Forces and Functions at the Nanoscale. Tom Charnock

Transcription

1 Forces and Functions at the Nanoscale Tom Charnock

2 Types of Forces 6 Gravity 6 Dispersion 6 Electrostatic Force 6 Scaling Laws 6 Types of Energy 6 Electrostatic Potential 7 Thermal Energy 7 Cohesive Energy 7 Born ( Self ) Energy 7 Thermal Scaling Laws 8 The Range of Gravitational Interactions 8 The Range of Dispersion Interactions 8 Range of Electrostatic Interactions 9 Solubility 9 Closest Distance of Approach 9 Polarity 0 Zwitterions 0 Dipoles 0 The Electric Field of a Dipole! 0 The Potential Energy of a Dipole 0 Polarisation of Atoms Lennard-Jones (6-) Potential Dispersion Interactions Between Bodies Gecko s Feet

3 Dispersion interaction between an atom and a surface. Dispersion interaction between a slab and a surface. 3 Dispersion interaction between a sphere and a surface. 4 Hamaker Constant 5 Other Interactions 6 Sphere-Sphere 6 Parallel Cylinders 6 Perpendicular Cylinders 6 AFM 6 Contact Force 6 Force Measurements using the AFM 7 Calibrating Spring Constants 8 Optical Tweezers 9 Wave Nature of Light 9 Intensity of Light 9 Optical Trapping 9 Point Dipole in an Electric Field 9 Trap Stiffness 9 Detection of Particles 0 Calibration of Optical Traps 0 Multiple Traps 0 Surface Energy Interfacial Energy Surface and Interfacial Tension Surface Energy and Latent Heat Surface Energy from Intersurface Potentials The Work of Adhesion 4 3

4 Wetting Interactions 4 Capillary Pressure 5 Capillary Rise 6 Nanoscale Manufacture 6 Colloidal Interactions 6 Osmotic Pressure Between Surfaces 9 Total Pressure Between Surfaces 9 Effects of Electrolytes 9 Colloidal Stability 30 The Boltzmann form of Entropy 30 Entropy of a Tethered Rod 30 Entropic Repulsion Force 3 Properties of Water 3 Hydrogen Bonding 3 Structure Of Water 3 Structure of Ice 3 Uses in Nanoscience 3 Uses in Nature 3 Hydrophobic Interactions 3 Interactions in Nature 33 Aggregation 33 One Dimensional Aggregates 33 Critical Aggregation Number 34 Two Dimensional Aggregates 34 Critical Aggregation Number 34 Three Dimensional Aggregates 34 Critical Aggregate Number 35 4

5 Formation of Aggregates 35 Self Assembly of Amphiphiles 35 Lipid Bilayers 37 Vesicle Formation 37 Curvature of Membranes 37 Biological Membrane 38 Membrane Curvature. 38 Membrane Proteins 38 Interactions Between Membranes 38 Charged Membranes (DLVO) 38 Uncharged Membranes 39 Entropic Repulsion 39 Undulation Forces 39 Lock and Key Mechanism 39 5

6 Types of Forces Gravity r. This law states that two masses eert a force towards each other with an inverse square with r. Gravity is a long range force which predominately acts over very large distances. The force due to gravity is given by Newtons law of Gravitation F G = Gm m Dispersion Dispersion forces are the interaction forces which act to hold, or repel atoms are molecules together. Dispersion forces are given by F Dis = 6C r 7. Due to the high power in the distance, the dispersion force only acts over an etremely small distance. For forces F >0 the electrostatic forces are repulsive. Electrostatic Force F = U r = q q 4 o r For force F <0 the electrostatic forces are attractive. In this case the force will be attractive if q q < 0 and repulsive if q q > 0. Scaling Laws F Dis Gravity and Dispersion can be compared to find out when either force takes over. Using to F G compare the ratio of the forces, a number greater than means that the Dispersion force is more predominant and vice versa. F Dis 6C r 6C = F G r 7 = Gm m Gm m r 5. 6C 5 This can be used to find the distance where the forces are equal. r cr =. Gm m Unfortunately this is incorrect due to the thermally driven motion of the atoms or molecules. Types of Energy When the energy U <0 it is called favourable and U >0 is called unfavourable. Z U = F r Gravity and Dispersion have an associated energy. 6

7 For Gravity, U G = Gm m r U Dis = C r 6. and for Dispersion, such as Van der Waals interactions, These energies can be compared with the thermal energy to find where thermal energies will take over the said interaction energy. Electrostatic Potential The electrostatic potential is U = q q 4 o r q are the amount of charge, o is the permittivity of free space and is the permittivity of the medium that the charges are in. Thermal Energy where r is the separation between charges, q and The thermal energy of a material is due to the movement of atoms when above a temperature of absolute zero. The thermal energy ceases to be significant when they are similar to the energy of interactions. To find thermal energy the relationship pv = nrt can be used. pv has the units of energy which means nrt is the thermal energy of the system. For an individual atom n = N N A where N is one. R N A = k which is Boltzmann s constant. This makes the thermal energy of a single atom U thermal kt. Cohesive Energy The cohesive energy is the energy used to hold a ionic crystal together. As each ion acts on the others ions in the crystal then this leads to a summing of the electrostatic energy. e U coh = 6 p + 8 Me p... = 4 o r 3 4 o r The reason for the series is due to the distance each surrounding ion is away. There a 6 attractive ions a distance of r away, then repulsive ions at p r away and so on. M is called the Madelung constant, which for NaCl is.748 Born ( Self ) Energy The Born energy is the energy required to bring a particle from infinity to make a new particle of radius r. If the charge on the stationary particle has a charge of ze then the Born energy is U Born = z e 8 o r. 7

8 If an ion is a sphere with a radius r has an initial charge of q, when a small amount of charge is added then the changing potential energy becomes U = the sphere is given: qdq 4 o r so the total energy associated with U Born = Z Q 0 Scaling Laws qdq 4 o r U Born = Q 8 o r U Born = Thermal 4 o r apple q Q 0 To find where the thermal energy will dominate over the gravitational or the dispersion interactions the energies can be compared. The Range of Gravitational Interactions The Gravitational potential energy of a system is U G = Gm m, and between two molecules, r such as methane, the mass is the same and can be given U G = Gm. r kt = Gm r = Gm r kt And so for two methane molecules r = ( ) r = m This is far smaller than the radius of which means that gravitational energies are much smaller than thermal energies until r becomes reasonably large. The Range of Dispersion Interactions an atomic nucleus ( 0 5 m), The energy of interaction between molecules or atoms is given by U Dis = C r 6. kt = C r And so for two methane molecules r = r = C 6 kt r =0.4nm This shows that the Dispersion energies are greater than the thermal energy at room temperature until distances larger than the diameter of a nucleus. This shows how gravity is negligible at room temperatures between individual atoms, on the other hand, dispersion interactions will have more effect than thermal energies when atoms are close. 8

9 Range of Electrostatic Interactions Like with the interaction range of gravity and dispersion, the range of the electrostatic interaction can be calculated. kt = q q r = 4 o r q q 4 o kt When >r then thermal energies are dominant, but when <r then the electrostatic potential energy is more dominant. When = 80 (that of water) then r is called the Bjerrum Length. Solubility The cohesive energy is different for different materials due the the different values of. The cohesive energy of NaCl in water is J in a vacuum, J in toluene and J in water. The thermal energy at room temperature 4 0 J. The reason NaCl dissolves easily in water is because its cohesive energy is small compared to that of air and toluene, with U coh 4kT. The thermal energy is high enough to completely disrupt the bonds in NaCl. If the dielectric constant (relative permittivity) is high, which waters is, then the ions are more soluble. Nanoscale Motion Atoms are constantly being bombarded. Individual molecules or atoms therefore do not move a great distance in a long amount of time. The mean squared displacement given by 6Dt where D is the diffusion coefficient from the Stokes-Einstein equation. D = kt where 6 a kt is the thermal energy of the atom, is the viscosity and a is the radius of the atom. Closest Distance of Approach v U = KE q q 4 o r + q 3q 4 o r = q o r The distance of closest approach is given by r = and so the change in potential energy is r min U = q o r o r min r o.the kinetic energy of the system is given by KE = mv m o. On closest approach, v =0, so KE = mv. To calculate the distance of r min : 9

10 mv = q o r min r o r min = mv o q + r o For two protons then r min nm. Polarity Some molecules have an electric dipole moment. More electronegative atoms have a higher affinity for electrons and so cause a permanent atomic dipole. Zwitterions Zwitterions are molecules where the polarity depends on the environment. For eample, some amino acids with a hydroyl group and an amine group. Dipoles The Electric Field of a Dipole The dipole moment is given by ep = qr epanded out then E Dipole = When r >>d then r d E e Dipole r = q 4 o q Parallel to the ais E + = d 4 o r total field is q 4 o " 4 = r apple rd r 4 i = qd e o r 3 i e E Dipole = r rd d 4 # i e. q 4 o i e and E = q 4 o r + d " r d r + d # i, so the e i. When this is e The Potential Energy of a Dipole There is no net force on the dipole but there is torque,. = F d sin = Fdsin 0

11 and because F = eq, = Eqdsin. The potential energy of a dipole in an electric field can be calculated by the work done in rotating the dipole. U = U = Z Eqdsin Eqdcos + c The dipole moment is given by p = qd and arbitrarily defining the energy scale so that c =0 means that U = E p e e Polarisation of Atoms U = Z If an electric field is applied to atom or a molecule then a dipole maybe induced, by separating the charge. This dipole moment of an atom or molecule is related to the eternal electric field by p where is the polarisability. e = E e Et To calculate the polarisability of an atom or molecule then the application of an electric field causes a dipole moment. If the charges remain at a fied separation of d then the force will balance the electrostatic force of the charges. F + F =0 e Et e Elec, F e Et = q q 4 o d ˆr e q F e = qe e Et And with qdˆr e = p e E e Et = 4 o d ˆr e = qe e Et p e 4 o d 3 And so the constant of polarisability =4 o d 3 p e =4 o d 3 E e Et

12 Lennard-Jones (6-) Potential The Lennard-Jones potential has no real physical interpretation, but fits the data of the separation of atoms and molecules to the potential energy etremely well. 6 U() =4 The U Repulsive relates to the dispersion energy which is given by Equilibrium Separation Attractive they are also used on the large scale. C r6. The dispersion energy is what attracts the atoms or molecules together. The accounts for the repulsive potential. U By using F =, the force eerted on atoms or molecules can be found from the Lennard-Jones Potential. 6 F = The equilibrium separation occurs when F =0 Dispersion Interactions Between Bodies On the small scale dispersion interactions are very visible. But Gecko s Feet Gecko s make use of dispersion forces to stick to sheer objects such as glass, etc.the foot has a large surface area and so a large interaction range. This shows the additive nature of the weak short range interactions. Dispersion interaction between an atom and a surface. If the number density of atoms in an annulus is n then the number of atoms in the annulus is N, where N = n y y. All the atoms in the annulus are at a distance of r = p + at =0, so each atom feels an interaction energy of U Disp = C r 6 If all the interaction between the atom and the annulus are summed then: U Annulus = NU Disp = ncy y ( +y ) 3 For an infinite disc then this needs to be integrated from 0 U Plane = R U Annulus ) nc R U Plane = nc = nc 4 0 h i 4 ( +y ) 0 y ( +y ) 3 y

13 Now a thin slice of infinite slice has been defined, a surface can be calculated where all the directions are summed between = D and = Z U Tot = = nc nc apple U Tot = D 3 3 nc 6D 3 This is much larger than the interatomic dispersion interaction potential. U Tot From this the force on the atom can be calculated by using F =. This gives: D F = nc D 4 4 D Dispersion interaction between a slab and a surface. 0 D The energy between an isolated atom and a surface at a separation is given by: U atom surf = n C 6 3 Now if the atom were now a thin slab of thickness, area A at a distance from a solid semiinfinite surface. This thin slab sits inside a larger slab whose end is a distance D for the surface. The total number of atoms in the thin slab N = n A. This means that the interaction energy between the thin slab and the semi-infinite solid is: U slab solid = NU atom surface 3

14 = n A n C 6 3 = n n CA 6 Summing all the interactions between all the thin slabs between = D and = gives the total interaction energy between the slabs and the solid. Z U Slab = n n AC 6 = n apple n AC 6 U Slab = 3 D n n CA D This has a yet greater range than the surface/atom interaction. Again the force can be found to be F Slab = n n CA 6D 3 Dispersion interaction between a sphere and a surface. 3 D R D r The energy between an isolated atom and a surface at a separation is given by: U atom surf = n C 6 3 Consider a small slice of a disk inside the sphere which is at a distance from the surface of the solid and which has a thickness The total number of atoms in the thin disk is: N = n V = n r Using Pythagoras R = r +(R ( D)) so r =R( D) ( D) so N = n [R( D) ( D) ] This means that the interaction energy between the thin disk and the semi-infinite solid is: U Tot = NU atom apple = n R( D) ( D) n C 6 3 4

15 = n n C 6 = n n C 6 R( D) ( D) 3 (R + D) D(R + D) + 3 Summing all the interactions between all the thin disks between = D and = D +R gives the total interaction energy between the sphere and the solid. U Tot = n n apple D+R C (R + D) D(R + D) ln + 6 = n n apple C (R + D) D(R + D) (R + D) D(R + D) ln(d +R) + +lnd + 6 D +R (D +R) D D Now cancel out to get over a common denominator: U Tot = n n apple C 4D(R + D)+D + 4(R + D)(R + D) (R + D) D +R ln 6 D(D +R) D U Tot = n n apple C R(R + D) D +R ln 6 D(D +R) D When D R then the separation is large compared to the radius of the sphere. D +R! D D + R! D This gives the dispersion interaction energy as: And therefore the force as: U Tot = F = n n CR 3D n n CR 3D D Hamaker Constant The strength of the dispersion interaction can be given in terms of the Hamaker Constant A. A = n n C This can be used in all the dispersion interaction energy and force equations such as: U Slab = n n CS D = A S D F Slab = A S 6 D 3 U Sphere = n n CR 3D = A R 3D 5

16 F Sphere = A R 3D Other Interactions Sphere-Sphere U = A R R 3D R + R Parallel Cylinders A L U = p R R D 3 R + R Perpendicular Cylinders U = A 6D p R R AFM Atomic Force Microscopy is a way of measuring nanoscale forces directly. It can be used in two forms. Contact Force The AFM tip is dragged along the surface to be studied. As the tip is moved up and down, the cantilever deflects. A laser is reflected off the cantilever and can be tracked using differential photodiodes. Using this form of detection allows for the range of the forces being studied to be changed by altering the length of the cantilever, or the distance from the cantilever to the photodiodes. The force can be interpreted using i = i i, where i is the signal strength of the photodiode. If i>0 then the force is upwards and if i<0 then the force is downwards. The photo diodes can detect movements of less than 0.nm and so forces in the nano-newton range can be measured. 6

17 Force Measurements using the AFM The force on the cantilever tip is given by F = kz, where k depends on the Youngs Modulus, the length and the geometric moment of inertia of the cantilever. l z F Neut R y y Consider a small along the beam. At introduces tensile bending induces This means there is a there are no stresses. element at a distance some points, bending forces, and at others compressive forces. neutral plane where At a distance y from fractional change in the neutral plane, the length (strain) is " l l = (R + y) R, where " = y. The stress in the thin slice is given by R R = E" = Ey R, where = F A leading to F = A = Ey R A. The fractional moment about the neutral plane is m = y F = Ey A, so the total geometrical R moment of inertia about the neutral plane is M = E Z y A = EI. As the curvature of the beam is R R I R so the moment can be given M = The moment which acts about the small element at distance along the beam is M = F (l Integrate to give: F (l ) z = F (l = F EI l + c ) so: 7

18 If the beam is fied horizontally at =0 and the displacement there are two conditions: z =0 and z =0 then c =0. Integrating again gives. The deflection at = l is: z() = F EI l 3 l 3 6 Z = F EI l 3 6 ) z = F EI l 3 3 F = 3EI l 3 Which gives the spring constant k = 3EI l 3. z The geometrical moment of inertia about the neutral plane for a rectangular beam is given by. b b a y y I = R y I = a R b b I = a h y 3 3 I = a 3 a y i b h y b b b3 8 i Calibrating Spring Constants I = ab3 There are two main ways of calibrating the spring constant of the cantilever. The first way is to hang small masses at the end of the cantilever and measure the value of k at different masses. This can lead to problems, which are, that the masses are out of the range of the atomic forces. Also the masses can contaminate the AFM tip which will change the properties of the cantilever. The second way of calibrating the cantilever is to use thermal tuning. The natural noise of the cantilever is measured. The mean squared displacement of this is comparable to the thermal energy provided. kz = k BT And so the spring constant is: k = k BT z 8

19 Optical Tweezers Optical tweezers provide a way of manipulating atoms and molecules as well as being able to measure the small forces present on them. Optical tweezers work by trapping the particles in a focussed laser beam. The particles get trapped because the dielectric particle eperience a force from the electric field of light. Wave Nature of Light The energy associated with an electromagnetic waves in the electric field: E = E o sin(!t k) Intensity of Light The intensity of light is related to the electric field by: Optical Trapping Point Dipole in an Electric Field hii = c"" on he i Consider a dielectric particle of polarisability, in an electric field, E. The dipole induced on the particle is given: p e / E e The energy of the dipole in this field is given: U = So the average energy is hui = he i. p E = E e e e The intensity of a light beam is I = c"" on he i, hence hui = U pulling the particle up the light gradient as F = r = c"" o n acts along the direction of increasing intensity gradient. c"" o n I. This gives the force I. This states that the force always A laser has a Gaussian shaped intensity, which means a particle will move to the centre of the light. This traps the particle in the y plane, but it still free to move in the z direction. If the laser beam is focussed then the particle can be fully trapped. The gradient gets stronger towards the focus point. This can be changed by altering the numerical aperture. The higher the NA the steeper the gradient. The numerical aperture is given by NA = n sin Trap Stiffness The Gaussian nature of the laser can be modelled by a quadratic function. This means that an effective spring constant can be found for the trap. The true intensity profile of the laser beam is: For r w o then: I = I o e r w o 9

20 And because r w o Using the potential energy: I o r w o + r 4 4w 4 o... then all higher order terms become vanishingly small. r I I o U = Hence the potential has a quadratic form in r. The force on the particle is then: U F = r = c"" o n I o I o c"" o n r F = w o I o c"" o n Which is in the form F = kr where k = I o c"" o nw o r w o r w o = c"" o n I o Detection of Particles As with AFM photodiodes can be used to detect the deflections of the particle. A quadrant photodiode is used for the optical trap to detect movement in the and y directions. For large particles then the shadow of the beam is detected, but for small particles then the fringe pattern can be detected on the photodiodes. Calibration of Optical Traps There are two ways of calibrating the value of k, the spring constant. The first is by applying a force and measuring the displacement. The second is the same way as with AFM, where the mean squared displacement h i is measured under the influence of thermal motion. Multiple Traps If traps are switched on and off faster than the particles can diffuse, then the same trap can trap many objects at the same time. r w o r w o 0

21 Surface Energy Bulk Atom Wi! a total interaction energy of U Bulk = N Bulk U Surface Atom Wi! an interaction energy of U Surf = N Surf u The bulk atom has N Bulk neighbours and the surface atom has N Surf neighbours, where N Surf <N Bulk. This leads to ecess energy with U = U Surf U Bulk =(N Bulk N Surf )u. Surface energy is the amount of energy required to create a surface in a vacuum or air. This is given by W = S, where is the surface energy and S is the surface area. Interfacial Energy The interfacial energy is the energy required to create an interface between two different materials. This is represented by. Surface and Interfacial Tension Tension is the force required to etend a surface. The surface energy and tension are the same. They both act to prevent the surface from increasing. The amount of energy required to increase the surface area of a liquid is given by U = L and so the force is given F = L. This means that = F L Nm.

22 Surface Energy and Latent Heat The Latent heat of vapourisation is the energy required per mole to break all the bonds in a substance and move the atoms apart. Latent heat of vapourisation = L V The energy require per atom= L V N A If the radius of the atom is R, then the area presented at the surface is A =(R) =4R. Hence the surface energy is the energy required to break half the bonds around an atom per unit area. This leads to the surface energy = L V = L V A 8N A R. N A The radius is related to the density and molar mass of the material. V = 4 3 R3 % = Mass V olume One atom has a mass= M r N A where M r is the molar mass in kg. This leads to: % = M r = 3M r N A N A 4 3 3Mr R = N A 4 % = L V 4 %NA 8N A 3M r Surface Energy from Intersurface Potentials AS From the value of the intersurface potential, U = Do 3 3 given by the movement of two surfaces from a distance D o to. This leads to = U S = A 4 Do., then the value of the surface energy is Consider an equation which describes the tip. It is assumed that the end of the tip is at the origin. y = m with m = y = tan. This leads to y = tan ) = 6.56 o = y =

23 The area in contact with the first liquid is a and the are in contact with the second liquid is a. The area of the liquidliquid interface displaced by the tip is a 3. The total energy of the tip = D L D AF Mtip 3

24 a + a a 3. The area of the small element is is A = y. a = Z D 0 y a = Z L Z D a = 0 a = D Z L y D a 3 = y =D a 3 = 4 =D apple a = 4 D 0 apple a = 4 L D a 3 = D 4 a = D a = (L D ) Hence the total energy of the tip is: So the force is given by: U = a + a a 3 U = apple D + (L D D ) U = apple( )D + L D F = [( ) ] D The Work of Adhesion The work of adhesion is the energy required to separate the surfaces of materials and in a third medium. W = Creation of Surfaces Breaking of Initial Bonds Wetting Interactions Surface and interfacial energies determine how macroscopic liquid droplets deform when they adhere to a surface. The angle of contact made between a droplet and a surface is determined by a balance between the interfacial energies and tensions. 4

25 R l + l p l + l l p R 3 l 3 This is given by: 3 = + 3 cos In reality the droplet deforms the surface slightly to compensate for the unresolved components of the force. If the Young s Modulus of the surface is high then only very small deformations occur. Capillary Pressure Capillary pressure is the pressure difference across a curved surface caused by the system resisting the deformation. In moving the interface by a small amount the change in area S is given by: S =(l + l )(l + l ) l l =(R + ) (R + ) R R = R R + R = R R apple + R + = R R apple + R + R + R + R R R R R 5

26 If R and R then and R R giving: S = R R apple + R R As the change in energy is W = S, then: apple W = R R + R R The work done in moving the interface is W = F, so: F = R R apple R + R And as force is also given by the pressure difference multiplied by the area of the interface. F =(p p )R R. Then equating the equations give: apple p p = + R R This shows that the pressure on the top side is larger than on the inside so the force acts to reduce the interface. This is the Capillary Pressure. Capillary Rise When a fine capillary is placed inside a liquid, a meniscus forms. The pressure drop across this interface causes the liquid to draw up into the capillary. The pressure is p >p due to curvature of the surface.this difference in pressure forces liquid up the capillary to form a column of height, h. The pressure p drop caused by forming the liquid column is: p atmosphere p h p = %Shg = %gh S Where %is the density of the liquid. At the equilibrium the pressure drop balances the pressure difference across the interface so: %gh = R ) h = R%g Nanoscale Manufacture Capillary rise can be used to manufacture nanoscale structures. The porous membranes in aluminium oide are easy to create and forms a very regular, well defined structures with pours whose radii can be controlled by the etching conditions. When this membrane is placed on a liquid then capillary rise occurs, and structures can be made. Colloidal Interactions Colloids are mitures, where small particles are suspended in a second substance. They are different that solutions because the particles are discrete, not dispersed. Colloidal interactions are used in quantum dots, ferrofluids and biomolecules from nano to micron size. The problem with making colloidal dispersions is that the systems have a very high surface area compared to the suspension material. This means there is a large energy penalty. 6

27 To stop the colloids flocculating (sticking together) then the particles are covered in charged groups, and often salt is added to make the suspension into an electrolyte. This is called the counterion cloud Stern (Helmhol#) Diffuse D%ble Layer Layer If a charged surface is added to the electrolyte then a layer of oppositely charged ions forms. This is called the Stern or Helmholtz layer. This layer causes a diffuse layer of repulsive charged particles. This is the diffuse double layer, which is filled with relatively mobile positive and negative counterions. If a second charged surface is added then an osmotic pressure is created, which pushes the surfaces apart to restore a uniform counterion concentration. Suppose the cloud of counterions as an ideal gas then the pressure due to the counterions is: 7

28 As R V = k B then p = nk B T p = nrt V If the ecess counterion concentration creates a net osmotic pressure: =p inside p outside For positive counterions: + =(n + n o )k B T For negative counterions: =(n n o )k B T Where n o is the number density outside the gap n+ and n are the number densities of the positive and negative counterions inside the gap. The total osmotic pressure between the surfaces is given by: = + + =(n + + n n o )k B T The values of n + and n can be calculated by assuming that concentrations of positive and negative counterions are different and so, therefore, is the electrostatic potential energy. The ions sit in a spatially varying electrostatic potential, so the energy is given by. E ± = ±zev (), where ze is the charge of the ions. The probability of finding a particle with a given energy state is given by Boltzmann statistics. p ± = p o e E ± k B T For the case of osmotic pressure then the number densities can be used. This gives: n ± = n o e zev () k B T Then inserting these values of the number densities into the epression for osmotic pressure to give: cosh ) e + e =k B Tn o e zev () k B T + e zev () k B T zev () =n o k B T cosh k B T zev () =n o k B T cosh k B T zev () When k B T then: z e =no (V ()) k B T To determine the value of V () the poisson equation V = % "" o must be used, where the charge density is calculated from the number densities of the counterions in the gap. The total charge density is 8

29 % = % + + % = ze(n + n ). As n ± = n o e zev () zev () k % = zen o e B T k e B T Then substituting this into the Poisson equation then: V = zen o. e zev () k B T "" o zev () k B T then! zev () e k B T In the limit of small V () when zev () k B T : V z e n o "" o k B T V () Because the Poisson equation is similar to the SHM equation, the solution can be similar with a negative eponential. The eponential must be negative because V () must tend to zero at large separations from the surface, so V () =V o e ( apple), where V o is the potential at the surface. Inserting this into the differential equation gives: V = V o apple e ( apple) = z e n o "" o k B T V oe ( apple) This gives apple = z e n o "" o k B T and the Debye Screening Length of apple Osmotic Pressure Between Surfaces Inserting the solution for the potential into the osmotic pressure gives: =n o z e V o Where D =. k B T e( appled) Total Pressure Between Surfaces The total pressure between two charged surfaces is given by summing up the repulsive osmotic pressure and the attractive dispersion forces. P tot = +P disp z P tot = n e V o appled) A o k B T e( 6 D 3 This is a simplified form of the DLVO theory of colloidal stability. This states that at short ranges (<0nm) then the repulsive pressures are high, but a around 4nm then the attractive forces begin to dominate.at larger ranges then the pressure is weakly attractive. Effects of Electrolytes As more electrolyte is added then the osmotic pressure forces will dominate, until too large an amount is added where the dispersion forces again dominate and the colloids stick together. 9

30 Colloidal Stability It is often not possible to use charges to keep particles apart when using organic solvents and so another method would be needed. This other method is to use entropic repulsion effects which involves decorating the surface with long polymers. As these polymers can move around freely they have a large number of configurations. If a surface is brought close to the polymer body then the number of configurations reduce and so the entropy reduces. As this conflicts with the second law of thermodynamics then entropic repulsion occurs to attempt to increase the entropy of the system. The Boltzmann form of Entropy S = k B ln W This states that the entropy of the system is given by the Boltzmann constant multiplied by the natural log of the number of microstates. Entropy of a Tethered Rod S = k B ln W = k B ln A rod has a length l and cross-sectional area a which is tethered to a surface. If the rod is unrestricted in its motion the top can sweep out an area, that of a hemisphere - (4 l )= l. If the area of the tip is a then the total number of orientations that the rod can adapt is: W = A l a a = l a Using the Boltzmann form of the entropy, the unrestricted motion of the rod is. When a solid surface approaches the rod its motion becomes restricted and so less space can be sampled. As a result the entropy D decreases as the system tries to resist this by generating a force. If the rod is at a some angle to the surface on which it sits, when it is rotated then the arc of the small strip of area is given by A (l ), and from the diagram, it can be seen that = l cos. This gives: A = l cos A = l [sin ] ma min A = l sin ma sin min In the case where min =0 and ma is determined by the separation of the surface then, if D is the ma height then D = l sin ma giving: A = ld The cross-sectional area is still a and so the number of microstates is given as: Hence the entropy of the rod is: And so the change in entropy is: l l ma a A a = ld a S = k B ln ld a S = k B ln ld a S = k B ln D l k B ln l a 30

31 Entropic Repulsion Force To calculate the energy associated with the change in entropy, and finally the force the Gibbs Free Energy must be calculated. U = H T S This states that the energy comes from the enthalpic energies, such as dispersion interactions and also from entropic energies. The force due to the Gibbs free energy is found by differentiating the equation with respect to D. F = D ( H)+T D ( S) As only the entropic force is being studied then this gives: F entr = T D k B ln D l F entr = k BT D The force is greater than zero and so states that it is a repulsive force. If the distance between the rods is d then each rod occupies an area of d and the repulsive pressure is: p = k BT d D If the dispersion forces are also calculated then the equilibrium position can be found at the point where: p entr + p disp =0 A 6 D 3 = k BT 4l D Properties of Water Water is irregular compared to most liquids on the earth due its its high dielectric constant and its high melting and boiling point. As well as this it has a high surface energy and latent heat of vapourisation. This is due the high electronegativity of oygen. Nitrogen, fluorine and chlorine are also highly electronegative. 3

32 + Hydrogen Bonding Electronegativity is the tendency for the atoms to pull electrons towards themselves. This electronegativity allows for oygen to pull the electron of hydrogen + + towards itself and therefore causing a polarity on the molecule. This polarity causes strong dipolar interactions, which are hydrogen bonds when hydrogen is involved. The hydrogen-bonds typically have an energy between 4 6k B T room. This is far lower than metallic or covalent bonds but is stronger than normal dispersion forces. Structure Of Water The directional nature of the hydrogen bonds between a certain hydrogen and a certain oygen leads to water being structured in a tetrahedral shape, even when it is a liquid at low temperatures. The molecules are quite mobile though, because the bonds are strong enough to cause permanent interaction Structure of Ice When water crystallises into ice the nearest neighbours reduces from 5 to 4 molecules and so the structure becomes a more open tetrahedral shape. Uses in Nanoscience The stickiness of water allows large networks of molecules to self assemble. Without the water then the molecules would flocculate and be useless. Even though the hydrogen bonds make the medium sticky, they are weak enough to be broken by thermal motion, so structures are able to find their lowest energy state. Molecules with polarity can also by controlled by the amount of hydrogen bonding which is allowed to occur. Uses in Nature Proteins and DNA use hydrogen bonding to produce their well defined structural elements, such as the double heli structure or flat beta-sheets. Hydrophobic Interactions 3

33 When a non-polar molecule comes into contact with a polar liquid then it can only interact by dispersion forces. Because these are weaker than hydrogen bonds then the liquid would prefer to hydrogen bond. This means that the non-polar molecule will rearrange to reduce its surface area. This reduces the entropy of the system and so the non-polar molecules will be forced together and form hydrophobic clusters. If a non-polar surface is placed in water then the interaction energy between the surface and the polar liquid is given by: U = Se D o Where S is the interfacial area, is the interfacial energy and o is the range of interaction nm The force of the hydrophobic force is therefore given by: F = S e Interactions in Nature Quite often proteins and other biological molecules have a hydrophilic and a hydrophobic end. When placed in a polar liquid then the molecules will cluster and fold into shapes so that the hydrophilic interactions are maimised and the hydrophobic interactions are minimised. This can be seen in structures like micelles where a ball is formed with all the hydrophilic ends protecting the hydrophobic ends on the inside. o D o Aggregation Many particles favour attraction of other particles when in a medium and so they will aggregate and can drop out of solution. Aggregates form when the energy per particle in an aggregate is less than the energy of the individual particle. Depending on the type of interaction that occur different shaped aggregates can form. If the attractions occur highly directionalised then long fibrils will form, which is like a one dimensional aggregate. For a two dimensional aggregate to form the interactions must occur around a plane of the particle, and for three dimensions the interactions are isotropic and therefore will form droplets or clusters. One Dimensional Aggregates A linear aggregate, in which all the particles are joined together in a line, has an energy dependent on the number of bonds. If each bond has an energy of u relative to the unbonded state then the total energy will be U agg = Nµ = (N )u when there average energy per molecule is given by (N )u µ = N = ( fracn) u. This means that as more molecules are added the more favourable the energy per molecule becomes. This tends to lead to the formation of etremely long bonds. 33

34 Critical Aggregation Number The critical aggregation number is the number of molecules needed aggregate to begin to form. This can only for the Nc u, which means that Nc = and aggregation happen when µ < 0 and so for aggregation to start 0 = will begin if there is more than one other particle in the medium. Two Dimensional Aggregates Two dimensional aggregates are generally disk-like. If the volume of a single particle is and the volume of the aggregate is R t then the number of particles in the aggregate is: N= R t If there are N particles in the disk then the bulk energy of the aggregate is UBulk = N µbulk where µbulk is the bulk interaction energy per particle. The aggregate has an unbounded perimeter of area Rt. If the surface energy is then the ecess surface energy of the aggregate is: USurf = Rt Hence the total energy of the aggregate is U = UBulk + USurf which is U = Because R = N t N µbulk + Rt. the interaction energy per particle is: µ= µbulk + ( t) N Critical Aggregation Number The aggregation number for two dimensions is given when µ = 0 this means that: 0= µbulk + f rac ( t) Nc µbulk = ( t) Nc ( t) Nc = µbulk ( t) Nc = µbulk Three Dimensional Aggregates The energy per molecule of a spherical aggregate can be found if the volume of each particle is and the volume of 3 then the number of particles per aggregate is: the aggregate is 4 3 R 3 N = 4 R 3 then the total bulk energy is UBulk = If the bulk energy per particle is µbulk N µbulk and the ecess surface energy of the aggregate is USurf = 4 R This gives the total energy of the aggregate is U = UBulk + USurf which is U = N µbulk + 4 R 34

35 The free energy per particle (chemical potential) is: As R = 3 N 4 3 then: µ = U N = µ Bulk + 4 R N µ = µ Bulk N 3 Critical Aggregate Number When the effects of the ecess surface energy just overcome the effects from bulk cohesion then µ =0 then: 0= µ Bulk µ Bulk =4 3 4 N 3 c = 4 3 µ Bulk 4 N c = 4 µ Bulk N 3 c 3 N 3 c Formation of Aggregates Aggregates can form in two ways. The first is due to thermal fluctuations, which is called Homogeneous nucleation. This takes a long time because it is very unlikely to get enough particles together via thermal fluctuations. The second is Heterogenous nucleation in which foreign surfaces act like catalysts to speed and increase the number of interactions. Self Assembly of Amphiphiles Amphiphiles are molecules with a water soluble head group and a hydrophobic tail group. Amphiphiles can self assemble to reduce unfavourable contact between the hydrocarbon tail and the water. These aggregates are called micelles. The critical micelle concentration is the number of amphiphiles needed to create a micelle. Micelles have a minimum number of amphiphiles necessary to form because the aggregate is unfavourable if there are not enough to seal the water out from the hydrocarbon tail groups. Likewise too many amphiphiles will lead to some being inside the structure and so there will be negative interactions between the head group and tail group. Because of this, micelles tend to have a well defined size, such that there are just enough to to shield the hydrophobic interior. Depending on the concentration of amphiphiles in the solution different structures will form. The shape depends on the hydrocarbon volume, which is the volume occupied by the hydrocarbon tail group, which is the larger part of the molecule; the critical chain length which is the length of the fully etended hydrocarbon chain and the optimum head group area, which is the effective area of the head group. The optimum head group area depends on the electrostatic or steric effects and the hydrophobic interactions. To make sure that the hydrocarbon chain is completely sealed from the water then the radius of the micelle cannot be longer than critical chain length. 3 35

36 The number of molecules along with the hydrocarbon volume and the optimum head group area can be used to determine the shape of the micelle. This is done by dividing the volume of the micelle by the surface area of the micelle gives the radial size. For spherical micelles the volume is given by 4 3 R3 = N and the surface area by 4 R = Na o. This means the radius is: 4 R 3 3(4 R ) = N Na o 3 a o = R Because R apple l c then the condition needed for spherical micelles is: l c a o apple 3 For cylindrical micelles the volume is R l = N and the surface area is Rl = Na o so the radius is: Again the chain length is l c R l Rl = N Na o a o = R R and so cylindrical micelles will form up to apple. As spherical micelles form when l c a o l c a o apple 3 then the cylindrical micelles form: 3 < l c a o apple To form a bi-layer then the volume is given by A(R) =N, the surface area is A = Na o. The radius is then: This means that A(R) A l c a o = and so bilayers will form when: = N Na o R = a o < l c a o apple 36

37 Lipid Bilayers Lipids have a small critical chain length but a large hydrocarbon volume which means that apple and so bilayers tend to form. Lipid bilayers are the main constituents of animal < l c a o cell membranes. Vesicle Formation As the hydrophobic hydrocarbons are eposed at the edges of bilayers then the bilayers have the tendency to fold in on themselves to form hollow spheres called vesicles. Vesicles can be used to contain highly toic drugs and only be allowed to break up under certain circumstances, forming a Magic Bullet. This could be used to target cancer cells effectively. Curvature of Membranes Vesicles can only have a limited number of constituent molecules. This is because too few would means there would be some eposure of hydrophobic hydrocarbons between the outer layer, and too many and a bilayer would have a more favourable energy. The minimum radius of curvature can be found by considering a spherical vesicle of radius R with a membrane thickness t. The total volume is given by V = 4 R3 3 frac4 (R t) 3 3 which, after rearranging and epanding is V = 4 3 [3R t 3Rt + t 3 ]. The area of the membrane is A =4 +4 (R t) and for t R and so A 8 R. Equating these give: a o = Using the quadratic formula the radius is: [3R t 3Rt +t 3 ] 8 R a o R = tr 3t R + t 3 6 0= 3t a o 3t R + t 3 R = l 3 ± 3 q a o l 3 This states that there are two possible solutions which satisfy this packing constraint. The system is likely to use only the largest radius, as this minimises the bending energy of the bilayer. R = q l 3+ 3 v o a o l a o l 3 a o l R decreases as l increases and so by using l = l c, the minimum radius can be found R c. 37

38 R c = r 4 l c a o l c a o l c Biological Membrane Biological membranes are made up of lipids, proteins and cholesterol. Due to lipids and amphiphiles making up to 80% of the membrane by mass then they can be approimated like lipid bilayers. Membrane Curvature. PC PE PC PE PC PE PE A membrane can be curved due to the shape of the phospholipid members. Phosphatidyl Choline (PC) has a truncated cone whereas the Phosphatidylethanol Amine (PE) has a a wedge shape cone. If these are alternated then a straight membrane can be made, but by packing them together gives a very tight radius of curvature. Membrane Proteins Membrane proteins allows the cell to interact with their environment. The proteins have adhesive contacts, which will allow some molecules entry to the cell, whilst stopping others. PE PC PE PC The proteins and lipids can form many structures. Some proteins have a hydrophobic centre, but hydrophilic ends. These sit within the membrane, with both ends outside, this is called integral proteins. Some proteins have a small hydrophobic end, and this will just attach to the outside of the membrane, this is called a peripheral protein. If two tight radius parts of the membrane are cut off from each other a pore forms. Interactions Between Membranes PE PE PE PC PC PC PE PC PC PC PC PE PC PE PC PE PE PC At long range then Van der Waals (dispersion forces) dominate the interaction between membranes, but at short range (less than nm) there could be repulsive double layer forces for charged lipids and steric repulsion forces for uncharged lipids. Charged Membranes (DLVO) The total pressure between two planar surfaces is a sum between dispersion forces and electrostatic double layer forces. 38

39 Where apple = ""o k B T n o z e z e Vo P = n o k B T e appled {z } electrostatic is the Debye screening length. A 6 D 3 {z } dispersion Uncharged Membranes Entropic Repulsion When membranes come close together then their movement becomes restricted and so entropic repulsion forces will arise. P = k BT d D {z } steric A 6 D 3 {z } dispersion Undulation Forces The dominant steric forces between membranes usually due to the confinement of the undulations of the membranes. The system resists the reduction in entropy by generating a pressure: P = (k BT ) YD 3 Where Y is the bending modulus and D is the distance between the membranes. This gives the total pressure as: P = (kb T ) Y Lock and Key Mechanism Some bonds can be made using highly oriented natural substances. This means that although most molecules could pass straight past the receptor, a certain ligand will have a lot of non-covalent interactions able to produce binding energies of 35kT. This is high enough to stick the ligand to the receptor and so form a bond. A 6 D 3 39