Phase Diagrams and Phase Equilibria

Chapter 4: Phase Diagrams

• A component is an element or compound in a mixture.
• A system is a series of alloys made of the same components, like the Iron-Carbon system.
• A phase is a portion of the system with uniform physical/chemical traits.
• Solid solutions can be substitutional (solute atoms replace solvent atoms, e.g., Cu-Zn in brass) or interstitial (solute atoms fit between solvent atoms, e.g., C-Fe in steel).
• A phase diagram indicates phases at equilibrium for differing compositions/temperatures.

Solubility Limit

• Maximum concentration that allows a single-phase solution.
• Solubility limit varies with temperature.
• In a sugar/water system at 20°C, the solubility limit is 65wt% sugar.
  • Less than that creates syrup, while more results in syrup plus sugar.

Phase Equilibria

• Equilibrium occurs when a system's free energy (G) is at a minimum under specific conditions.
• At equilibrium, a system's macroscopic characteristics are stable over time.
• Phase equilibrium is defined by phase constitution: number of phases, phase compositions, and phase amounts.
• A metastable state is a non-equilibrium state that lasts indefinitely.

Binary Isomorphous Systems

• "Isomorphous" describes complete solubility of one component in another.
• An alpha phase field spans 0 to 100wt%.
• Examples: Cu-Ni and H2O-Sugar.
• A binary phase diagram includes two components.
• Temperature and composition are variables.
• Phase diagrams show phases as a function of T, C & P.
• Independent variables for binary systems: T & C, at P = 1 atm.
• The phase diagram is graphed as T vs. C.

Interpretation of Phase Diagrams

• Knowing T & C reveals present phases, the composition of each phase, and the weight percentage of each phase.
• A tie line connects phases in equilibrium and is also called an isotherm.
• Example: At T = 1300°C, $CL = 31 wt%$ Ni and $C\alpha = 43 wt%$ Ni.
• Formula for weight percentage of liquid phase: $W_L = \frac{S}{R + S}$
• Formula for weight percentage of alpha phase: $W\alpha = \frac{R}{R + S}$
• The Lever Rule uses the tie line as a lever, with the fulcrum at overall composition.
• Overall composition is calculated as: $C_o = W_LC_L + W\alpha C\alpha$

Development of Microstructure in Isomorphous Alloys

• Equilibrium cooling occurs with slow cooling.
• Non-equilibrium cooling happens with fast cooling, leading to cores with varied compositions.
• Cored structures and solidification result in initial solid forming with higher melting point element concentration and later solid forming has a lower concentration.

Binary Eutectic Systems

• "Eutectic" is derived from the Greek εύτηκτος, meaning "easily melted".
• A eutectic system consists of two components and exhibits a specific composition that has a minimum melting temperature.
• Example: Cu-Ag system,
• Alpha is a copper-rich solid solution.
• Beta is a silver-rich solid solution.
• Alpha and beta phases have solubility limit lines.
• Eutectic isothermal reaction is defined by：$L \xrightarrow{cooling} \alpha + \beta$
• Eutectic composition is represented by $C_E$.
• Eutectic temperature is represented as $T_E$.

Development of Microstructure in Eutectic Alloys

• With alloy composition at $C_E$, the alloy is in liquid phase at temperatures above the eutectic point.
• As the temperature cools to the eutectic temperature, the liquid transforms into alpha and beta solid phases, forming a lamellar eutectic structure.
• With alloy composition less than $C_E$, the alloy transitions to a mixed-phase region of liquid and alpha phase, where the alpha phase precipitates, and the remaining liquid solidifies.
• With Alloy Composition greater than $C_E$, the alloy forms primary beta crystals, then becomes surrounded by a eutectic structure of alpha and beta phases during solidification.
• Hypoeutectic is "left of eutectic".
• Hypereutectic is "right of eutectic".

Equilibrium Diagrams Having Intermediate Phases or Compounds

• An intermetallic compound is a chemical compound between two metals.
• Intermetallic compounds have definite stoichiometry, e.g., $Mg_2Pb$.
• The Mg-Pb system depicts a binary system with terminal solid solutions (alpha and beta).
• $Mg_2Pb$ is an intermediate phase.
• Also includes eutectic points and regions.

Eutectoid and Peritectic Reactions

• In a eutectoid reaction, one solid phase transforms into two other solid phases $S_1 \xrightarrow{cooling} S_2 + S_3$
• Example: In steel, $\gamma (0.76 wt% C) \xrightarrow{cooling} \alpha (0.022 wt% C) + Fe_3C (6.70 wt%C)$.
• In a peritectic reaction, solid phase + liquid phase becomes a new solid phase: $S_1 + L \xrightarrow{cooling} S_2$.

Congruent Phase Transformations

• Congruent transformation has no compositional change during phase transformation
  • This includes allotropy and melting of a pure material.
• Incongruent transformations have compositional changes during phase transformation.
  • These include transformations like Eutectic, Eutectoid and Peritectic.

Ceramic Phase Diagrams

• In the $Al_2O_3-Cr_2O_3$ system $Al_2O_3$ and $Cr_2O_3$ are completely soluble in each other in the solid state.
• In the $ZrO_2-CaO$ system there are various stable phases at different temperatures and compositions.
  • Including stabilization of cubic $ZrO_2$ phase.

The Iron-Carbon System

• The Iron-Carbon (Fe-C) phase diagram is vital in metallurgy.
• Used in steel and cast iron production.
• Alpha-ferrite (BCC) is stable at room temperature.
• Gamma-austenite (FCC) dissolves a considerable amount of carbon.
• Delta-ferrite (BCC) is stable at high temperatures.
• $Fe_3C$ (iron carbide or cementite) is a hard, brittle intermetallic compound.
• L stands for Liquid.
• In the eutectic reaction in the Fe-C system, $\gamma + Fe_3C$ is Ledeburite.
• In the eutectoid reaction in the Fe-C system, $\alpha + Fe_3C$ is Pearlite.
• Also defines $\gamma \xrightarrow{cool} \alpha + Fe_3C$ , defining steel versus cast iron variations
• Hypoeutectoid steel has a composition of $C_0 < 0.76wt% C$ and features proeutectoid alpha-ferrite grains along with pearlite.
• Hypereutectoid steel has a composition of $C_0 > 0.76wt% C$ and consists of proeutectoid cementite, which forms austenite grain boundaries with pearlite.

Example Problem

• Given a $99.6 wt% Fe-0.40 wt% C$ steel just below the eutectoid temperature:
  • The compositions are: $C\alpha = 0.022 wt% C$ and $CFe_3C = 6.70 wt% C$.
  • The amounts are: $W\alpha = \frac{6.70 - 0.40}{6.70 - 0.022} = 94.3wt%$ and $WFe_3C = \frac{0.40 - 0.022}{6.70 - 0.022} = 5.7wt%$.
  • The proeutectoid ferrite and pearlite amounts are: $W\alpha' = \frac{0.76 - 0.40}{0.76 - 0.022} = 48.8wt%$ and $W_P = \frac{0.40 - 0.022}{0.76 - 0.022} = 51.2wt%$.

Influence of Other Alloying Elements

• Alloying elements alter $T_E$, $C_E$ and also affect mechanical properties.

Chapter 5: Diffusion

• Diffusion is mass transport by atomic motion.
• Interdiffusion is when atoms migrate from high concentration regions in an alloy
• Self-diffusion occurs when atoms migrate in an elemental solid.

Lecture 10: October 26, 2023

• Content: Stability of feedback loops (gain and phase margins)
• Content Next week: Robustness and Loop Shaping.

Review of controllers design

• Developed controller design tools.
• Meets specs on steady-state error and transient response.
• Currently focuses on frequency-domain techniques to evaluate stability.
• Lacks controllers that are designed for a specific plant.
• Also need to consider the plants real-world variation.
• Need to understand robustness to plant variations.

Robustness

• Cope with uncertainty through stability margins.
• Addresses effects of model uncertainty on Stability.
• Uses strategy to deal with uncertainty in design.

Robust Stability

• Design involves the nominal system meeting performance specs.
• Checks for the system to remain stable to plant changes and variations.

Stability Margins

• Gain margin indicates how much the gain can increase before instability.
• Considering the system $\frac{Y}{R}=\frac{K G(s)}{1+K G(s)}$, closed loop is stable if the roots of $1 + KG(s) = 0$ are in the LHP, $KG(s)$ does not equal $-1$ and $G(s)$ does not equal $-\frac{1}{K}$.
• Plot $G(j\omega)$ (Nyquist Plot) on the complex plane to show the distance from the origin ($|G(j\omega)|$) and the angle from the real axis ($\angle G(j\omega)$).
• K is real if $-\frac{1}{K}$ is a point on the real axis.
• The system is stable if $G(j\omega)$ does not encircle the point $-\frac{1}{K}$.

Gain Margin

• Gain margin is the measure of how much the open-loop gain can increase before instability.
• With $\angle G(j\omega) = -180^{\circ}$ at frequency $\omega_0$ instability occurs if $K |G(j\omega_0)| > 1$.
• Defined as $G.M. = \frac{1}{|G(j\omega_0)|}$ or $G.M. = -20 \log_{10}(|G(j\omega_0)|)$ in db.

Gain Margin Example

• Given $G(s) = \frac{1}{s(s+1)(s+5)}$, $G(j\omega) = \frac{1}{j\omega(j\omega+1)(j\omega+5)}$, $|G(j\omega)| = \frac{1}{\omega\sqrt{\omega^2+1}\sqrt{\omega^2+25}}$ and $\angle G(j\omega) = -90^{\circ} - \tan^{-1}(\omega) - \tan^{-1}(\frac{\omega}{5})$
• Need to find $\omega_0$ where $\angle G(j\omega) = -180^{\circ}$ i.e. $\tan^{-1}(\omega) + \tan^{-1}(\frac{\omega}{5}) = 90^{\circ}$.
• $\frac{\omega + \frac{\omega}{5}}{1 - \omega \cdot \frac{\omega}{5}} = \tan(90^{\circ}) = \infty$, therefore, $1 - \frac{\omega^2}{5} = 0$ indicates $\omega = \sqrt{5}$.
• With instability at $\omega = \sqrt{5}$, $|G(j\omega)| = \frac{1}{\sqrt{5} \sqrt{6} \sqrt{30}} =.0274$ and $G.M. = \frac{1}{.0274} = 36.4$.
• G.M. in decibels is $G.M. = 20 \log_{10}(36.4) = 31.2 \mathrm{db}$.

Phase Margin

• Phase margin indicates how much the phase can be changed before instability.
• With $|G(j\omega)| = 1$ at frequency $\omega_1$, instability happens when $\angle KG(j\omega_1) = -180^{\circ}$.
• Phase margin equals $P.M. = 180 + \angle G(j\omega_1)$

Phase Margin Example

• Given $G(s) = \frac{1}{s(s+1)(s+5)}$, and need to find $\omega_1$ where $|G(j\omega_1)| = 1$ or $\omega_1\sqrt{\omega_1^2+1}\sqrt{\omega_1^2+25} = 1$.
• Solving numerically gives solution $\omega_1 =.195$.
• At $\omega =.195$, $\angle G(j\omega) = -90^{\circ} - \tan^{-1}(.195) - \tan^{-1}(\frac{.195}{5}) = -101.2^{\circ}$ and $P.M. = 180 - 101.2 = 78.8^{\circ}$.

Phase and Gain Margin Values

• $G.M. \geq 6\mathrm{db}$
• $P.M. \geq 30^{\circ}$
• Phase margin is a better indicator of robustness than gain margin.

Bode Plot

• Can determining the gain and phase margins of a transfer function from the Bode Plots
• Can find the frequency where the phase plot crosses -180 degrees.
• the gain margin is the difference between the magnitude at this frequency and 0 dB.
• Find the frequency where the magnitude plot crosses 0 dB.
  • the phase margin is the difference between the phase at this frequency and -180 degrees.

Matlab Commands

• The margin(G) command can be used to compute the gain margin, phase margin, and associated frequencies.

Matlab Example

• Evaluate the gain and phase margins (using Matlab) of $G(s) = \frac{1}{s^3 + s^2 + s + 1}$.
• The Nyquist stability criterion states that a closed-loop system is stable if and only if the Nyquist plot of the open-loop transfer function does not encircle the -1 point
• Is unstable in this example

Example Bode Plot

• Evaluating $G(s) = \frac{1}{s^2 + 0.2s + 1}$
• the phase drops rapidly near $\omega = 1$.
  • Small changes in the plant can cause instability.
• Gain Margin $= \infty \mathrm{db}$
• Phase Margin $= 11.4 \mathrm{deg}$ at $w = 0.904 \mathrm{rad} / \mathrm{sec}$

Lecture 10: Summary

• Stability margins measure robustness.
• Gain margin measures the gain possible before instability and reasonable values are $G.M. \geq 6 \mathrm{db}$.
• Phase margin measures phase change possible before instability and reasonable values are $P.M. \geq 30^{\circ}$.
• The phase margin is a better indicator of robustness than the gain margin.

Evidence for the Big Bang Theory

• 1 - Expansion of the Universe
  • Redshift of galaxies caused by Hubble's Law: $v = H_0D$ where $v$ represents recession velocity, $H_0$ is the Hubble constant, and $D$ is the distance.
• 2 - Cosmic Microwave Background (CMB)
  • The afterglow of the Big Bang with a temperature of 2.725 K, discovered by Penzias and Wilson in 1965.
• 3 - Abundance of Light Elements
  • Big Bang Nucleosynthesis (BBN) predicts the abundance of light elements H, He, and Li.

The Big Bang Theory Open Questions

• Cause of the Big Bang.
• Nature of dark matter and dark energy
• Fate of the Universe.

Universe Evolution Timeline

• Universe began at $t = 0$ with the Big Bang.
• Planck Era occurred $t = 10^{-43} s$ later.
• Inflation occurred at $t = 10^{-36} s$.
• Nucleosynthesis began $t = 3 min$ later.
• Recombination occurred $t = 380,000 years$ later.
• Formation of first stars began $t = 1 billion years$ later.
• Is in its current form $t = 13.8 billion years$ after origination.
• The prevailing cosmological model for the Universe describes the Universe as expanding from an extremely hot, dense state.