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Material Synthesis

Field: Solid State Chemistry

The creation of materials with new or improved properties.

Sequence of Expressions

Definition

Stoichiometry

Let a chemical reaction be represented by the general form: ∑i=1NνiAi→∑j=1MηjBj\sum_{i=1}^{N} \nu_i A_i \rightarrow \sum_{j=1}^{M} \eta_j B_j, where AiA_i and BjB_j are chemical species, and νi\nu_i and ηj\eta_j are the stoichiometric coefficients. The principle of mass conservation dictates that for every element EE present in the system, the total number of atoms of EE consumed must equal the total number of atoms of EE produced. Formally, for any element EE: \(\sum_{i=1}^{N} \nu_i \cdot (\text{atoms of } E \text{ in } A_i) = \sum_{j=1}^{M} \eta_j \cdot (\text{atoms of } E \text{ in } B_j)\.
Define the mole (nn) as the amount of substance. Let NAN_A be Avogadro's number. The relationship between the number of particles (NN) of a substance and the amount of substance (nn) is defined by: Nn=NA\frac{N}{n} = N_A. Specifically, for any particle type (atoms, molecules), NA≈6.022×1023 particles/molN_A \approx 6.022 \times 10^{23} \text{ particles/mol}. This constant serves as the conversion factor between the particle count and the molar quantity.
Define a solid solution S\mathcal{S} as a material composed of elements A and B, where the local atomic arrangement is homogeneous. The composition is defined by the mole fraction xAx_A and xB=1−xAx_B = 1-x_A. The stability of the solid solution is determined by the minimization of the Gibbs Free Energy of mixing, ΔGmix\Delta G_{mix}, relative to the pure components:\nΔGmix(xA,T,P)=∑ixiΔGi∘+RT∑ixiln⁡(xi)+ΩxAxB\Delta G_{mix}(x_A, T, P) = \sum_{i} x_i \Delta G_{i}^{\circ} + R T \sum_{i} x_i \ln(x_i) + \Omega x_A x_B
A system is at thermodynamic equilibrium if and only if the total Gibbs Free Energy GG is minimized with respect to changes in state variables (T,P,xT, P, \mathbf{x}), such that the differential change in GG is zero under constant constraints:\ndG=0subject to dT=0,dP=0, and ∑xidN~i=0dG = 0 \quad \text{subject to } dT=0, dP=0, \text{ and } \sum x_i d\tilde{N}_i = 0
Define the Gibbs free energy change (ΔG\Delta G) for a reaction proceeding from reactants RR to products PP at constant temperature TT and pressure PP: ΔG=∑jνjμj∘+RT∑jνjln⁡(ajaj∘)\Delta G = \sum_{j} \nu_j \mu_j^{\circ} + R T \sum_{j} \nu_j \ln\left(\frac{a_j}{a_j^{\circ}}\right). Alternatively, using the standard thermodynamic definition: ΔG=ΔH−TΔS\Delta G = \Delta H - T\Delta S. The spontaneity criterion requires ΔG<0\Delta G < 0 for a reaction to proceed spontaneously under the given conditions.
Let τ\tau be the average crystallite size (in nm), λ\lambda be the X-ray wavelength, θ\theta be the Bragg angle (in radians), and β\beta be the peak broadening (in radians). The Scherrer equation provides an estimate for τ\tau based on the peak width β\beta via the relationship:\nτ≈Kλβcos⁡θ\tau \approx \frac{K\lambda}{\beta \cos\theta}
Consider a system at equilibrium, defined by the reaction quotient QQ and the equilibrium constant KK: Q=KQ = K. If a change in external conditions (concentration CC, pressure PP, or temperature TT) is applied, the system shifts to restore equilibrium. For a change in concentration, let ΔC\Delta C be the change in concentration of a species. The system adjusts such that the rate of reaction rr changes to maintain the equilibrium condition: ddt(Q−K)=0\frac{d}{dt} (Q - K) = 0. For temperature changes, the dependence of KK on TT is governed by the van't Hoff equation: dlnKd(1/T)=−ΔH∘R\frac{d\text{ln}K}{d(1/T)} = -\frac{\Delta H^{\circ}}{R}.
Define a crystal lattice Λ\Lambda in dd-dimensional space as the set of all points r\mathbf{r} generated by integer linear combinations of dd linearly independent basis vectors a1,…,ad\mathbf{a}_1, \dots, \mathbf{a}_d: Λ={∑i=1dniai∣ni∈Z}\Lambda = \left\{ \sum_{i=1}^{d} n_i \mathbf{a}_i \mid n_i \in \mathbb{Z} \right\}. The periodicity is defined by the lattice vectors R=∑i=1dniai\mathbf{R} = \sum_{i=1}^{d} n_i \mathbf{a}_i. The reciprocal lattice Λ∗\Lambda^* is defined by the basis vectors bi\mathbf{b}_i such that ai⋅bj=2πδij\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi \delta_{ij}. The structure factor F(k)F(\mathbf{k}) for a unit cell containing atoms at positions rj\mathbf{r}_j is given by: ∑jfjeik⋅rj\sum_{j} f_j e^{i \mathbf{k} \cdot \mathbf{r}_j}.
Principle

Vegard's Law

Consider a solid solution AxB1−x\mathrm{A}_x\mathrm{B}_{1-x} where xx is the atomic fraction of element A. If the solid solution exhibits ideal mixing and follows Vegard's Law, the lattice parameter a(x)a(x) is a linear interpolation of the pure component lattice parameters aAa_A and aBa_B:\na(x)=xaA+(1−x)aBa(x) = x a_A + (1-x) a_B
For a multi-component system at constant pressure PP, the stable phases ϕ\phi are determined by minimizing the total Gibbs Free Energy GtotalG_{total} with respect to temperature TT and composition xx. The phase boundaries are defined by the condition that the chemical potential μi\mu_i of each component ii must be equal in all coexisting phases ϕj\phi_j and ϕk\phi_k:\nμi(ϕj)=μi(ϕk)for all i∈components, and ϕj,ϕk coexisting\mu_i(\phi_j) = \mu_i(\phi_k) \quad \text{for all } i \in \text{components}, \text{ and } \phi_j, \phi_k \text{ coexisting}