bridge derivation: atoms to solid state material properties

[bhpayne]

prompt for Gemini 2.0 Flash Thinking Experimental on https://aistudio.google.com/prompts/new_chat

Atoms are comprised of electrons, protons, and neutrons.

Material properties at macroscopic scale include latent heat, stiffness, hardness, and opacity.

In mathematical Physics a derivation is comprised of steps and equations. Is there a derivation that bridges the physics of atoms with the material properties observed at macroscopic scales? If yes, provide the steps of the derivation. Format the answer using Latex as a single file.

Latex is a software system for typesetting documents. LaTeX is widely used for the communication and publication of scientific documents and technical note-taking in many fields, owing partially to its support for complex mathematical notation.

Do not have equations inline with the text. Equations should be separate and inside a math environment for Latex:

\begin{equation}
\end{equation}

Each equation should have a left-hand side and right-hand side. Specify mathematical equations explicitly. Define the variables used in equations.

thoughts

The user is asking if there is a derivation in mathematical physics that connects the microscopic world of atoms (electrons, protons, neutrons) to macroscopic material properties like latent heat, stiffness, hardness, and opacity. If such a derivation exists, the user wants to see the steps, formatted in LaTeX, with equations separated and variables defined.

This is a very broad question. Connecting the microscopic to the macroscopic is a central goal of condensed matter physics and materials science. A single, simple derivation spanning all these properties is unlikely to exist. Instead, there are many derivations and theoretical frameworks that link atomic properties to specific macroscopic properties.

Let's consider each macroscopic property and think about the link to atomic properties:

• Latent Heat: Latent heat is related to phase transitions (solid to liquid, liquid to gas). This is governed by interatomic forces (bonding) and statistical mechanics. A derivation would involve calculating the energy difference between phases based on atomic interactions and then relating this to macroscopic latent heat.

• Stiffness: Stiffness (e.g., Young's modulus) is related to the resistance of a material to deformation. This is directly tied to the interatomic potential and the arrangement of atoms in a solid (crystal structure). A derivation would likely involve considering the forces between atoms when they are displaced from their equilibrium positions and relating this to macroscopic stress and strain.

• Hardness: Hardness is related to the resistance to indentation or scratching. This is more complex than stiffness and involves plastic deformation and material failure at the atomic level. While related to interatomic forces and crystal structure, a direct "derivation" in the same sense as stiffness might be more challenging. Perhaps a more phenomenological approach based on bonding strength and defect formation would be more appropriate.

• Opacity: Opacity is related to how light interacts with a material. This depends on the electronic structure of atoms and their collective behavior in a material. A derivation would involve quantum mechanics and electromagnetism, considering how photons interact with electrons in the material and whether they are absorbed or transmitted.

Given the complexity and the fact that a single unifying derivation is unlikely, a more realistic and helpful approach would be to:

1. Acknowledge that a single, simple derivation for all properties is not feasible.
2. Choose one or two properties as examples and provide simplified derivations or outlines of how such derivations would proceed. Stiffness and perhaps latent heat seem like good candidates, as they have relatively straightforward connections to atomic properties.
3. For each chosen property:
   • Briefly explain the macroscopic property in terms of atomic behavior.
   • Outline the steps involved in a derivation, mentioning key concepts and equations.
   • Provide simplified equations if possible, defining variables. Recognize that rigorous derivations might be very complex and beyond the scope of a brief answer.
   • Emphasize the approximations and simplifications made.

Let's focus on Stiffness (Young's modulus) as a primary example, and perhaps briefly touch on Latent Heat.

Stiffness (Young's Modulus):

• Macroscopic Stiffness: Resistance to stretching or compression. Defined as stress/strain.

• Microscopic Origin: Interatomic forces. Atoms are held together by bonds (e.g., covalent, ionic, metallic). Stretching the material means stretching these bonds. The stronger the bonds and the more atoms per unit area, the stiffer the material.

• Simplified Derivation Outline:

  1. Model interatomic potential (e.g., Lennard-Jones, Morse). For simplicity, maybe even a harmonic potential near equilibrium.
  2. Relate interatomic potential to force between atoms.
  3. Consider a crystalline solid (e.g., simple cubic) for simplicity.
  4. Calculate the force required to displace atoms from their equilibrium positions.
  5. Relate this force to macroscopic stress.
  6. Relate atomic displacement to macroscopic strain.
  7. Young's modulus is then stress/strain, derived from atomic parameters.

Latent Heat (of Fusion):

• Macroscopic Latent Heat: Energy required to change phase (e.g., solid to liquid) at constant temperature.

• Microscopic Origin: Breaking of interatomic bonds. In a solid, atoms are in fixed positions due to strong bonds. In a liquid, bonds are weaker, allowing atoms to move more freely. Latent heat is the energy needed to break enough bonds to transition from solid to liquid.

• Simplified Derivation Outline:

  1. Model the energy of a solid and a liquid phase based on interatomic potentials.
  2. Calculate the difference in energy per atom between the solid and liquid phases.
  3. Relate this energy difference per atom to the macroscopic latent heat per mole or per unit mass.
  4. Statistical mechanics might be needed for a more rigorous treatment, considering entropy changes.

Opacity: This is more complex and requires quantum mechanics. Briefly mention that it arises from the interaction of photons with electrons and depends on electronic band structure, energy levels, and transition probabilities. No simple "derivation" in the same sense as stiffness.

Hardness: Acknowledge that hardness is complex and not directly derivable in the same way. It is related to bonding strength, but also to plastic deformation mechanisms at the atomic level, such as dislocation motion.

Structure of the LaTeX answer:

1. Introduction: Acknowledge the broad question, state that a single derivation for all properties is unlikely.
2. Focus on Stiffness (Young's Modulus):
   • Explain macroscopic stiffness and its atomic origin.
   • Provide a simplified derivation with steps and equations.
3. Briefly discuss Latent Heat:
   • Explain macroscopic latent heat and its atomic origin.
   • Outline a derivation approach (less detailed equations than stiffness).
4. Briefly discuss Opacity and Hardness: Explain their atomic origins but state that simple derivations are not readily available or are very complex.
5. Conclusion: Reiterate that connecting micro to macro is complex, and simplified models are often used.

Let's proceed with writing the LaTeX answer based on this thought process.

response

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}

\title{Bridging Atomic Physics and Macroscopic Material Properties}
\date{\today}

\begin{document}
\maketitle

\section{Introduction}

The question of how the microscopic world of atoms, comprised of electrons, protons, and neutrons, gives rise to the macroscopic properties of materials such as latent heat, stiffness, hardness, and opacity is a central theme in condensed matter physics and materials science.  While a single, universal derivation that elegantly bridges all these properties from first principles is not feasible, there are theoretical frameworks and derivations that connect atomic physics to specific macroscopic properties.  These derivations often involve approximations and simplifications, but they provide crucial insights into the microscopic origins of macroscopic phenomena.

This document will illustrate, using simplified examples, how one can begin to bridge the gap between atomic properties and macroscopic material behavior. We will focus primarily on the derivation of stiffness (specifically Young's modulus) from interatomic potentials, and briefly discuss how similar principles can be applied to other properties.

\section{Derivation of Stiffness (Young's Modulus) from Interatomic Potentials}

Stiffness, as quantified by Young's modulus ($E$), describes a material's resistance to elastic deformation under tensile or compressive stress. At the atomic level, stiffness arises from the interatomic forces that hold atoms together in a solid. When a material is stretched, the interatomic distances increase, and the restoring forces between atoms resist this change, leading to macroscopic stiffness.

Let's consider a simplified model of a solid where atoms are arranged in a regular lattice and interact through a pairwise potential.  We will focus on a one-dimensional chain of atoms for simplicity, but the principles extend to three-dimensional solids.

\subsection{Step 1: Interatomic Potential}

We assume that the interaction between two adjacent atoms can be described by a potential energy function $U(r)$, where $r$ is the distance between the atoms.  A common model potential is the Lennard-Jones potential, but for simplicity in this derivation, we can consider a general potential $U(r)$ that has a minimum at the equilibrium distance $r_0$.

The force $F(r)$ between two atoms is the negative derivative of the potential energy with respect to the distance:
\begin{equation}
F(r) = - \frac{dU(r)}{dr}
\end{equation}
At equilibrium distance $r_0$, the force is zero, i.e., $F(r_0) = 0$, and the potential energy is at a minimum.

\subsection{Step 2: Restoring Force for Small Displacements}

When we stretch or compress the material slightly, the interatomic distance deviates from the equilibrium distance $r_0$. Let's consider a small displacement $\delta r = r - r_0$. For small displacements, we can approximate the force using a Taylor expansion around $r_0$:
\begin{equation}
F(r) \approx F(r_0) + \left. \frac{dF}{dr} \right|_{r=r_0} (r - r_0) + \mathcal{O}((r-r_0)^2)
\end{equation}
Since $F(r_0) = 0$ and we consider only linear elasticity (small displacements), we can neglect higher-order terms. Also, $\frac{dF}{dr} = -\frac{d^2U}{dr^2}$. Let $k = \left. \frac{d^2U}{dr^2} \right|_{r=r_0}$ be the second derivative of the potential energy at the equilibrium distance. This $k$ represents the "spring constant" for the interatomic bond. Then, the restoring force for small displacements is approximately:
\begin{equation}
F(r) \approx -k (r - r_0) = -k \delta r
\end{equation}
Here, $k$ represents the stiffness of the interatomic bond.

\subsection{Step 3: Macroscopic Stress and Strain}

Consider a one-dimensional chain of atoms with equilibrium spacing $r_0$.  Let $A$ be the effective cross-sectional area per chain (in 3D, this would be the area per atom in a plane perpendicular to the tensile force). When we apply a macroscopic tensile force $\mathcal{F}$ to stretch the material, each interatomic bond experiences a force approximately equal to $\mathcal{F}$. The macroscopic stress $\sigma$ is the force per unit area:
\begin{equation}
\sigma = \frac{\mathcal{F}}{A}
\end{equation}
The macroscopic strain $\epsilon$ is the fractional change in length. If we consider a length $L_0 = N r_0$ consisting of $N$ atoms in equilibrium, and the length becomes $L = N r$ after stretching, the strain is:
\begin{equation}
\epsilon = \frac{L - L_0}{L_0} = \frac{N r - N r_0}{N r_0} = \frac{r - r_0}{r_0} = \frac{\delta r}{r_0}
\end{equation}

\subsection{Step 4: Young's Modulus}

From Hooke's Law, the macroscopic stress is related to the macroscopic strain by Young's modulus $E$:
\begin{equation}
\sigma = E \epsilon
\end{equation}
Substituting the expressions for stress and strain from atomic considerations, we have:
\begin{equation}
\frac{\mathcal{F}}{A} = E \frac{\delta r}{r_0}
\end{equation}
Also, from Step 2, we have $\mathcal{F} \approx k \delta r$ (magnitude of restoring force equals applied force in equilibrium). Substituting this into the previous equation:
\begin{equation}
\frac{k \delta r}{A} = E \frac{\delta r}{r_0}
\end{equation}
We can cancel out $\delta r$ (for small non-zero strain), and solve for Young's modulus $E$:
\begin{equation}
E = \frac{k r_0}{A} = \frac{r_0}{A} \left. \frac{d^2U(r)}{dr^2} \right|_{r=r_0}
\end{equation}
Thus, Young's modulus $E$ is related to the second derivative of the interatomic potential at the equilibrium distance, the equilibrium interatomic distance $r_0$, and the effective cross-sectional area per chain $A$.  For a given material and crystal structure, $r_0$ and $A$ are determined by atomic packing, and $k$ is determined by the strength and shape of the interatomic potential.

\subsection{Example: Lennard-Jones Potential}

For the Lennard-Jones potential:
\begin{equation}
U(r) = 4 \epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right]
\end{equation}
where $\epsilon$ is the depth of the potential well and $\sigma$ is the distance at which $U(r) = 0$. The equilibrium distance $r_0 = 2^{1/6} \sigma$. Calculating the second derivative and evaluating it at $r_0$ will give a specific expression for $k$ in terms of $\epsilon$ and $\sigma$, which can then be used to estimate Young's modulus.

\section{Brief Discussion of Other Material Properties}

\subsection{Latent Heat}

Latent heat is the energy absorbed or released during a phase transition at constant temperature. For example, the latent heat of fusion is the energy required to melt a solid into a liquid. At the atomic level, this energy is associated with changes in the interatomic bonding and potential energy.  In a solid, atoms are tightly bound in a relatively ordered structure. Melting involves breaking some of these bonds, allowing atoms to move more freely in the liquid phase. The latent heat can be related to the difference in potential energy between the solid and liquid phases, considering also changes in entropy.  A derivation might involve calculating the internal energy difference between phases based on statistical mechanics and interatomic potentials.

\subsection{Hardness}

Hardness is a material's resistance to permanent deformation, such as indentation or scratching.  It is a more complex property than stiffness and involves plastic deformation, which is related to the movement of defects (like dislocations) in the crystal lattice.  While interatomic bonding strength plays a role, hardness also depends on the microstructure, grain boundaries, and mechanisms of plastic deformation at the atomic scale. A direct derivation from first principles is more challenging compared to stiffness and often involves more phenomenological models and simulations of atomic-scale deformation processes.

\subsection{Opacity}

Opacity is a material's ability to block the transmission of light.  At the atomic level, opacity arises from the interaction of photons with the electrons in the material.  The electronic structure of atoms and how they are arranged in the material (electronic band structure in solids) determines how light is absorbed or transmitted.  For example, metals are opaque because their free electrons readily absorb photons across a wide range of frequencies. In contrast, transparent materials have a band gap such that visible light photons do not have enough energy to excite electrons across the gap, and thus light is transmitted.  Derivations of opacity involve quantum mechanics and electromagnetic theory, considering transitions between electronic energy levels and the absorption and scattering of photons by electrons.

\section{Conclusion}

Bridging the physics of atoms to macroscopic material properties is a complex endeavor.  While a single, simple derivation for all properties is not available, we can derive relationships for specific properties like stiffness by starting from interatomic potentials and using approximations.  For properties like latent heat, hardness, and opacity, the connections to atomic physics are more intricate and require different theoretical frameworks, including statistical mechanics, defect theory, and quantum mechanics.  The simplified derivation of Young's modulus presented here illustrates the general approach of connecting macroscopic behavior to microscopic interactions, highlighting the importance of interatomic potentials and atomic arrangements in determining material properties.

\end{document}

[bhpayne]

Results from this issue have been opened as new issues:

• bridge derivation: Stiffness (Young’s Modulus) from Interatomic Potentials #30
• bridge derivation: latent heat from atomic scale interactions #31
• bridge derivation: opacity from atomic-scale phenomena #32