Chemical Thermodynamics of Molecular Aggregates (Lecture Notes)

Overview

  • Topic: Chemical thermodynamics of molecular aggregates, focusing on how aggregation, solvation, partitioning, and transport are governed by thermodynamic and statistical-mechanical principles.
  • Key idea: Solvation free energy and chemical potential balance determine partitioning between regions, phases, or regions within a phase; this balance controls dissolution, recognition, sorption, and reactions.
  • Scales and examples: Aggregates range from soft, molecular-scale assemblies to interfaces (micelles, lipid membranes, protein/polymer assemblies) up to nano- and micron-scale structures; functions depend on order and fluctuations from molecular to µm scales.
  • Framework: Solvation as a unified framework to analyze partitioning, transport, and aggregation via free energy and transport coefficients; MD simulations and statistical-mechanical theory underpin the approach.

Functions of Molecular Aggregates and Solvation

  • Aggregates enable partitioning and transport of molecules by forming environments with distinct solvation properties.
  • Examples of relevant assemblies: micelles, lipid membranes, protein and polymer aggregates, interfaces, and supercritical fluids/ionic liquids.
  • Core framework components:
    • Intermolecular interaction and mode of aggregation
    • Solvation as a framework to analyze partitioning
    • Quantification via free energy (partitioning) and transport coefficients (diffusion, conduction, viscosity, heat transport)
  • Unified principle: A single framework connects dissolution/partitioning, recognition, sorption, and transport through thermodynamics and statistical mechanics.

Disciplines and Generality

  • Physical chemistry and thermodynamics underpin diverse fields:
    • Inorganic chemistry: inorganic compounds
    • Organic chemistry: organic compounds
    • Biochemistry: bio-related molecules
    • Physical chemistry: collections/aggregates of atoms and molecules, essential for engineering problems
  • Key characteristics of thermodynamics for aggregates:
    • High generality; abstract and mathematical formulation
    • Focus on macroscopic properties: dissolution, mixing, phase behavior, etc.

Boundary Conditions and Mechanical Equilibrium

  • Pressure-driven boundary movement (two-phase boundary):
    • If p1 > p2, wall moves to the right; if p1 < p2, wall moves to the left; if p1 = p2, wall is stationary.
    • Mechanical equilibrium occurs when the boundary does not move and no work is exchanged between regions.
  • Equations: notated as p1, p2 with wall position controlled by pressure differences; at equilibrium, the boundary is stationary and work between regions ceases.

Temperature, Heat, and Thermal Equilibrium

  • Zeroth law: If energy flows from A to B and B to C, energy flows from A to C; defines a transitive notion of thermal contact and temperature.
  • Temperature T describes hotness/coldness and the direction of thermal energy flow.
  • Concepts:
    • Heat transfer allowed: dq (positive when energy enters the system)
    • Adiabatic: no flow of thermal energy
  • States:
    • T1 > T2 implies energy flows from T1 region to T2 region across a boundary; T1 = T2 implies thermal equilibrium with no net flow.
  • Notation: general temperature fields T, T1, T2 used to describe around-the-boundary energy exchange.

System, Surroundings, and First Law

  • System vs surroundings: system is the body of matter of interest; surroundings are everything else.
  • Basic energy accounting:
    • dq: heat transferred from surroundings to system
    • dw: work done on/by the surroundings to the system
    • dU = dq + dw: change in internal energy; U is a state function (depends only on state, not path).
    • Heat and work are path functions; only the end states determine dU.
  • Sign conventions (system viewpoint):
    • dq positive when heat is added to the system
    • dw positive when work is done on the system
  • Isolated system: dq = 0, dw = 0; U constant.

Reversible Work and Equilibrium Pathways

  • Work of the surroundings on the system in a reversible process:
    • dwrev = - pex dV, where p_ex is the external (surroundings) pressure.
    • In a reversible process, p_ex = p (system pressure).
  • For a reversible process: dw_rev = - p dV.
  • Equilibrium and reversibility:
    • Equilibrium: same T and p between system and surroundings.
    • Reversible process: process kept at equilibrium (quasi-static).

Spontaneity, Path-Dependence, and Heat Exchange

  • The direction of spontaneous change is tied to the path chosen, but state functions (U, S, V) determine end states.
  • Path dependence:
    • Work dw depends on the path; dq is an inexact differential (path-dependent heat transfer).
    • Q(final) − Q(initial) equals the integral of dq along the chosen path; different paths give different dq even between the same end states.
  • Spontaneity in terms of heat is best described using state functions like S and T; heat alone is not a state function.

Entropy and the Second Law

  • Entropy S is introduced via reversible paths; dS is a state function, independent of the reversible path chosen.
  • Second law: dS ≥ 0 for processes without external constraints; in differential form with heat transfer along a reversible path: dS = dq_rev / T.
  • Third law sets the zero of the entropy scale.
  • Summary: Entropy provides a practical, state-function description for changes in heat and irreversibility.

Helmholtz and Gibbs Free Energies

  • Helmholtz free energy: A = U − T S
    • A is a state function: A = A(T, V)
    • Spontaneity at constant volume: dA ≤ 0; equilibrium at constant T and V: dA = 0
    • Thermodynamic identity: dU = TdS − p dV; and dA = − S dT − p dV
  • Gibbs free energy: G = U − T S + p V
    • G is a state function: G = G(T, p)
    • Spontaneity at constant T and p: dG ≤ 0; equilibrium: dG = 0
    • Differential forms: dG = − S dT + V d p
  • Remarks:
    • G and A are convenient potentials for constrained conditions (constant T,p) or (constant T,V), respectively.
    • These potentials quantify the maximum useful work (non-PV work) obtainable under the given constraints.

Practical Inequalities and Maximum Work

  • Constant temperature and volume (T, V): maximum non-PV work is associated with changes in Helmholtz free energy A; the maximum obtainable work under these constraints is related to ΔA (sign conventions depend on whether work is extracted or supplied).
  • Constant temperature and pressure (T, p): maximum non-PV work is associated with changes in Gibbs free energy G; the magnitude of ΔG bounds the maximum useful work under these constraints.
  • General form (conceptual):
    • Under appropriate constraints, the maximum non-PV work W_max is related to the decrease in the corresponding thermodynamic potential (A for fixed V, G for fixed p).

State Functions and Partial Derivatives in U, A, G

  • Fundamental thermodynamic relation with U(S, V):
    • dU = T dS − p dV
    • T = ∂U/∂S|_V
    • p = − ∂U/∂V|_S
  • Helmholtz free energy A(S, V) or A(T, V):
    • dA = − S dT − p dV
    • S = − ∂A/∂T|_V
    • p = − ∂A/∂V|_T
  • Gibbs free energy G(T, p):
    • dG = − S dT + V d p
    • S = − ∂G/∂T|_p
    • V = ∂G/∂p|_T

Temperature, Pressure, and Chemical Potential

  • Chemical potential µ is the partial molar Gibbs free energy:
    • For multiple species, total G = G(T, p, nA, nB, …)
    • dG = − S dT + V d p + ∑ μi d ni
  • At fixed T and p, mass transfer between phases yields: dG = − μ1 d n + μ2 d n; at equilibrium dG = 0 ⇒ μ1 = μ2
  • Concept of partitioning and chemical potential:
    • Distribution of species between phases or regions is governed by equality of chemical potentials: µ(in phase A) = µ(in phase B)

Concentration Dependence and Ideal/Excess Terms

  • Chemical potential dependence on concentration c:
    • µ = µ^iso(c) + µ^ex(c)
    • Ideal term: µ^iso ≈ RT log(c/w) (with standard state w)
    • Real (excess) term: µ^ex depends on concentration and interactions
  • Common form (ideal + excess):
    • µ = RT log(c/w) + µ^{ex}
  • Activity and standard state:
    • Activity a = γ c / w, where γ is an activity coefficient and w is the standard concentration
    • At infinite dilution (c → 0), γ → 1, a ≈ c / w

Partitioning and Chemical Equilibrium

  • Partitioning between two phases (A and B):
    • At equilibrium: µi(in phase A) = µi(in phase B) for each species i
    • Partition coefficient K relates concentrations c(A) and c(B) via K = ∏ c(i)^{νi(product)}/∏ c(i)^{νi(reactant)} with stoichiometric coefficients ν_i
  • Decomposition of µ into ideal and excess terms (for partitioning of a single species):
    • µ(in region A) = µ(in region B) with same standard states for A and B leads to a partitioning description controlled by µ^ex terms (solvation effects)
  • Partition constant and solvation:
    • K ∝ exp[−(µ^ex(B) − µ^ex(A)) / RT]

Solvation Free Energy and Solubility

  • Solvation free energy Dµ (transfer free energy from vacuum):
    • Dµ = − R T
      abla \,?
    • In simple form: Dµ = − RT \, \log S, where S is solubility relative to vacuum
    • Higher solubility S corresponds to more negative Dµ (more favorable solvation)
  • Solubility S is the reference measure for how easily a solute dissolves in a solvent.
  • Relationship: Solvation effect reflects how surroundings alter the energy of a solute.
  • In many contexts: Solvent shifts the reaction equilibrium by altering solubilities S of reactants/products, thereby modifying K.
  • Basic equation for solvation-driven solubility influence on equilibrium:
    • K = K_0 \, \frac{S(B) S(C)}{S(A)}
    • For a reaction A → B + C in solution, solvent can shift equilibrium toward species with higher solubility in the solvent.

Solvation and Structure Determination

  • All-atom consideration: structural determination and solvation contribute to stability of biomolecules and complexes.
  • Example: Structure determination of protein complexes using all-atom energy Eintra plus solvation contribution Dµ:
    • Eintra + Dµ determines relative stability and RMSD to crystal structure.

Hydration, Solvent Effects, and Density Dependence

  • Hydration effects in water can be tuned by density and temperature, including supercritical regimes:
    • Room temperature water density ~1 g/cm^3; supercritical water exists above 374 °C and at densities from ~0.2 to ~0.5 g/cm^3.
    • Hydration number can be designable via solvent conditions.
  • Solvation free energy Dµ changes with solvent density and temperature; at higher density/temperature, Dµ can vary by several kcal/mol, comparable to electronic effects, enabling tuning of equilibrium and reaction pathways by solvent conditions.
  • Data context (illustrative): Dµ/RT curves vs solvent density (ρ) and temperature illustrate solvent-strength changes.

C1 Chemistry in Hot Water and Hydration-Controlled Pathways

  • In hot water, many reaction pathways are accessible with modest catalysts, enabling C1 chemistry evolution toward C2 compounds.
  • Key reactions include:
    • CH3OH + HCOOH → HCHO + HOCH2COOH (glycolic acid) via acid catalytic pathways
    • Dehydration, hydration, decarbonylation, hydration-driven decarboxylation, and related equilibria
  • Hydration shifts reaction selectivity:
    • In hot water, hydration environment can favor CO generation, H2 production, and water-gas shift behavior where H2 generation is enhanced under certain hydration conditions.
  • Example representation: as solvent conditions change, the relative weights of reaction paths (e.g., CO path vs CO2 path) shift, enabling selective product formation.

Hydration, Solvation, and Solubility in Non-Ideal Systems

  • Ideal vs real solutions:
    • Ideal: a ≈ c / w, with µ ≈ RT log(c/w) + constant
    • Real: µ = RT log(a/w) + µ^{ex}
  • Activity and standard state concepts are extended to non-ideal solutions to accommodate interactions and finite concentrations.
  • For polymers and solvents, solvation free energy and activity coefficients govern dissolution and partitioning behavior.

Protein Aggregation and Environment Effects

  • Protein aggregation is influenced by solvent environment:
    • Amyloids (e.g., in neurodegenerative diseases) vs random inclusion bodies
    • Solvent conditions and temperature/pressure can modulate aggregation propensity and stability
  • Engineering aggregation: tuning protein–solvent interactions via temperature, pressure, and cosolvents can promote or inhibit aggregation for desired outcomes (e.g., expression, crystallization).

All-Atom Analysis of Peptide Aggregates

  • Example: NACore segment of α-synuclein (11 residues, 148 atoms) used to study aggregation tendencies.
  • Observables:
    • Intra-solute energy Eintra and solvation free energy Dµ, combined as Eintra + Dµ to describe aggregate stability.
    • Aggregation states: monomer, 8-mer, 16-mer, 24-mer, with varying energetics per monomer.
  • Cosolvent effects: cosolvents like urea and DMSO stabilize or destabilize aggregates by altering Dµ per monomer and per aggregate.
  • Key result: Dµ per monomer and aggregation energy per monomer depend on aggregation number n, with cosolvents generally reducing aggregation tendency (increasing dissolution).

Cosolvent Effects on Aggregation

  • Cosolvents (e.g., urea, DMSO) stabilize solutes and can inhibit aggregation, with stronger effects at lower aggregation numbers (smaller n).
  • Quantitative measure: Dµ/n (per monomer) and ES (intrinsic intra-solute energy) vs ES + Dµ per monomer show cosolvent-induced changes.

High-Performance Computing and Modern Resources

  • Fugaku supercomputer project (Japan): development from K to Fugaku (K = 10^16 operations, Fugaku today ~10^4+ Pflops peak per node)
  • Purpose: support a wide range of science and engineering with strong connection to industry and societal needs.

Societal and Sustainability Context

  • SDGs (Sustainable Development Goals) provide a framework for aligning scientific advances with social and environmental outcomes, including energy efficiency, clean energy, water treatment, and sustainable industry.
  • Relevance to chemical thermodynamics: design of separations, membranes, and materials to reduce energy consumption and enable sustainable processes.

Separation Membranes and Polymers

  • Separation membranes rely on polymer media; energy savings arise from membrane-based separations vs traditional distillation.
    • In Japan, roughly one-sixth of total energy use is for energy; about 80% of that is for distillation, so membrane separations can cut energy use by ~30% or more.
  • Principles for polymer-based separation:
    • Diffusion/ Mobility: diffusion coefficient (D) governs transport through the membrane.
    • Partition coefficient (K): affinity of solute to the membrane.
    • Permeability (P) is given by the product K × D; thus, the key parameter is K, which is governed by solvation free energy Dµ.
    • Design goal: predict and tailor Dµ to optimize K and P for target separations.

All-Atom MD and Polymer Solvation

  • All-atom molecular dynamics (MD) provides molecular-level insight into solvation in polymers:
    • Focus on solvation free energy of water Dµ in polymer environments to understand hydrophilicity/hydrophobicity.
    • Key atomic-level contributions include hydrogen bonding, excluded-volume effects, polymer chain flexibility, and local amorphous structure.
  • MD studies demonstrate correlation between computed Dµ and experimental water solubility and partitioning in various polymers (e.g., PE, PP, PPS, PVDF, PMMA, PET, Nylon6, etc.).
  • Example results show strong correlation (e.g., correlation coefficient ~0.98) between computed and experimental ΔG values for water in different polymers.

Polymer Structures and Water Dissolution

  • Copolymer composition strongly influences Dµ for water dissolution:
    • Copolymers: periodic (block) and graft copolymers with varying l:m ratios control the dissolved water environment.
    • The overall composition l:m largely determines Dµ; topology (periodic vs graft) has limited impact on water dissolution.
  • Structural features influencing Dµ:
    • Degree of fragmentation of hydrophilic/hydrophobic moieties
    • Composition balance and architecture (block vs graft) affects dissolution energetics more than topology alone.
  • Examples of copolymer structures studied: ethylene vinylidene difluoride, vinyl acetate, acrylamide copolymers, with varying l:m and n (degree of polymerization).

Solvation Theory: From Thermodynamics to Statistical Mechanics

  • Core idea: A theoretical framework to analyze molecular aggregates (solutions, micelles, membranes, polymers, proteins) using solvation free energy and statistical mechanics.
  • Key concepts: solvophilic vs solvophobic tendencies, the balance of enthalpic and entropic contributions to solvation, and how solvent conditions tune aggregation and partitioning.
  • Outcome: A practical bridge from macroscopic thermodynamics to molecular-level structures and interactions, enabling prediction and design in engineering and medicine.

Mathematical Highlights (Key Equations)

  • First Law: dU = dq + dw
  • Reversible work: dw{ ext{rev}} = -p{ ext{ex}} \, dV
  • In reversible processes, p{ ext{ex}} = p and dw{ ext{rev}} = -p \, dV
  • Helmholtz free energy: A = U - T S
    • State function: A = A(T, V)
    • Spontaneity at constant volume: dA \,\le\, 0
    • Differential: dA = -S \, dT - p \, dV
  • Gibbs free energy: G = U - T S + p V
    • State function: G = G(T, p)
    • Spontaneity at constant temperature and pressure: dG \le 0
    • Differential: dG = -S \, dT + V \, dp
  • Fundamental relations (U as function of S,V):
    • dU = T \, dS - p \, dV
    • T = \left(\frac{\partial U}{\partial S}\right){V}, p = -\left(\frac{\partial U}{\partial V}\right){S}
  • Helmholtz differential (S,V):
    • dA = -S \, dT - p \, dV
  • Gibbs differential (T,p):
    • dG = -S \, dT + V \, dp
  • Chemical potential (partial molar Gibbs free energy):
    • For a reaction or transfer with species i: dG = -S \, dT + V \, dp + \sumi \mui \, d n_i
  • Equilibrium condition for partitioning: at equilibrium, for each species i, \mui^{(A)} = \mui^{(B)}
  • Solvation free energy (transfer from vacuum):D\mu = - R T \ln S
  • Overall chemical potential with ideal and excess terms:\mu = \mu^{\text{iso}} + \mu^{\text{ex}} = RT \ln\left(\frac{c}{w}\right) + \mu^{\text{ex}}\n- Activity: a = \gamma \; c / w\quad\text{and}\quad g = a / c
  • Partitioning coefficient in terms of solvation: K = K_0 \; \frac{S(B) \; S(C)}{S(A)}
  • Relation between solvation and structure in polymers (conceptual): P \propto K \times D$$ where K is partition coefficient and D is diffusion coefficient; Dµ governs K via solvation energetics.

Examples and Practical Implications

  • Hydration control in hot water enables selection of reaction pathways and generation of hydrogen via water–gas shift and related reactions.
  • Solvation tuning via density and temperature can significantly alter Dµ (and thus solubility and reaction equilibria) by several kcal/mol, enabling reversible control of aggregation, dissolution, and product distributions.
  • In separation technology, understanding Dµ and K in polymer membranes enables rational design of materials with targeted permeability (P = K D) to reduce energy consumption in industrial separations.

Connection to Real-World Context

  • Sustainability and industry:
    • Membrane separations can drastically reduce energy usage compared with distillation, contributing to energy efficiency and SDGs.
    • High-throughput MD and solvation energetics guide the design of polymers and mixed solvents to optimize separation performance.
    • Understanding solvation and aggregation enables better control of protein expression, drug formulation, and materials design.

Quick Recap of Core Concepts

  • Aggregates and solvation are governed by a balance of enthalpy and entropy, captured by chemical potentials and thermodynamic potentials (A, G).
  • Equilibrium between phases or regions is achieved when chemical potentials align across phases: µ’s are equal.
  • Solvation free energy Dµ encodes solvent effects and determines partitioning, dissolution, and aggregation tendencies; it is central to predicting and tuning reactions and transport.
  • The framework connects microscopic interactions (through MD) with macroscopic observables (partitioning, solubility, diffusion, permeability).
  • Practical design leverages the tunability of solvent density, temperature, cosolvents, and polymer composition to control aggregation, dissolution, and reaction pathways for engineering applications.