/gel/squeeze film

Hlavacek et al.

1995

39.

Surface-active phospholipid

Schwarz and Hills

1998

40.

Interstitial fluid pressurization

Ateshian et al.

1998

A special note should be made concerning the doctoral thesis by Lawrence Malcom in 1976 [36]. This is an excellent study of cartilage friction and deformation, in which a device resembling a rotary plate rheometer was used to investigate the effects of static and dynamic loading on the frictional behavior of bovine cartilage. The contact geometry consisted of a circular cylindrical annulus in contact with a concave hemispherical section. It was found that dynamically loaded specimens in bovine synovial fluid yielded the more efficient lubrication based on friction measurements. The Malcom study is thorough and excellent in its attention to detail (e. g., specimen preparation) in examining the influence of type of

Loading and time effects on cartilage friction. It does not, however, consider cartilage wear and damage except in a very preliminary way. And it does not consider the influence of fluid biochemistry on cartilage friction, wear, and damage. In short, the Malcom work represents a superb piece of systematic research along the lines of mechanical, dynamic, rheological, and viscoelastic behavior—one important dimension of synovial joint lubrication.

In Vitro Cartilage Wear Studies

Over the past fifteen years, studies aimed at exploring possible connections between tribology and mechanisms of synovial joint lubrication and degeneration (e. g., osteoarthritis) have been conducted by the author and his graduate and undergraduate students in the Department of Mechanical Engineering at Virginia Polytechnic Institute and State University. The basic approach used involved in vitro tribo – logical experiments using bovine articular cartilage, with an emphasis on the effects of fluid composition and biochemistry on cartilage wear and damage. This research is an outgrowth of earlier work carried out during a sabbatical study in the Laboratory for the Study of Skeletal Disorders, The Children’s Hospital Medical Center, Harvard Medical School in Boston. In that study, bovine cartilage test specimens were loaded against a polished steel plate and subjected to reciprocating sliding for several hours in the presence of a fluid (e. g., bovine synovial fluid or a buffered saline reference fluid containing biochemical constit­uents kindly provided by Dr. David Swann). Cartilage wear was determined by sampling the test fluid and determining the concentration of 4-hydroxyproline—a constituent of collagen. The results of that earlier study have been reported and summarized elsewhere [37-40]. Figure 21.4 Shows the average hydroxyproline contents of wear debris obtained from these in vitro experiments. These numbers are related to the cartilage wear which occurred. However, since the total quantities of collected fluids varied somewhat, the values shown in the bar graph should not be taken as exact or precise measures of fluid effects on cartilage wear.

The main conclusions of that study were as follows:

Normal bovine synovial fluid is very effective in reducing cartilage wear under these in vitro conditions as compared to the buffered saline reference fluid.

There is no significant difference in wear between the saline reference and distilled water.

The addition of hyaluronic acid to the reference fluid significantly reduces wear; but its effect depends on the source.

Under these tests conditions, Swann’s LGP-I (Lubricating Glycoprotein-I), known to be extremely effective in reducing friction in cartilage-on-glass tests, does not reduce cartilage wear.

Joint Lubrication

FIGURE 21.4 Relative cartilage wear based on hydroxyproline content of debris (in vitro tests with cartilage on stainless steel).

Joint Lubrication

FIGURE 21.5 Friction and wear are different phenomena

However, a protein complex isolated by Swann is extremely effective in reducing wear—producing results similar to those obtained with synovial fluid. The detailed structure of this constituent is complex and has not yet been fully determined.

Last, the lack of an added fluid in these experiments leads to extremely high wear and damage of the articular cartilage.

In discussing the possible significance of these findings from a tribological point of view, it may be helpful first of all to emphasize once again that friction and wear are different phenomena. Furthermore, as suggested by Fig. 21.5, certain constituents of synovial fluid (e. g., Swann’s Lubricating Glycoprotein) may act to reduce friction in synovial joints while other constituents (e. g., Swann’s protein complex or hyaluronic acid) may act to reduce cartilage wear. Therefore, it is necessary to distinguish between biochemical anti-friction and anti-wear compounds present in synovial fluid.

In more recent years, this study has been greatly enhanced by the participation of interested faculty and students from the Virginia-Maryland College of Veterinary Medicine and Department of Biochem­istry and Animal Science at Virginia Tech. One major hypothesis tested is a continuation of previous work showing that the detailed biochemistry of the fluid-cartilage system has a pronounced and possibly controlling influence on cartilage wear. A consequence of the above hypothesis is that a lack or deficiency of certain biochemical constituents in the synovial joint may be one factor contributing to the initiation and progression of cartilage damage, wear, and possibly osteoarthritis. A related but somewhat different hypothesis concerns synovial fluid constituents which may act to increase the wear and further damage of articular cartilage under tribological contact.

To carry out continued research on biotribology, a new device for studies of cartilage deformation, wear, damage, and friction under conditions of tribological contact was designed by Burkhardt [33] and later modified, constructed, and instrumented. A simplified sketch is shown in Fig. 21.6. The key features of this test device are shown in Table 21.2. The apparatus is designed to accommodate cartilage-on – cartilage specimens. Motion of the lower specimen is controlled by a computer-driven x-y table, allowing simple oscillating motion or complex motion patterns. An octagonal strain ring with two full semi­conductor bridges is used to measure the normal load as well as the tangential load (friction). An LVDT, not shown in the figure, is used to measure cartilage deformation and linear wear during a test. However, hydroxyproline analysis of the wear debris and washings is used for the actual determination of total cartilage wear on a mass basis.

In one study by Schroeder [41], two types of experiments were carried out, i. e., cartilage-on-stainless steel and cartilage-on-cartilage at applied loads up to 70N—yielding an average pressure of 2.2 MPa in the contact area. Reciprocating motion (40 cps) was used. The fluids tested included: (1) a buffered saline solution, (2) saline plus hyaluronic acid, and (3) bovine synovial fluid. In cartilage-on-stainless steel tests, scanning electron microscopy, and histological staining showed distinct effects of the lubricants on surface and subsurface damage. Tests with the buffered saline fluid resulted in the most damage, with large wear tracks visible on the surface of the cartilage plug, as well as subsurface voids and cracks. When hyaluronic acid, a constituent of the natural synovial joint lubricant, was added to the saline reference fluid, less severe damage was observed. Little or no cartilage damage was evident in tests in which the natural synovial joint fluid was used as the lubricant.

Joint Lubrication

FIGURE 21.6 Device for in vitro cartilage-on-cartilage wear studies

TABLE 21.2 Key Features of Test Device Designed for Cartilage Wear Studies [33]

Contact system Cartilage-on-cartilage

Contact geometry Flat-on-flat, convex-on-flat, irregular-on-irregular

Cartilage type Articular, any source (e. g., bovine)

Specimen size Upper specimen, 4 to 6 mm diam., lower specimen, ca. 15 to 25 mm diam.

Applied load 50-660 N

Average pressure 0.44-4.4 MPa

Type of motion Linear, oscillating; circular, constant velocity; more complex patterns

Sliding velocity 0 to 20 mm/s

Fluid temperature Ambient (20°C); or controlled humidity

Environment Ambient or controlled humidity

Measurements Normal load, cartilage deformation, friction; cartilage wear and damage, biochemical analysis of

Cartilage specimens, synovial fluid, and wear debris; sub-surface changes

These results were confirmed in a later study by Owellen [42] in which hydroxyproline analysis was used to determine cartilage wear. It was found that increasing the applied load from 20 to 65N increased cartilage wear by eight-fold for the saline solution and approximately three-fold for synovial fluid. Fur­thermore, the coefficient of friction increased from an initial low value of 0.01 to 0.02 to a much higher value, e. g., 0.20 to 0.30 and higher, during a normal test which lasted 3 hours; the greatest change occurred during the first 20 minutes. Another interesting result was that a thin film of transferred or altered material was observed on the stainless steel disks—being most pronounced with the buffered saline lubricant and not observed with synovial fluid. Examination of the film with Fourier Transfer Infrared Microspectrom­etry shows distinctive bio-organic spectra which differs from that of the original bovine cartilage. We believe this to be an important finding since it suggests a possible bio-tribochemical effect [43].

Joint Lubrication

Joint Lubrication

FIGURE 21.7 Cartilage damage produced by sliding contact

In another phase of this research, the emphasis is on the cartilage-on-cartilage system and the influence of potentially beneficial as well as harmful constituents of synovial fluid on wear and damage. In cartilage- on-cartilage tests, the most severe wear and damage occurred during tests with buffered saline as the lubricant. The damage was less severe than in the stainless steel tests, but some visible wear tracks were detectable with scanning electron microscopy. Histological sectioning and staining of both the upper and lower cartilage samples show evidence of elongated lacunae and coalesced voids that could lead to wear by delamination. An example is shown in Fig. 21.7 ( Original magnification of 500X on 35mm slide). The proteoglycan content of the subsurface cartilage under the region of contact was also reduced. When synovial fluid was used as the lubricant, no visible wear or damage was detected [44]. These results demonstrate that even in in vitro tests with bovine articular cartilage, the nature of the fluid environment can have a dramatic affect on the severity of wear and subsurface damage.

In a more recent study carried out by Berrien in the biotribology program at Virginia Tech, a different approach was taken to examine the role of joint lubrication in joint disease, particularly osteoarthritis. A degradative biological enzyme, collagenase-3, suspected of playing a role in a cartilage degeneration was used to create a physiologically adverse biochemical fluid environment. Tribological tests were performed with the same device and procedures described previously. The stainless steel disk was replaced with a 1 in. diameter plug of bovine cartilage to create a cartilage sliding on cartilage configuration more closely related to the in vivo condition. Normal load was increased to 78.6 N and synovial fluid and buffered saline were used as lubricants. Prior to testing, cartilage plugs were exposed to a fluid medium containing three concentrations of collagenase-3 for 24 hr. The major discovery of this work was that exposure to the collagenase-3 enzyme had a substantial adverse effect on cartilage wear in vitro, increasing average wear values by three and one-half times those of the un exposed cases. Figure 21.8 Shows an example of the effect of enzyme treatment when bovine synovial fluid was used as the lubricant. Scanning electron microscopy showed disruption of the superficial layer and collagen matrix with exposure to collagenase-3, where unexposed cartilage showed none. Histological sections showed a substantial loss of the superficial layer of cartilage and a distinct and abnormal loss of proteoglycans in the middle layer of collagenase-treated cartilage. Unexposed cartilage showed only minor disruption of the superficial layer [45].

Joint Lubrication

FIGURE 21.8 Effect of collagenase-3 on cartilage wear

подпись: figure 21.8 effect of collagenase-3 on cartilage wear

Ou/ml 50 u/ml 100u/ml

Collagenase-3 Cartilage Treatment

подпись: ou/ml 50 u/ml 100u/ml
collagenase-3 cartilage treatment
This study indicates that some of the biochemical constituents that gain access to the joint space, during normal and pathological functions, can have a significant adverse effect on the wear and damage
Of the articular cartilage. Future studies will include determination of additional constituents that have harmful effects on cartilage wear and damage. This research, using bovine articular cartilage in in vitro sliding contact tests, raises a number of interesting questions:

Has ‘Nature’ designed a special biochemical compound which has as its function the protection of articular cartilage?

What is the mechanism (or mechanisms) by which biochemical constituents of synovial fluid can act to reduce wear of articular cartilage?

Could a lack of this biochemical constituent lead to increased cartilage wear and damage?

Does articular cartilage from osteoarthritic patients have reduced wear resistance?

Do any of the findings on the importance of synovial fluid biochemistry on cartilage wear in our in vitro studies apply to living or in vitro systems as well?

How does collagenase-3 treatment of cartilage lead to increased wear and does this finding have any significance in the in vivo situation? This question is addressed in the next section.

Biotribology and Arthritis: Are There Connections?

Arthritis is an umbrella term for more than 100 rheumatic diseases affecting joints and connective tissue. The two most common forms are osteoarthritis (OA) and rheumatoid arthritis (RA). Osteoarthritis—also referred to as osteoarthrosis or degenerative joint disease—is the most common form of arthritis. It is sometimes simplistically described as the “wear and tear” form of arthritis. The causes and progression of degenerative joint disease are still not understood. Rheumatoid arthritis is a chronic and often pro­gressive disease of the synovial membrane leading to release of enzymes which attack, erode, and destroy articular cartilage. It is an inflammatory response involving the immune system and is more prevalent in females. Rheumatoid arthritis is extremely complex. Its causes are still unknown.

Sokoloff defines degenerative joint disease as “an extremely common, noninflammatory, progressive disorder of movable joints, particularly weight-bearing joints, characterized pathologically by deteriora­tion of articular cartilage and by formation of new bone in the sub-chondral areas and at the margins of the “joint” [46]. As mentioned, osteoarthritis or osteoarthrosis is sometimes referred to as the “wear and tear” form of arthritis; but, wear itself is rarely a simple process even in well-defined systems.

It has been noted by the author that tribological terms occasionally appear in hypotheses which describe the etiology of osteoarthritis (e. g., “reduced wear resistance of cartilage” or “poor lubricity of synovial
Fluid”). It has also been noted that there is a general absence of hypotheses connecting normal synovial joint lubrication (or lack thereof) and synovial joint degeneration. Perhaps it is natural (and unhelpful) for a tribologist to imagine such a connection and that, for example, cartilage wear under certain circumstances might be due to or influenced by a lack of proper “boundary lubrication” by the synovial fluid. In this regard, it may be of interest to quote Swanson [12] who said in 1979 that “there exists at present no experimental evidence which certainly shows that a failure of lubrication is or is not a causative factor in the first stages of cartilage degeneration.” A statement made by Professor Glimcher [52] may also be appropriate here. Glimcher fully recognized the fundamental difference between friction and wear as well as the difference between joint lubrication (one area of study) and joint degeneration (another area of study). Glimcher said that wearing or abrading cartilage with a steel file is not osteoarthritis; and neither is digesting cartilage in a test tube with an enzyme. But both forms of cartilage deterioration can occur in a living joint and in a way which is still not understood. It is interesting that essentially none of the many synovial joint lubrication theories consider enzymatic degradation of cartilage as a factor whereas practically all the models of the etiology of degenerative joint disease include this as an important factor.

It was stated earlier that there are at least two main areas to consider, i. e., (1) mechanisms of synovial joint lubrication and (2) the etiology of synovial joint degeneration (e. g., as in osteoarthrosis). Both areas are extremely complex. And the key questions as to what actually happens in each have yet to be answered (and perhaps asked). It may therefore be presumptuous of the present author to suggest possible con­nections between two areas which in themselves are still not fully understood.

Tribological processes in a movable joint involve not only the contacting surfaces (articular cartilage), but the surrounding medium (synovial fluid) as well. Each of these depends on the synthesis and transport of necessary biochemical constituents to the contact region or interface. As a result of relative motion (sliding, rubbing, rolling, and impact) between the joint elements, friction and/or wear can occur.

It has already been shown and discussed—at least in in vitro tests with articular cartilage—that compounds which reduce friction do not necessarily reduce wear; the latter was suggested as being more important [10]. It may be helpful first of all to emphasize once again that friction and wear are different phenomena. Furthermore, certain constituents of synovial fluid (e. g., Swann’s Lubricating Glycoprotein) may act to reduce friction in synovial joints while other constituents (e. g., Swann’s protein complex or hyaluronic acid) may act to reduce cartilage wear.

A significant increase in joint friction could lead to a slight increase in local temperatures or possibly to reduce mobility. But the effects of cartilage wear would be expected to be more serious. When cartilage wear occurs, a very special material is lost and the body is not capable of regenerating cartilage of the same quality nor at the desired rate. Thus, there are at least two major tribological dimensions involved—one concerning the nature of the synovial fluid and the other having to do with the properties of articular cartilage itself. Changes in either the synovial fluid or cartilage could conceivably lead to increased wear or damage (or friction)asshown in Fig. 21.9.

Joint LubricationA simplified model or illustration of possible connections between osteoarthritis and tribology is offered in Fig. 21.10 Taken from Furey [53]. Its purpose is to stimulate discussion. There are other pathways to the disease, pathways which may include genetic factors.

Reduced Wear

Resistance of

Articular

Cartilage

Increased

Cartilage

Wear and

Reduced

Damage

Anti-Wear

Properties of

Synovial Fluid

FIGURE 21.9 Two tribological aspects of synovial joint lubrication

Joint Lubrication

FIGURE 21.10 Osteoarthritis-tribology connections?

In some cases, the body makes an unsuccessful attempt at repair, and bone growth may occur at the periphery of contact. As suggested by Fig. 21.10, This process and the generation of wear particles could lead to joint inflammation and the release of enzymes which further soften and degrade the articular cartilage. This softer, degraded cartilage does not possess the wear-resistance of the original. It has been shown previously that treatment of cartilage with collagenase-3 increases wear significantly, thus sup­porting the idea of enzyme release as a factor in osteoarthritis. Thus, there exists a feedback process in which the occurrence of cartilage wear can lead to even more damage. Degradative enzymes can also be released by trauma, shock, or injury to the joint. Ultimately, as the cartilage is progressively thinned and bony growth occurs, a condition of osteoarthritis or degenerative joint disease may exist. There are other pathways to the disease, pathways which may include genetic factors. It is not argued that arthritis is a tribological problem. However, the inclusion of tribological processes in one set of pathways to osteo­arthrosis would not seem strange or unusual.

A specific example of a different tribological dimension to the problem of synovial joint lubrication (i. e., third-body abrasion), was shown by the work of Hayes et al. [54]. In an excellent study of the effect of crystals on the wear of articular cartilage, they carried out in vitro tests using cylindrical cartilage sub­chondral bone plugs obtained from equine fetlock joints in sliding contact against a stainless steel plate. They examined the effects of three types of crystals (orthorhombic calcium pyrophosphate tetrahydrate, monoclinic calcium pyrophosphate dehydrate, and calcium hydroxyapatite) on wear using a Ringer’s solution as the carrier fluid. Concentration of cartilage wear debris in the fluid was determined by analyzing for inorganic sulphate derived from the proteoglycans present. Several interesting findings were made, one of them being that the presence of the crystals roughly doubled cartilage wear. This is an important contribution which should be read by anyone seriously contemplating research on the tribol – ogy of articular cartilage. The careful attention to detail and potential problems, as well as the precise description of the biochemical procedures and diverse experimental techniques used, set a high standard.

Recapitulation and Final Comments

It is obvious from the unusually large number of theories of synovial joint lubrication proposed, that very little is known about the subject. Synovial joints are undoubtedly the most sophisticated and complex tribological systems that exist or will ever exist. It will require a great deal more research—possibly very different approaches—before we even begin to understand the processes involved.

Some general comments and specific suggestions are offered—not for the purpose of criticizing any particular study but hopefully to provide ideas which may be helpful in further research as well as in the re-interpretation of some past research.

Terms and Definitions

First of all, as mentioned earlier in this chapter, part of the problem has to do with the use and misuse of various terms in tribology—the study of friction, wear, and lubrication. A glance at any number of the published papers on synovial joint lubrication will reveal such terms and phrases as “lubricating ability,” “lubricity,” “lubricating properties,” “lubricating component,” and many others, all undefined. We also see terms like “boundary lubricant,” “lubricating glycoprotein,” or “lubricin”. There is nothing inherently wrong with this but one should remember that lubrication is a process of reducing friction and/or wear between rubbing surfaces. Saying that a fluid is a “good” lubricant does not distinguish between friction and wear. And assuming that friction and wear are correlated and go together is the first pitfall in any tribological study. It cannot be overemphasized that friction and wear are different, though sometimes related, phenomena. Low friction does not mean low wear. The terms and phrases used are therefore extremely important. For example, in a brief and early review article by Wright and Dowson [55], it was stated that “Digestion of hyaluronate does not alter the boundary lubrication,” referring to the work of Radin, Swann, and Weisser [56]. In another article, McCutchen re-states this conclusion in another way, saying “… the lubricating ability did not reside in the hyaluronic acid” and later asks the question “Why do the glycoprotein molecules (of Swann) lubricate?” [57] These statements are based on effects of various constituents on friction, not wear. The work of the present author showed that in tests with bovine articular cartilage, Swann’s Lubricating Glycoprotein LGP-I which was effective in reducing friction did not reduce cartilage wear. However, hyaluronic acid—shown earlier not to be responsible for friction-reduction—did reduce cartilage wear. Thus, it is important to make the distinc­tion between friction-reduction and wear-reduction. It is suggested that operational definitions be used in place of vague “lubricating ability,” etc. terms in future papers on the subject.

Experimental Contact Systems

Secondly, some comments are made on the experimental approaches that have been reported in the literature on synovial joint lubrication mechanisms. Sliding contact combinations in in vitro studies have consisted of (1) cartilage-on-cartilage, (2) cartilage-on-some other surface (e. g., stainless steel, glass), and (3) solids other than cartilage sliding against each other in X-on-X or X-on-Y combinations.

The cartilage-on-cartilage combination is of course the most realistic and yet most complex contact system. But variations in shape or macroscopic geometry, microtopography, and the nature of contact present problems in carrying out well-controlled experiments. There is also the added problem of acquiring suitable specimens which are large enough and reasonably uniform.

The next combination—cartilage-on-another material—allows for better control of contact, with the more elastic, deformable cartilage loaded against a well-defined hard surface (e. g., a polished, flat solid made of glass or stainless steel). This contact configuration can provide useful tribological information on effects of changes in biochemical environment (e. g., fluids), on friction, wear, and sub-surface damage. It also could parallel the situation in a partial joint replacement in which healthy cartilage is in contact with a metal alloy.

The third combination, which appears in some of the literature on synovial joint lubrication, does not involve any articular cartilage at all. For example, Jay made friction measurements using a latex-covered stainless steel stud in oscillating contact against polished glass [31]. Williams et al., in a study of a lipid component of synovial fluid, used reciprocating contact of borosilicate glass-on-glass [58]. And in a recent paper on the action of a surface-active phospholipid as the “lubricating component of lubricin,” Schwarz and Hills carried out friction measurements using two optically flat quartz plates in sliding contact [59]. In another study, a standard four-ball machine using alloy steel balls was used to examine the “lubricating ability” of synovial fluid constituents. Such tests, in the absence of cartilage, are easiest to control and carry out. However, they are not relevant to the study of synovial joint lubrication. With a glass sphere sliding against a glass flat, almost anything will reduce friction—including a wide variety of chemicals, biochemicals, semi-solids, and fluids. This has little if anything to do with the lubrication of synovial joints.

Fluids and Materials Used as Lubricants in In Vitro Biotribology Studies

Fluids used as lubricants in synovial joint lubrication studies have consisted of (1) “normal” synovial fluid (e. g., bovine), (2) buffered saline solution containing synovial fluid constituents (e. g., hyaluronic acid), and (3) various aqueous solutions of surface active compounds neither derived from nor present in synovial fluid. In addition, a few studies used synovial fluids from patients suffering from either osteoarthritis or rheumatoid arthritis.

The general comment made here is that the use of synovial fluids—whether derived from human or animal sources and whether “healthy” or “abnormal”—is important in in vitro studies of synovial joint lubrication. The documented behavior of synovial fluid in producing low friction and wear with articular cartilage sets a reference standard and demonstrates that useful information can indeed come from in vitro tests.

Studies that are based on adding synovial fluid constituents to a reference fluid (e. g., a buffered saline solution) can also be useful in attempting to identify which biochemical compound or compounds are responsible for reductions in frictions or wear. But if significant interactions between compounds exist, then such an approach may require an extensive program of tests. It should also be mentioned that in the view of the present author, the use of a pure undissolved constituent of synovial fluid, either derived or synthetic, in a sliding contact test is not only irrelevant but may be misleading. An example would be the use of a pure lipid (e. g., phospholipid) at the interface rather than in the concentration and solution form in which this compound would normally exist in synovial fluid. This is basic in any study of lubrication and particularly in the case of boundary lubrication where major effects on wear or friction can be brought on by minor, seemingly trivial, changes in chemistry.

The Preoccupation with Rheology and Friction

The synovial joint as a system—the articular cartilage and underlying bone structure as well as the synovial fluid as important elements—is extremely complex and far from being understood. It is noted that there is a proliferation of mathematical modeling papers stressing rheology and the mechanics of deformation, flow, and fluid pressures developed in the cartilage model. One recent example is the paper “The Role of Interstitial Fluid Pressurization and Surface Properties on the Boundary Friction of Articular Cartilage” by Ateshian et al. [21]. This study, a genuine contribution, grew out of the early work by Mow and connects also with the “weeping lubrication” model of McCutchen. Both McCutchen and Mow have made significant contributions to our understanding of synovial joint lubrication, although each approach is predominantly rheological and friction-oriented with little regard for biochemistry and wear. This is not to say that rheology is unimportant. It could well be that, as suggested by Ateshian, the mechanism of interstitial fluid pressurization that leads to low friction in cartilage could also lead to low wear rates [60].

The Probable Existence of Various Lubrication Regimes

In an article by Wright and Dowson, it is suggested that a variety of types of lubrication operate in human synovial joints at different parts of a walking cycle stating that, “At heel-strike a squeeze-film situation may develop, leading to elastohydrodynamic lubrication and possibly both squeeze-film and boundary lubrication, while hydrodynamic lubrication may operate during the free-swing phase of walking” [55].

Joint Lubrication

FIGURE 21.11 (a) Hip joint forces and angular velocities at different parts of the walking cycle (after Graham and Walker [61]). (b) Calculated ratio of velocity to force for the hip joint (from Figure 22.11a)

In a simplified approach to examining the various regimes of lubrication that could exist in a human joint, it may be useful to look at Fig. 21.11a wHich shows the variation in force (load) and velocity for a human hip joint at different parts of the walking cycle (taken from Graham and Walker [61]). As discussed earlier in this chapter, theories of hydrodynamic and elastohydrodynamic lubrication all include the hydrodynamic factor (n U/W) as the key variable, where n = fluid viscosity, U = the relative sliding velocity, and W = the normal load. High values of (n U/W) lead to thicker hydrodynamic films—a more desirable condition if one wants to keep surfaces apart. It can be seen from Fig. 21.11a That there is considerable variation in load and velocity, with peaks and valleys occurring at different parts of the cycle. Note also that in this example, the loads can be quite high (e. g., up to three times body weight). The maximum load occurs at 20% of the walking cycle illustrated in Fig. 21.11a, With a secondary maximum occurring at a little over the 50% point. The maximum angular velocity occurs at approxi­mately 67% of the cycle. If one now creates a new curve of relative velocity/load or (U/W) from Fig. 21.11a, the result obtained is shown in Fig. 21.11b. We see now a very different and somewhat simplified picture. There is a clear and distinct maximum in the ratio of velocity to load (U/W) at 80% of walking cycle, favoring the formation of a hydrodynamic film of maximum thickness. However, for most of the cycle (e. g., from 0 to 60%), the velocity/load ratio is significantly lower, thus favoring a condition of minimum film thickness and “boundary lubrication”. However, we also know that synovial fluid is non-Newtonian;

At higher rates of shear, its viscosity decreases sharply, approaching that of water. The shear rate is equal to the relative velocity divided by fluid film thickness (U/h) and is expressed in s-1. This means that at the regions of low (U/W) ratios or thinner hydrodynamic films, the viscosity term in (n U/W) is even lower, thus pushing the minima to lower values favoring a condition of boundary lubrication. This is only a simplified view and does not consider those periods in which the relative sliding velocity is zero at motion reversal and where squeeze-film lubrication may come into play. A good example of the complexity of load and velocity variation in a human knee joint—including several zero-velocity peri­ods—may be found in the chapter by Higginson and Unsworth [62] citing the work of Seedhom et al., which deals with biomechanics in the design of a total knee replacement [63].

The major point made here is that (1) there are parts of a walking cycle that would be expected to approach a condition of minimum fluid film thickness and boundary lubrication and (2) it is during these parts of the cycle that cartilage wear and damage resulting from contact is more likely to occur. Thus, approaches to reducing cartilage wear in a synovial joint could be broken down into two categories (i. e., promoting thicker hydrodynamic films and providing special forms of “boundary lubrication.”)

Recent Developments

Recent developments in addressing some of the problems that involve cartilage damage and existing joint replacements include (1) progress in promoting cartilage repair [64], (2) possible use of artificial cartilage materials (e. g., synthetic hydrogels) [65,66], and (3) the development and application of more compliant joint replacement materials to promote a more favorable formation of an elastohydrodynamic film [67]. Although these are not strictly “lubricant-oriented” developments, they do and will involve important tribological aspects discussed in this chapter. For example, if new cartilage growth can be promoted by transplanting healthy chondrocytes to a platform in a damaged region of a synovial joint, how long will this cartilage last? If a hydrogel is used as an artificial cartilage, how long will it last? And if softer, elastomeric materials are used as partial joint replacements or coatings, how long will they last? These are questions of wear, not friction. And although the early fundamental studies of hydrogels as artificial cartilage measured only friction, and often only after a few moments of sliding, we know from recent work that even for hydrogels, low friction does not mean low wear [68].

Conclusions

The following main conclusions relating to the tribological behavior of natural, “normal” synovial joints are presented:

An unusually large number of theories and studies of joint lubrication have been proposed over the years. All of the theories focus on friction, none address wear, many do not involve experimental studies with cartilage, and very few consider the complexity and detailed biochemistry of the synovial-fluid articular-cartilage system.

It was shown by in vitro tests with bovine articular cartilage that the detailed biochemistry of synovial fluid has a significant effect on cartilage wear and damage. “Normal” bovine synovial fluid was found to provide excellent protection against wear. Various biochemical constituents isolated from bovine synovial fluid by Dr. David Swann, of the Shriners Burns Institute in Boston, showed varying effects on cartilage wear when added back to a buffered saline reference fluid. This research demonstrates once again the importance of distinguishing between friction and wear.

In a collaborative study of biotribology involving researchers and students in Mechanical Engi­neering, the Virginia-Maryland College of Veterinary Medicine, and Biochemistry, in vitro tribo­logical tests using bovine articular cartilage demonstrated among other things that (1) normal synovial fluid provides better protection than a buffered saline solution in a cartilage-on-cartilage system, (2) tribological contact in cartilage systems can cause subsurface damage, delamination, changes in proteoglycan content, and in chemistry via a “biotribochemical” process not under­stood, and (3) pre-treatment of articular cartilage with the enzyme collagenase-3—suspected as a factor in osteoarthritis—significantly increases cartilage wear.

It is suggested that these results could change significantly the way mechanisms of synovial joint lubrication are examined. Effects of biochemistry of the system on wear of articular cartilage are likely to be important; such effects may not be related to physical/rheological models of joint lubrication.

It is also suggested that connections between tribology/normal synovial joint lubrication and degenerative joint disease are not only possible but likely; however, such connections are undoubt­edly complex. It is not argued that osteoarthritis is a tribological problem or that it is necessarily the result of a tribological deficiency. Ultimately, a better understanding of how normal synovial joints function from a tribological point of view could conceivably lead to advances in the pre­vention and treatment of osteoarthritis.

Several problems exist that make it difficult to understand and interpret many of the published works on synovial joint lubrication. One example is the widespread use of non-operational and vague terms such as “lubricating activity,” “lubricating factor,” “boundary lubricating ability”, and similar undefined terms which not only fail to distinguish between friction (which is usually measured) and cartilage wear (which is rarely measured), but tend to lump these phenomena together—a common error. Another problem is that a significant number of the published exper­imental studies of biotribology do not involve cartilage at all—relying on the use of glass-on-glass, rubber-on-glass, and even steel-on-steel. Such approaches may be a reflection of the incorrect view that “lubricating activity” is a property of a fluid and can be measured independently. Some suggestions are offered.

Last, the topic of synovial joint lubrication is far from being understood. It is a complex subject involving at least biophysics, biomechanics, biochemistry, and tribology. For a physical scientist or engineer, carrying out research in this area is a humbling experience.

Acknowledgments

The author wishes to acknowledge the support of the Edward H. Lane, G. Harold, and Leila Y. Mathers Foundations for their support during the sabbatical study at The Children’s Hospital Medical Center. He also wishes to thank Dr. David Swann for his invaluable help in providing the test fluids and carrying out the biochemical analyses as well as Ms. Karen Hodgens for conducting the early scanning electron microscopy studies of worn cartilage specimens.

The author is also indebted to the following researchers for their encouraging and stimulating discus­sions of this topic over the years and for teaching a tribologist something of the complexity of synovial joints, articular cartilage, and arthritis: Drs. Leon Sokoloff, Charles McCutchen, Melvin Glimcher, David Swann, Henry Mankin, Clement Sledge, Helen Muir, Paul Dieppe, Heikki Helminen, as well as his colleagues at Virginia Tech—Hugo Veit, E. T. Kornegay, and E. M. Gregory.

Last, the author expresses his appreciation for and recognition of the valuable contributions made by students interested in biotribology over the years. These include graduate students Bettina Burkhardt, Michael Owellen, Matt Schroeder, Mark Freeman, and especially La Shaun Berrien, who contributed to this chapter, as well as the following summer undergraduate research students: Jean Yates, Elaine Ashby, Anne Newell, T. J. Hayes, Bethany Revak, Carolina Reyes, Amy Diegelman, and Heather Hughes.

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Further Information

For more information on synovial joints and arthritis, the following books are suggested: The Biology of Degenerative Joint Disease [46], Adult Articular Cartilage [5], The Joints and Synovial Fluid: I [6], Textbook of Rheumatology [47], Osteoarthritis: Diagnosis and Management [48], Degenerative Joints: Test Tubes, Tissues, Models, and Man [49], Biology of the Articular Cartilage in Health and Disease [50], and Crystals and Joint Disease [51].

Lieber, R. L., Burkholder, T. J. “Musculoskeletal Soft Tissue Mechanics.” The Biomedical Engineering Handbook: Second Edition.

Ed. Joseph D. Bronzino

Boca Raton: CRC Press LLC, 2000

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