Surface Energy PDF

2.1.1. Surface Energy The surface energy is a fundamental physics parameter of the solid surface. Its importance be in the understanding the surface phenomenon, as the behavior of the catalysis, adsorption, surface distinguishing, surface corrosion, growth rate, lattice constants, grain bound...

2.1.1. Surface Energy The surface energy is a fundamental physics parameter of the solid surface. Its importance be in the understanding the surface phenomenon, as the behavior of the catalysis, adsorption, surface distinguishing, surface corrosion, growth rate, lattice constants, grain boundaries formation, and thermal stability. The surface energy can be defined as “ the additional work to overstretch or tear an elastic continuum”, or the additional energy to create a new surface area unit, then, the work to form two boundaries is; 𝑎𝑌 ≈ 2𝛾 … … … … … … … …. (2 − 17) 2 where 𝑎 is the particle radius, 𝑌 is Young’s modulus and (𝛾) is the surface energy. So, we can consider the surface energy as a free surface energy (Gibbs's energy) (G) of area, then from the thermodynamic looking, the surface energy is one of the parameters in Gibbs's energy which is explained by; 𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 + 𝛾𝑑𝐴 … … … … … … … (2 − 18) where 𝑆, 𝑇, 𝑉, 𝑃 , and 𝐴 are entropy, temperature, volume, pressure and area respectively. At the pressure and the temperature are constant, the surface energy may be written as; 𝜕𝐺 𝛾=( ) … … … … … …. (2 − 19) 𝜕𝐴 𝑇,𝑃 Example (2-2): Calculate the resulted boundary volume of system that has a 𝑁 surface energy (1 𝐉/𝑚2 ) and Young’s modulus (50𝐺𝑝𝑎). (Note: 1𝑃𝑎 = = 𝑚2 𝐉/𝑚2 )  Solution: the work to form two boundaries is; 𝑎𝑌 ≈ 2𝛾 2 4𝛾 4(1) 𝐉/𝑚2 𝑎≈ = 9 2 = 8 × 10−11 𝑚 ≈ 0.08 𝑛𝑚 𝑌 50 × 10 𝐉/𝑚 𝐷 = 0.1 𝑛𝑚 Back to thermodynamic, the surface energy can be estimated by calculate the bonding energy of an atom on the surface, to estimated it can using the dimensionality approach. Consider the surface energy of atoms in the crystal surface is closer to the sublimation energy (∆𝐻𝑠 ), which be found between 80 𝐾𝐽/𝑚𝑜𝑙 for cesium and 900 𝐾𝐽/𝑚𝑜𝑙 for tungsten, the surface energy can be written as; ∆𝐻𝑠 𝛾≈ … … … … … … … ….. (2 − 20) 𝑁𝑎 𝑆𝑎 Where 𝑁𝑎 is Avogadro’s number and 𝑆𝑎 is the surface area of atom. Suppose that, the sublimation of atom is 200 𝐾𝐽/𝑚𝑜𝑙 and its surface area is 10−19 𝑚2. The surface energy can be estimated as; 𝐾𝐽 ∆𝐻𝑠 200 𝛾≈ ≈ 𝑚𝑜𝑙 ≈ 3.33 𝑱/𝑚2 𝑁𝑎 𝑆𝑎 23 −1 −19 6 × 10 (𝑚𝑜𝑙 ) × 10 𝑚 2 For more accuracy in the calculations of the inner bonds, that require using the pair potential method, which depends on the distances between the inner atoms, where the reaction energy of the pair is (𝜑(𝑟) ) as shown in figure (2-5).  Figure (2-5): the reaction energy of the pair as a function of the distance between two atoms. As example for estimate the surface energy of {100} in body- centered cubic (BCC). Generally, the total energy of each atom in the bulk is the sum of bonds pair of the neighboring atoms, as; 𝑘 1 𝜀𝑐 = ∑ 𝑛𝑖 𝜑(𝑟) … … … … …. (2 − 21) 2 𝑖=1 where 𝑛𝑖 is the number of atoms in the 𝐾 𝑡ℎ sphere coordinate. Supposing there are two atoms (𝐾 = 2) into bulk, as shown in figure (2-6).  Figure (2-6): The energy per one atom in the bulk of (BCC). The energy per one atom in the bulk of (BCC) is; 1 𝜀𝑐 = [8𝜑1 + 6𝜑2 ] … … … ….. (2 − 22) 2 = 4𝜑1 + 5𝜑2 … … … … …... (2 − 23) where 𝜑1 and 𝜑2 are 𝜑(𝑟1 ) and 𝜑(𝑟2 ) respectively. The energy per one atom on the surface {100} of BCC is; 1 𝜀𝑠 = [4𝜑1 + 5𝜑2 ] … … ….. (2 − 24) 2 = 2𝜑1 + 2.5𝜑2 … … … ….. (2 − 25)  Figure (2-7): The energy per one atom on the surface {100} of BCC. The surface energy is the difference between the two energies of atom in the bulk and on the surface divided per surface area for one atom. 𝜀𝑐 − 𝜀𝑠 𝛾= … … … … ….. (2 − 26) 𝑎2 4𝜑1 + 3𝜑2 − 2𝜑1 − 2.5𝜑2 𝛾= 𝑎2 2𝜑1 + 0.5𝜑2 𝛾= … … … … …. (2 − 27) 𝑎2 Now, let us estimate the surface energy of {100} in face- centered cubic (FCC); using figures (2-8) and (2-9). The energy per one atom in the bulk of (FCC). 1 𝜀𝑐 = [12𝜑1 + 6𝜑2 ] … … … ….. (2 − 28) 2 = 6𝜑1 + 3𝜑2 … … … … …... (2 − 29) The energy per one atom on the surface {100} of FCC is; 1 𝜀𝑠 = [8𝜑1 + 5𝜑2 ] … … ….. (2 − 30) 2 = 4𝜑1 + 2.5𝜑2 … … … ….. (2 − 31) Using equation (2-27), again, the surface energy of {100} in (FCC) can be written as; 6𝜑1 + 3𝜑2 − 4𝜑1 − 2.5𝜑2 𝛾= 𝑎2 2𝜑1 + 0.5𝜑2 𝛾= … … … … …. (2 − 32) 𝑎2  Figure (2-8): The energy per one atom in the bulk of (FCC). Figure (2-9): The energy per one atom on the surface {100} of (FCC).  If the approximation of the values used is taken into consideration, (𝜑1 ≈ 𝜑2 ) and (𝑟1 ≈ 𝑟2 ≈ 𝑎) ,and work on the surface only, we get to approximate uniform relationship of equation (2-28) written as; 1 𝛾 = 𝑁𝑏 𝜀𝜌𝑎 … … … … …. (2 − 33) 2 where 𝑁𝑏 is the number of the broken bonds and 𝜌𝑎 is the surface intensity of the surface atoms. In this modal, the reaction between the atoms of molecule is neglected, and the length of bonds are equaled. The energy of both bulk and surface is same, and does not include the size effect, the pressure and entropy. It is applied with the rigid solid matters, because the surface is not relaxed (if the surface is relaxed, the surface atoms move or restructure, then, the surface energy will reduced). This relation is given only an ordinal estimate of the actual surface energy of the solid surface. For example, if FCC lattice has a lattice constant (a), we can calculate the surface energy of both {100} and {110}. For {100} plane: There are eight bonds of four atoms were broken as shown in figure (2- 10). {100} Figure (2-10): The broken bonds of the surface atom in {100} of (FCC).  The surface energy of {100}can be found using equation (2-34); 18 𝜀 4𝜀 𝛾{100} = = … … … … … … ….. (2 − 34) 2 𝑎2 𝑎2 For {110} plane: There are ten bonds of five atoms were broken as shown in figure (2-11). Figure (2-11): The broken bonds of the surface atom in {110} of (FCC). 1 10 𝜀 5𝜀 𝛾{110} = = … … … … … (2 − 35) 2 √2𝑎2 √2𝑎2 One result of the reduction in the size (an increase of the area/ volume ratio) of the matter is the increased surface energy,. A reduction in the nanoparticles size can congregation of atoms or molecules on the surface, forming regular structures. The main causes of this formation are forces that tend to decrease the surface area of particle and therefore its surface energy,, that is an important factor of particles formation. The surface energy of particle becomes an increasing factor with the reduction of size as shown in table (2-2), which indicates how the specific surface area and the total surface energy of one gram of (𝑁𝐶𝑙) vary with particle size. Table (2-2) Variation of surface energy with particle size.  To understand this phenomenon, its effects and what the mechanisms of surface energy reduction, let take a closer look on the surface of matter. 1.3.3.1 Surface Energy Reduction Mechanisms Both solids and liquids tend to have a strong tendency to reduce its total surface energy. There are several mechanisms to do it. 1- Surface Relaxation When the surface atoms or surface ions los its bonds, the surface of solids has unsaturated atoms or molecules, these atoms and molecules bring under direct inner forces of subsurface layers because of the unsaturated bonding. The bonds’ length of surface particles are shorter than that between interior particles ( see figure (2- 12)). The reduction in the particle size leads to contraction in the bonds length between the surface atoms and that inside be so important, lattice constants bring a tangible reduction, and the extra energy of surface atoms or molecules appears as a free surface energy or surface tension. Figure (2-12) demonstrates the change in the distance between the surface layer and other layers inside substance. The shift in the surface atoms may be directly (inward) shift, or diagonally (lateral) shift depending on the structure of layers as shown in figure (2-13). Figure (2-13) type of shifting in the surface atoms to inside layer. Let discuss this phenomenon in {100} of sample cubic as an example. The coordination number of each atom is 6, as demonstrated in figure (2-14). Spirited atom Surface atoms 𝐹റ Subsurface atom Figure (2-14) type of shifting in the surface atoms to inside layer. Each surface atom bounds with one atom in the subsurface atom and four neighboring atoms in the its layer, any chemical bond be supposed as attractive force. All surface atoms influenced by normal attractive forces to bottom directly on the surface.  The distance between the surface atomic layer and the sublayer will decrease (less than other) because of the influenced force between the two layers without change in the atomic structure of the surface, as shown in figure (2-13). Moreover, the distance between the inner layers under the surface will be decreasing too. This is one image of the surface relaxation images. Additionally, in the bulk materials the decrease in the lattice constants is unnoticeable and neglected. But in the nanomaterials may be clearly observed and more influential, this leads to reduce in the bond length in the nanoparticles. 2- Restructure A new created a tenser bond is the probable bond to create, if it had more than one broken bond. For example, the {100} plane of silicon crystal, as shown in figure (2-15). The restructure in both silicon or diamond need less surface energy. This process is important influence on crystal growth. Figure (2-15) the {100} plane of silicon crystal, 3- Surface Adsorption Some materials tend to consist chemical and physical bonds with other materials in the surrounding environment by the chemical adsorption and the weak electrical attraction (electrostatic) or Van devil’s force to the reduction of the surface energy of solids. For example, the bounding of hydrogen atoms with surface atoms of the diamond and hydroxyl groups as shown in figure (2-15).  Figure (2-16) the chemical adsorption in the silicon and diamond surfaces. 4- Composition Segregation Impurity The segregation impurity mechanism is like the chemical adsorption, that is active in the liquid phases more than the solids because of the solid state need to high diffusion and large distances for the diffusion. In nanostructures, the separation may play an important role in the reduction of the surface energy, due to the significant impact of surface energy and distances for the diffusion. Although there is not sufficient evidence directly at the experimental level of the effect of the segregation impurities in nanomaterials to reduce surface energy. But, the difficulty of doping and gating of the perfect crystalline structure indicate to the removal of impurities and defects form in the surface of the nanomaterials. Figure (2-17) illustrates the segregation of (a) Cr into Ti(CN) allay at 800 oC and Au2Ti allay on the surface of TiO2 thin film. b a Figure (2-17) the segregation of (a) Cr into Ti(CN) allay at 800oC and Au2Ti allay on the surface of TiO2 thin film. Thermodynamically, the designation of the geometrical shape depending on the kinetic factors, that have been dependent on the conditions of both processing and growth. The kinetic factors illustrate the variation in the morphological structure in same crystal under deferent conditions. Generally, in this phenomenon, the reduction of the surface energy includes two mechanisms. 1. The aggregation of the individual nanoparticles to reduce the surface energy level (if the activation energy is sufficient to continue this process). 2. The individual nanoparticles aggregate without change in the crystalline structure. Figure (2-17) the aggregation of the individual nanoparticles to reduce the surface energy level with and without change in the crystalline structure. 1.3.3.2 Surface Curvature The solids have finite surfaces that are lead to the difference in the properties of the surface atoms and inner atoms because of the perpendicular bonding of the surface atoms with the surface of the subsurface layer. To understand the concept of the surface curvature, let us assume the diagram in figure (2-18). Figure (2-18): The Curvature in the Surface  The circle with radius (𝑟) touches the curve at point C, 𝑟 is called the radius of the curvature at C, and the inverted of 𝑟 is called the curvature which may change with the position of C along the curve. The curvature is positive with the convex surfaces and negative with the concave surfaces. Gibbs’ free energy influence by with change in both pressure and curvature as shown in the expression; 2γV ∆G = ∆PV = … … … … … (2 − 36) r According to equation (2-36), the increase of pressure with decreasing in the particle size results to a rising in the curvature. The curvature effect magnitude is mostly unimportant, but is so important in the nanoscale. The equilibrium number of the vacancies is one of the parameters that determine the ∆G of the particles, this effect can be explained by; ∆Gvtotal = ∆Gvb + ∆Gvex … … … … … (2 − 37) Where ∆𝐺𝑣𝑏 is the equilibrium Gibbs’ free energy for the vacancies formation in the bulk, and ∆𝐺𝑣𝑒𝑥 is the Gibbs’ free energy for the formed vacancies due to the effect of the curvature. Ωγ where; ∆Gvex = … … … … … (2 − 38) r and Ω is the atomic volume, then ∆𝐺𝑣𝑡𝑜𝑡𝑎𝑙 can be written as; Ωγ ∆Gvtotal = ∆Gvb +… … … … … (2 − 39) r The total equilibrium concentration of vacancies in the nanoparticles can be given by; ∆Gvtotal Xvtotal = exp [− ] … … … (2 − 30) kBT where 𝑘𝐵 is Boltzmann constant and 𝑇 is the temperature. By substituting of eq. (2-29) into (2-30), we obtained;  ∆Gvb Ωγ Xvtotal = exp [− ] exp [− ] … … … … (2 − 31) kBT rk B T b ∆Gvb Xv = exp [− ] … … … … (2 − 32) kBT Ωγ Xvtotal = Xvb exp [− ] … … … … (2 − 33) rk B T 𝑥2 𝑥3 𝑥𝑛 Using the expansion of exponential function ( 𝑒 𝑥 = 1 + 𝑥 + + + …….+ ) , we can 2! 3! 𝑛! obtained; Ωγ X vtotal = X vbulk [1 − ] … … … … (2 − 34) 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑐𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 rk B T Ωγ X vtotal = X bulk v [1 + ] … … … … (2 − 35) 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑐𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 rk B T This means that the concentration under the plane surfaces is lass than under the concave surfaces and greater than under the convex surfaces. This plays an important role in the determination of the physical properties of nanoparticles as the electrical properties, thermal capacitance, diffusion and catalytic activity. Figure (2-19) shows the size effect in the diffusivity of some metallic nanoparticles. Clearly, we note disappearance this effect above 10 nm of size, and decreasing in the diffusivity of the concave surfaces, while appearance of an opposite behavior in the convex surfaces. Figure (2-19): Diffusivity at 900° in silver, gold, and platinum nanoparticles of different sizes normalized with respect to bulk diffusivities  When two particles are sintering to each other as shown in figure (2-20). A neck region is formed between the two particles, has a concave surface due to the reduction in the pressure, lead to peregrination of atoms from the positive convex surface ( high positive energy) to the negative concave surface ( high negative energy), that leads to fusion the two particles to gather and disappearance of the neck region. In other ward, the nanoparticles tend to assemble even at room temperature due to the curvature effect. Figure (2-20): Aberration-corrected STEM image of two nanoparticles sintering at room temperature, and schematic showing the sintering process of two nanoparticles. To understand the effect of nanoparticle size on the lattice constant, the Gauss – Laplace formula will assumed for nanodrop, which is given by; 4γ ΔP = … … … …. (2 − 36) d Where Δ𝑃 is the difference in the pressure inside drop and the outer environment and 𝑑 is the radius of drop. If the drop is a solid cubic crystalline structure which has a lattice constant(𝑎). The compressibility(𝐾), that measures the variation in the size due to the change in the pressure with constant temperature is given by; 1 ∂V K= [ ] … … … … (2 − 37) Vo ∂P T where 𝑉𝑜 = a3 by some mathematical processes, we can obtained;  1 𝜕a3 4𝑎𝐾 𝐾 = 3[ ] ⟹d=𝛾 … … … (2 − 38) a 4𝛾 3∆𝑎 𝑑 𝑇 We can see the proportional of surface energy and lattice constant with the particle size as shown in figure (2- 21), that shows the change in the lattice constant of aluminum lattice constant with the size of Al nanoparticles. Figure(2- 21): Lattice parameter of Al (aluminum) as a function of particle size. 1.3.3.3 strain Confinement Planar defects, such as dislocations are also affected when present in a nanoparticle. Dislocations play a crucial role in plastic deformation, thereby controlling the behavior of materials when subjected to a stress above the yield stress. In the case of an infinite crystal, the strain energy of a perfect edge dislocation loop is given by;  μb2 r Ws = ln { } … … … (2 − 39) 4π C where µ is the shear modulus, 𝑏 is the Burgers vector, 𝑟 is the radius of the dislocation stress field, and 𝑐 is the core cutoff parameter. If the crystal size is reduced to the nanometer scale, the dislocation will be increasingly affected by the presence of nearby surfaces. As a consequence, the assumption associated with an infinite crystal size becomes increasingly invalid. Therefore, in the nano-scale regime, it is vital to take into account the effect posed by the nearby free surfaces. In other words, there are image forces acting on the dislocation half-loop. As a consequence, the strain energy of a perfect edge dislocation loop contained in a nanoparticle of size 𝑅 is given by μb2 R − rd Ws ≅ ln { } … … … … … … (2 − 40) 4π R where 𝑟𝑑 is the distance between the dislocation line and the surface of the particle and the other symbols have the same meaning as before. A comparison of eq.s (2- 39) and (2- 40) reveals that for small particle sizes, the stress field of the dislocations is reduced. In addition, the presence of the nearby surfaces will impose a force on the dislocations, causing dislocation ejection toward the nanoparticle’s surface. The direct consequence of this behavior is that nanoparticles below a critical size are self- healing as defects generated by any particular process are unstable and ejected. 2.2. Quantum Effect In your previous studies, you know the energy levels of bulk crystalline materials are divided into valence bands that are fully filled with electrons at minimum temperatures and may be separated from the conduction band by the energy gap. But in the reduction of size, the material behavior will be different completely in this aspect. An electron will suffer many different types of confinements depending on size, shape and type of particle that will be installed. An electron in 0D will be confined by three dimensions as in quantum dots, in 1D will confined by dual space as nano(wires, tubes and rods). Using quantum mechanics.