Macro-crystals

Here, M is the reduced mass for the optic oscillation in the cell, or the mass of crystal cell for the acoustic phonons mk the mode frequency and N the number of cells in the macro-crystal. It is easy to see that the sum in Eq. (11) converges for all types of the displacements 8Rt due to the rapid decrease in 8Rt with the increasing distance from the center Rt. Therefore, the lattice particles located near the center only give the real contribution to the sum. The number N is very large, so the displacement 8qg for each mode is very small. Then, one may take into account the first few terms only in the expansion of the final phonon wave function on displacement 8q ... 

Macro crystal science and technology. Manual of STC12C5412 AD series MCU device. China, tvtvtv. MCU-memory.com.2006 7-%. 

The crystal is a fractal structure and the organization of the primitive unit cells can often be seen in the shape of the macroscopic crystal. A classic example of the fractal repetition of the unit cell is the crystalline structure of a snowflake. The unit cells have the ability to build into a variety of complex shapes, yet each unit cell retains its perfect structure. The primary unit cell structure in the case of a snowflake is hexagonal and undergoes dendritic growth to produce an array of different macro crystals (Figure 2.4). The final shape of the snow crystal will depend on the conditions used in the growth process (temperature, humidity, etc), which leads to a wide variety of observed morphologies. 

There are a great number of metal films which are thermodynamically unstable in this sense. Such fihns can only be maintained as glassy deposits if the temperature is so low that no migration can occur. The predilection which various sputtered or evaporated metallic deposits have for aggregation into micro- or macro-crystals is illustrated by the collection of observations in Table 84. The evidence from X-ray, electron diffraction, and optical experiments reveals that the three-dimensional crystalline state is very readily formed. One notes also that a rise in temperature can cause an orientation of crystallites to agree with that of the underlying solid lattice. Sometimes alloy systems may occur, evidencing mobility and interdiffusion of atoms, or certain crystal parameters may be... 

Macro Crystals.—Nearly defect-free crystals of polymers have been produced by direct polymerization of monomer at gas-solid or liquid-solid interface. Poly-(diacetylene) single crystals have been produced from irradiation of the crystalline monomer and involves the instantaneous polymerization and crystallization of... 

Liquid crystal polymers (LCP). Their molecules are macro-crystals which orientate themselves dnring flow in the direction of the greatest shear. As the shear rate rises, there is a steep fall in the viscosity of the fluid melt. Because of this, when an HR system is nsed one shonld [8] ... 

The internal structure of crystals, the crystal structure itself, determines habit, growth and dissolution of the visible, macro crystal. But up to now we know very little about this. " ... 

Solving the nucleation mechanism is important to understand structures and physical properties of any materials. To our best knowledge, no one has succeeded in observing directly the nucleation from the melt, because the number density of small nuclei on the order of nanometers (which we will henceforth call a nano-nucleus ) is too small to detect [4, 5]. Hence, only alternative experimental studies have been performed on macroscopic crystals (macro-crystals) by means of optical microscopy (OM) or bubble chamber [2]. Recent simulation studies performed on colloid systems [6, 7] also fail to provide a direct observation of nano-nucleation because the thermal fluctuation of nano-nuclei should be much more significant than that of macro-crystals or macro-colloids. This chapter introduces CNT, describes experimental approaches, and discusses the results of direct observation of nano-nucleation. 

Figure 4.4 Plots of logf(Af, t) against ogN as a function of t at r< = 129.0°C. Five plots of f N, t) are decomposed by the extended Guinier plot method. N = 450rep. unit.decreases with the increase of N for any f.This means that a small fraction of nano-nuclei survive and grow up to macro-crystals.

Step 1 Start from an appropriate set of kinetic parameters (in this study, the kinetic parameters of the macro-crystals in Reference [8]) and calculate Pb(A0 using Equation (4.22) and Equation (4.23). Obtain/st(A0 using Equation (4.21). 

We obtain the kinetic parameters (Te(nano) and Act of the nano-nucleus Acr(nano). They are cTe(nano) = 18.5 X 10 J/m and Ac nano) = 1.0 x 10 J/m . Previously, CTe(macro) = 88 x 10" J/m was obtained using AT dependence of /of the macro-crystals (Gibbs-Thomson plot) by means of OM [8]. Therefore, cTe(nano) is much smaller than cTe(macro),... 

We fit the fst(N) obtained in Rgure 4.4 against N with Pb(N) (PB(nano)) for N < ATusing the boundary condition of Step 2, as shown in Rgure 4.6a. PB(nano) fits fst(N) very well for A < AT. For comparison. Figure 4.6a shows the Pb(A0 of the macro-crystals (PB(niacro)) using the kinetic parameters of macro-crystals using the... 

On the other hand, the macro-crystal has less fluctuation with respect to its shape and size. Actually, we can neglect the disappearance of crystal, and f(N, i) cannot decrease with the increase of N for N AT, as optical observation usually shows. Therefore, PB(ntacro) cannot fit. 

In the case of macro-crystals, it is well known that surface particles are thermodynamically reconstructed into smooth and flat surface by surface diffusion, which is a kind of Ostwald ripening and results in large <7e(macro). 

It is interesting to clarify what kind of molecular structure of end surface of polymer crystals corresponds to small and large <7eS of nano-nucleus and macro-crystal of polymers, respectively. However, the size of also affects the end structures. Figure 4.7 shows that the nano-nucleus will form a loosely folded or bundle nucleus as Price predicted [12], In the case of a loosely folded or bundle type nano-nucleus (small N), the chain density on the end surface is small and not overcrowded. In the case of a fold type nano-nucleus, the energy required to form sharp folds is very large because one fold of PE has to have three gauche bonds [28]. Therefore, (Te (loose fold or bundle) is much smaller than... 

Macro-crystals tend to form sharp folds [29]. If the nucleus remains bundle-like throughout the construe-... 

Figure 4.7 Schematic illustration of against N. The nano-nucleus should form a loose fold or bundle nucleus as Price showed [12] because the energy to form sharp folds is very large. The macro-crystal tends to form sharp folds [29]. Therefore, cTe(expected) shows the size dependence of expected a. (See color insert.)...

Relationship between Nano-Nudeation and Macro-Crystallization... 

Figure 4.8 AT dependence of nano-nucleation and macro-crystallization, (a) Plots of AT dependence of f(N, t) against f for = 2.2 X lO rep. unit > At. ff AT = 10.5 K) = 450 rep. unit, which is the maximum N" in this study. f N, t) of AT = 13.9 and 11.5 K are shown in the left and top axes. f N, t) of AT = 10.5 K is shown in the right and bottom axes. It was impossible to observe the saturation of f N, t) for larger AT due to onset of lamellar stacking, r of AT = 10.5 K and rs for each AT are also shown, (b) Plots of 7, and theoretical j against and comparison of these slopes. The paraUel lines confirm the same AT dependence.

AG (AT) is given by AG (A ) 4aaJAg for Ag Act, the 2D nucleus [12]. We define them here per one particle or repeating unit. If critical nano-nucleation mainly controls both nano-nucleation and macro-crystallization, j can be rewritten as... 

Figure 4.9 Illustration of AG against N. A is the surface area of the nucleus. The nano-nucleus shows significant fluctuation with respect to its size and shape. AG (nano) corresponds to critical nano-nucleation, as shown in Reference [16]. According to Eyring s theory of absolute reaction rate, critical nano-nucleation should become the activation barrier of nucleation. The macro-crystal has smooth and flat surfaces and does not disappear. (See color insert.)...

It is impossible to observe j directly, but some observable quantities should correspond to it. The inverse of in nano-nucleation should be directly related to j, as well as I in macro-crystallization [2], Equation (4.17) defines I by the rate of macro-crystaUization per unit volume and time. If critical nano-nucleation mainly controls nano-nucleation and macro-crystalUzation, both and I should be proportional to j, respectively, that is, Tf j. We verify the direct correspondence between nano-nucleation and macro-crystaUization by obtaining this proportionality experimentally. 

The time and Af evolutions of the observed /(Af, t) confirm that many nano-nuclei appear and disappear frequently for Af < AT. It shows that only a very small part of them survive to form larger nuclei and macro-crystals. After growing into larger nuclei and macro-crystals, they do not disappear. 

The free energy of the end surface (<7e) of the nanonucleus (cTe(nano)) is much smaller than that of the macro-crystal (cTe(macro)), that is, <7e(nano) (1/5) (76(macro). Here, macro-crystal means the macroscopic crystal. We conclude that the nano-nucleus shows significant fluctuation with respect to size and shape and repeats frequent generation and disappearance. In the case of macro-crystals, it is well known that surface particles are thermodynamically reconstructed into smooth and flat surfaces by surface diffusion, which is a type of Ostwald ripening that results in large (7e(macro). 

The relationship between nano-nucleation and macro-crystallization has been studied. We obtain the AT dependence of nucleation rate (I) of a macro-crystal whose size is more than 1 pm by optical microscopy. We describe the empirical formula by /(AT) exp[-C7AT], where C is a constant. We obtain t and I by using the AT dependence of fiN, t) proportional to the net flow of nucleation (/), that is, t °c / as the zero-th approximation. This shows that the critical nano-... 

The attachment and detachment frequencies are closely related to the quantity electric current density L Thus the net current to the surface of a macro-crystal with /cm kink sites is ... 

---