Partial Molar Volumes Research Articles

Hydrogen bonding in glassy trehalose-water system: Insights from density functional theory and molecular dynamics simulations.

We report a detailed density functional theory and molecular dynamics study of hydrogen bonding between trehalose and water, with a special emphasis on interactions in the amorphous solid state. For comparison, water-water interactions in water dimers and tetramers are evaluated using quantum calculations. The results show that the hydrogen bonding energy is dependent not only on the geometry (bond length and angle) but also on the local environment of the hydrogen bond. This is seen in quantum calculations of complexes in vacuum as well as in amorphous solid states with periodic boundary conditions. The temperature-induced glass transition in the trehalose-water system was studied using molecular dynamics simulations with varying cooling and heating rates. The obtained parameters of the glass transition are in good agreement with the experiments. Moreover, the dehydration of trehalose in the glassy state was investigated through a gradual dehydration with multiple small steps under isothermal conditions. From these simulations, the values of water sorption energy at different temperatures were obtained. The partial molar enthalpy of mixing of water value of -18 kJ/mol found in calorimetric experiments was accurately reproduced in these simulations. These findings are discussed in light of the hydrogen bonding data in the system. We conclude that the observed exothermic effect is due to different responses of liquid and glassy matrices to perturbations associated with the addition or removal of water molecules.

Volumetric-acoustic investigations of antihypertensive drug in aqueous acetaminophen system

Captopril is an antihypertensive drug used to lower the blood pressure in living organisms. In order to arbitrate and understand the synergy between antihypertensive drug viz., captopril and antipyretic drug acetaminophen in aqueous system, the density and sound velocity at different temperatures (293.15 to 318.15) K with temperature interval of 5 K were measured using Anton Paar density sound velocity meter (DSA5000M). The experimental measured data were further used to calculate volumetric and acoustic parameters such as apparent and partial molar volume, apparent and partial molar isentropic compressibility, partial molar expansibility, Hepler’s constant, compressibility hydration number, and intermolecular free length. Positive values of apparent molar volume (ϕv) and negative values of apparent and partial molar compressibility (ϕk and ϕko) of captopril in water and aqueous acetaminophen solution has been shown strong intermolecular interactions in solutions. Negative values of Hepler's constant indicate the structure breaking behavior of drug in water.

Local composition-regular solution theory for analysis of pharmaceutical solubility in mixed-solvents

Local composition regular solution theory (LC-RST) model was developed by introducing non-equal i–j and j–i parameters into regular solution theory (RST) to account for specific molecular interactions. The LC-RST model was applied to the correlation of solubilities of active pharmaceutical ingredients (API) in hydrogen bond donor (HBD) – hydrogen bond acceptor (HBA) mixed-solvent systems. In the assessment of 32 API-mixed-solvent systems that contained 11 different APIs and 1947 data, it was determined that the LC-RST model had a logarithmic average relative deviation (ARDln) in solubility of 0.028, the Wilson model had an ARDln value of 0.24 and the purely predictive RST model had an ARDln value of 1.29. Partial molar excess enthalpies and entropies of the API in the mixed-solvents determined from the LC-RST model allowed estimation of API solubility regions where specific interactions occur between API and HBD-HBA complex molecules. In the correlation scheme, component numbers were designated as API molecule (1) – HBD molecule (2) - HBA molecule (3) and systems were grouped according to HBD molecule interactions with the API (pro-solvent, anti-solvent, synergistic with HBA molecule). Phenomenological parameter sensitivity analysis of the LC-RST model showed that dominant specific interactions tended to be 1–2 or 2–1 molecules for pro-solvent groupings, 2–3 or 2–3 molecules for anti-solvent groupings and all i–j and j-–i molecular pairs for synergistic HBD-HBA molecule groupings. Based on parameters determined from a given API-mixed-solvent system, the LC-RST model allows qualitative prediction of solubilities for other systems. The LC-RST model is simple to apply and it retains inherent predictive characteristics of regular solution theory.

Calorimetric study of the enthalpy of mixing in the liquid systems Li-Zn and Li-Sn-Zn

Partial and integral molar enthalpies of mixing of binary liquid Li-Zn alloys and ternary liquid Li-Sn-Zn alloys were determined by drop calorimetry at 823 K. The binary system was investigated up to xZn = 0.85 and the integral mixing enthalpy was described with a Redlich-Kister-polynomial. Li-Zn shows an exothermic behavior and a minimum molar mixing enthalpy of about ΔHmix = -11.7 kJ⋅mol−1 at xZn = 0.6. The mixing enthalpy along nine ternary sections of Li-Sn-Zn (A: xLi/xSn ≈ 9:11, B: xLi/xSn ≈ 1:4, C: xSn/xZn ≈ 9:1, D: xSn/xZn ≈ 7:3, E: xSn/xZn ≈ 1:1, F: xLi/xZn ≈ 1:3, G: xLi/xZn ≈ 1:1, H: xLi/xZn ≈ 3:2, I: xLi/xZn ≈ 7:3) was investigated. Solid pieces of Li, Sn, and Zn respectively were dropped to the liquid binary starting alloys in the calorimeter at 823 K. The integral values at the crossing points of the various sections show excellent agreement. The course of the partial and integral enthalpy values along the sections A, D, E, F, G, H, and I indicate multiple-phase regions. The occurrence of such partially liquid and de-mixed liquid regions is discussed and compared with a calculated version of the isothermal section at 823 K which exhibits a large ternary liquid miscibility gap. The experimental ternary data allocated to the fully liquid monophasic regions was numerically fitted based on a Redlich-Kister-Muggianu polynomial for substitutional solutions including ternary interactions. In addition, a comparison with enthalpy values from the extrapolation methods based on binary data according to the Muggianu and Toop model, respectively, is shown. Both models fail to accurately reproduce the experimental values. Iso-enthalpy plots of calculated integral mixing enthalpy for stable and metastable fully homogeneous liquid alloys for the entire ternary system are shown.

Investigation of the shapes and sizes of soft molecular aggregates in solution via partial molar volume and dynamic light scattering measurements

Soft molecular aggregates are naturally dynamic and deformable nanoscale objects of various shapes and sizes. However, soft nanoparticle shape can be difficult to measure and often requires advanced instrumentation or methods. Yet, foreknowledge of soft aggregate overall shape is often critically important in many experiments that measure the aggregate’s internal structure and properties. Small-angle neutron and x-ray scattering are often used to measure soft nanoparticle shape. Molecular dynamics simulations can also give an indication of soft molecular shape but, like small-angle scattering methods, requires considerable expense and training. To complement these approaches a more straightforward experimental method of determining soft molecular aggregate shape using a combination of partial molar volume and dynamic light scattering measurements (PMV-DLS) is presented. PMV-DLS is inexpensive, requires modest instrumentation, and can be performed in any laboratory. In this article, the PMV-DLS technique is demonstrated on two separate soft molecular aggregates: AOT reverse micelles and Triton X-100 micelles. The shapes of these aggregates were investigated over a range of solution properties. In both cases, the penetration of continuous phase solvent molecules into the aggregate was found to play an important role in shape determination. AOT reverse micelles were observed to transition from ellipsoidal to spherical as the solution water content increased. Triton X-100 micelle shape was found to be critically dependent upon the amount of water that partitions into the surfactant polyethylene oxide head group region.

Investigations on the thermo physical properties of the binary mixtures of 2-methyl-pentanediol-2,4 with n-alkanols at various temperatures (hexylene glycol with 1-octanol, 1-nonanol, 1-decanol)

ABSTRACT Densities (ρ), speeds of sound (u) and viscosities (η) for three binary mixtures of hexylene glycol (2-methyl-2,4-pentanediol or MPD) with 1-octanol (D1), 1-nonanol (D2) and 1-decanol (D3) are calculated at temperatures ranging from 303.15 to 313.15 K and atmospheric pressure over the entire composition range. Experimental measurements are used to calculate the excess molar volume, excess isentropic compressibility, deviation in viscosity and excess Gibbs-free energy of activation of viscous flow. Partial molar properties and infinite dilution molar partial properties are also calculated for each binary system. The results obtained can be discussed in terms of weaker interactions and dispersion forces with the absence of specific intermolecular interactions in binary mixtures. Finally, the Prigogine-Flory-Patterson (PFP) theory is applied to identify the most predominant molecular interaction. Jouyban-Acree model results are discussed in terms of mean relative deviation (MRDs) and individual relative deviation (IRD) between calculated and experimental densities, speeds of sound and viscosities as an accuracy criterion.

Effect of the charge rate on the mechanical response of composite graphite electrodes: in situ experiment and mathematical analysis.

The cycling lifespan and coulombic efficiency of lithium-ion batteries are crucial to high C-rate applications. The Li-ion concentration is crucial in determining the mechanical integrity and structural stability of electrodes. In this work, graphite is selected as the working electrode due to its widespread use in the electric vehicle industry. The experimental data have shown that the electrodes with a mass loading of 6.54 mg cm-2 exhibited poor cycling performance and high charge transfer resistance at high charge rates. To explain this phenomenon, an in situ stress measurement system and a C-rate-dependent stress model are established to study the mechanical properties of the composite graphite electrode during the electrochemical process at various C-rates. Moreover, the effect of the Li-ion concentration-dependent modulus and C-rate-dependent partial molar volume is taken into account in the mathematical model. The computational curvature data fit well with the corresponding experimental data, highlighting the importance of considering lithium-ion concentration in mechanical stress. It has been found that stresses along the thickness of the active layer switch between compressive and tensile stresses due to the competition between bending stress and diffusion-induced stress. The stress at the outer surface of the composite graphite electrodes reaches a maximum magnitude of 27.5 MPa at a 1.5C-rate. In contrast, the stress at the interface of the active layer is maximum at a 0.5C-rate due to the existence of more lithium ions. Our study provides a direct insight into the quantitative analysis of electrode stresses at different C-rates.

Thermophysical investigation of glycol ethers in mannitol solutions at various temperatures

Ultrasonic analysis can be very helpful to comprehend the molecular dynamics and interactions in liquid systems. Employing the Anton-Paar (DSA 5000 M) at concentrations and 0.1 MPa, sound speed as well as density of glycol ethers, i.e., phenoxyethanol (PE) and butoxyethanol (BE) in solutions of a well-known sugar alcohol (D-mannitol) were measured at 288.15-303.15 K. A variety of advanced acoustic-thermodynamic parameters, including apparent molar parameters, partial molar parameters and transfer molar properties, were estimated using the experimentally attained the velocity and density values. These derived values are used to express the interactions between solutes and their solvents. The propensity of the solute to generate or destroy structures in a solvent is also a subject of research. Analysis was done on the interactions between the molecules in the ternary mixture of D-mannitol and glycol ethers in aqueous medium.

Enthalpies and entropies of hydration from Monte Carlo simulations.

The changes in free energy, enthalpy, and entropy for transfer of a solute from the gas phase into solution are the fundamental thermodynamic quantities that characterize the solvation process. Owing to the development of methods based on free-energy perturbation theory, computation of free energies of solvation has become routine in conjunction with Monte Carlo (MC) statistical mechanics and molecular dynamics (MD) simulations. Computation of the enthalpy change and by inference the entropy change is more challenging. Two methods are considered in this work corresponding to direct averaging for the solvent and solution and to computing the temperature derivative of the free energy in the van't Hoff approach. The application is for neutral organic solutes in TIP4P water using long MC simulations to improve precision. Definitive results are also provided for pure TIP4P water. While the uncertainty in computed free energies of hydration is ca. 0.05 kcal mol-1, it is ca. 0.4 kcal mol-1 for the enthalpy changes from either van't Hoff plots or the direct method with sampling for 5 billion MC configurations. Partial molar volumes of hydration are also computed by the direct method; they agree well with experimental data with an average deviation of 3 cm3 mol-1. In addition, the results permit breakdown of the errors in the free energy changes from the OPLS-AA force field into their enthalpic and entropic components. The excess hydrophobicity of organic solutes is enthalpic in origin.

Density and volume properties of MF-K2ZrF6 (M = Li, Na, and K)

Critical evaluation of volume properties (based on density measurements using Archimedean method) of the zirconium-based fluorides melts MF-K2ZrF6 (M = Li, Na, and K) revealed the different properties of KF system in contrast to LiF and NaF systems. Excess molar volume adopts negative values in the entire concentration range for all three investigated systems. Partial molar volumes of K2ZrF6 were obtained by using two different approaches - simple polynomial regression analysis and multicomponent polynomial regression in the form of the Redlich-Kister equation. Finally, the excess partial molar volume of K2ZrF6 in KF shows smaller overall volume contraction that adopts 60% of volume contraction in LiF or NaF systems. The obtained data of density, molar volumes and partial molar volumes were compared with accessible literate data after recalculation to the MF-ZrF4 coordinates.

An innovative method for calculating partial molar volume from a standard molecular trajectory

In this research, we present a novel method for calculating the partial molar volume (PMV) of system components through post-processing the output trajectory of a molecular dynamic simulation. The method involves employing linear regression and sampling small volumes from the system during the simulation time, to obtaining the PMV of its components. To validate our approach, simulations were conducted on two systems previously studied in the Journal of Molecular Liquids (386, 2023, 122498). A comparison between our results and experimental values revealed average relative deviations as minimal as 2.49 %, and specific sampling methods achieved even lower relative deviations, down to 1.21 % and 1.18 %. The average relative deviation across all components for the two systems was calculated at 7.24 %, translating to average deviations of 6.19 cm^3/mol for mesitylene and 7.80 cm^3/mol for isopropanol. To demonstrate the method's implications, we conducted simulations on a series of systems comprising model asphaltene, an organic solvent, and a chemical inhibitor. By analyzing the PMV results for the precipitant phase (asphaltene and inhibitor), we offer insights into the inhibition mechanism, showcasing how it enhances the solubility of the precipitants in the employed solvents. Finally, we aim to hypothesize a molecular-level explanation of the inhibition process through quantitative measures by comparing the size and number of nanoaggregates and exploring the molecular positioning within these aggregates. Our method introduces a valuable approach to calculate PMVs to comprehend the behavior of self-associating solutes and their interactions with solvents.

PH dependent physicochemical signatures of amino acids in aqueous buffers: Thermodynamic analysis

The physicochemical signatures such as apparent molar volume (V2,Φ), isoentropic compressibilities (KS,2,φ), and dilution enthalpies of a series of amino acids (AAs) glycine (Gly), L-alanine (Ala), DL-α-amino-n-butyric acid (ABA), L-valine (Val) and L-leucine (Leu) have been measured at various pH values (pH = 2, 5 7 and 9) at a buffer molality of 0.1 mol kg−1 of water at 298.15 K. The data obtained from high sensitivity densimeter and sound analyser along with ultrasensitive isothermal titration calorimetry has been analyzed to evaluate the values of standard partial molar quantities such as for volume at infinite dilution (V2,m0), compressibility KS, 2,m0 and dilution enthalpy (ΔdilH0) of the AAs together with standard partial molar quantities of transfer of these solutes from water to aqueous buffer systems at these pH values. The linear correlation of V2,m0 for this series of AAs against number of carbon atoms in their alkyl chain has been used to evaluate the contribution of zwitterionic end groups (NH3+, COO–), the CH2 group, and other parts of alkyl chains of the AAs to the overall value of V2,m0. The results of standard partial molar transfer properties on volumes, compressibilities and enthalpies of dilution from water to aqueous buffer systems at various pH values have been connected and explained in terms of ion-polar, ion-hydrophilic, hydrophobic-hydrophobic and ion-hydrophobic interactions of the participating groups. These thermodynamic signatures when compared at different pH values suggest that at electrostriction of water dominates at pH 2. Such physicochemical properties of constituents of proteins such as amino acids have not been reported earlier. Thus, the results demonstrate that the interaction of individual amino acid residues with the buffer components does not lead to significant change in their physicochemical properties thereby permitting their use in obtaining mechanistic insights into systems of biological interest consisting of these constituents.

Thermodynamic study of choline chloride-based deep eutectic solvents with dimethyl sulfoxide and isopropanol

The thermodynamic properties of four binary mixtures of deep eutectic solvent (DES), i.e., ChCl/Gly (choline chloride + glycerol at a molar ratio of 1:2) and ChCl/EG (choline chloride + ethylene glycol at a molar ratio of 1:2) with two common co-solvents, i.e., dimethyl sulfoxide (DMSO) and isopropanol (IPA), were studied. Density and viscosity were determined at temperatures from 288.15 to 323.15 K, while measuring the enthalpy of mixing was performed at 308.15 and 318.15 K. The volumetric properties (i.e., excess molar volume and excess partial molar volume) and excess viscosities (i.e., viscosity deviation and logarithmic excess viscosity) were further calculated to investigate the effects of temperature, types of co-solvent and DES, and their contents on the non-ideal behavior of these systems, where the influence of treating the DES as a mixture of two components or a pseudo-component was discussed. The results of volumetric properties indicate that the combined influence of packing effects and H-bonding interactions between the DES and co-solvents led to the contraction of mixture volume, and the results of excess viscosities show H-bonding interactions play an important role in their variations. The results of enthalpy of mixing show that the mixing of DES with IPA is endothermic, while the mixing of DES with DMSO is exothermic. Furthermore, the nonrandom two-liquid (NRTL) model and Gibbs-Helmholtz equation were combined to represent the experimental results of the enthalpy of mixing, and the total average relative deviation (ARD) of all studied systems is 5.43%.

Volumetric and acoustic studies of binary mixtures of DEC with isomeric chlorobenzene at various temperatures

ABSTRACT Densities, ρ, and speeds of sound, u, of binary liquid mixtures of diethyl carbonate with 1,2-dichlorobenzene, 1,3-dichlorobenzene and 1,2,4-trichlorobenzene have been measured over the entire range of composition at 303.15 K, 308.15 K, 313.15 K and 318.15 K and at atmospheric pressure. From the experimental data of the density and speed of sound, the excess properties such as excess molar volume, V m E , excess isentropic compressibility, κ s E , excess molar isentropic compressibility, K s , m E , excess speed of sound, u E , and excess isobaric thermal expansion, α p E , have been calculated. For better knowledge of the solute and solvent interaction, excess partial molar volumes, V ˉ m , 1 E and V ˉ m , 2 E , and excess partial molar volumes, V ˉ m , 1 ∘ E and V ˉ m , 2 ∘ E of the components at infinite dilution have been measured. All the excess parameter values have been fitted using the Redlich–Kister polynomial smoothing equation. The calculated excess values have been analysed in terms of molecular interactions and structural effects.

Comparing Experimentally-Measured Sand’s Times with Concentrated Solution Theory Predictions in a Polymer Electrolyte

We compare the electrochemically measured Sand’s time, the time required for the cell potential to diverge when the applied current density exceeds the limiting current, with theoretical predictions for a 0.47 M poly(ethylene oxide) (5 kg mol−1)/LiTFSI electrolyte. The theoretical predictions are made using concentrated solution theory which accounts for both concentration polarization and polymer motion, using independently measured parameters that depend on concentration, c: conductivity (κ), salt diffusion coefficient (D), cationic transference number with respect to the solvent velocity (), thermodynamic factor and partial molar volume of the salt (); is the mean molar activity coefficient of the salt. We find quantitative agreement between experimental data and theoretical predictions. We derive a generalized analytical expression for Sand’s time for electrolytes based on dilute solution theory. This expression correctly predicts the divergence of the Sand’s time at the limiting current, in agreement with experimental data and concentrated solution theory predictions. When the applied current is large compared to the limiting current, the analytical expression approaches the standard expression for Sand’s time used in the literature.

Acoustic and volumetric study of binary mixtures containing Ethyl propionate with amides at various temperatures

The development of biofuels to replace fossil fuels has recently increased at an exponential rate. Many chemical molecules with a wide range of functional groups make up biofuels. There is a strong correlation between the interactions between these molecules and the performance of biofuels. In this paper, the authors presented the new findings related to interactions existing in ethyl propionate and Formamide(F), N-Methylformamide (NMF), and N, N-Dimethylformamide (DMF) in a pure state as well as in their mixtures. Densities and speeds of sound at various temperatures (298.15–323.15 K) and at ambient pressure 0.1 MPa have been measured for these mixtures. From experimental data, the excess properties such as excess molar volume, VmE, excess isentropic compressibility, κsE, excess molar isentropic compressibility, Ks,mE, excess speed of sound, uE, and excess isobaric thermal expansivity, αpE, of the investigated mixtures were calculated. All the excess parameter values were fitted using the Redlich-Kister polynomial smoothing equation. The results were analysed in terms of molecular interactions. The excess partial molar volume, V¯m,1E, V¯m,2E , excess partial molar isentropic compressibility, K¯s,m,1EK¯s,m,2E over the whole composition range together with partial molar volume, V¯m,10, V¯m,20, partial molar isentropic compressibility K¯s,m,10, K¯s,m,20, excess partial molar volume, V¯m,10E, V¯m,20E and excess partial molar isentropic compressibility, K¯s,m,10E,K¯s,m,20E at infinite dilution have also been calculated from the experimental density measurements and analytically obtained using Redlich-Kister polynomials. The variations of these parameters with changes in composition and temperature have been discussed in terms of intermolecular interactions prevailing in these mixtures.

Osmotic and Gibbs–Donnan equilibrium for ions and neutral solutes

The general set of equations for the equilibrium of two solutions with a mixture of non-permeating and permeating ions and neutral solutes at each side of a permselective membrane is formulated using the principles of electroneutrality and mass conservation law for each solution, and equilibrium conditions: equality of electrochemical potentials at both sides of the membrane for each permeating solution component. There is at least one permeating neutral chemical species (solvent) in the system. The theory is in general valid for non-ideal solutions. The generalized Gibbs–Donnan (G–D) equilibrium coefficients depend on activities/fractions of all species at one side of the membrane, and charges of ions and partial molar volumes of all species. The equilibrium osmotic pressure across the membrane is also provided by the theory and can be calculated using the ratio of activities (or equivalently the G–D factor) of any permeating neutral solute (including solvent) or the ratios of activities (or equivalently the G–D factors) of any two permeating ions.

The miscibility gap between the rock salt and wurtzite phases in the MgO–ZnO binary system to 3.5 GPa

Abstract. At ambient pressure, MgO crystallizes in the rock salt (B1) structure, whereas ZnO crystallizes in the wurtzite structure (B4). The asymmetric miscibility gap between these two structures in the MgO–ZnO binary system narrows with increasing pressure, terminating at the wurtzite-to-rock-salt phase transition in pure ZnO, which occurs at approximately 5 GPa at 1000 ∘C. Despite their essential simplicity, the pressure–temperature–composition (P–T–X) relations in the MgO–ZnO binary system have been sparsely studied experimentally, with disparate results that are inconsistent with available thermodynamic data. Here we report the experimental determination of the P–T–X relations of the miscibility gap from 940 to 1500 ∘C and 0 to 3.5 GPa, which we combine with calorimetric and equation-of-state data from the literature and on the transition in endmember ZnO, to build a thermodynamic model that resolves many of the inconsistencies. The model treats the rock salt phase as an ideal solution (no excess Gibbs free energy of mixing), while in the wurtzite phase the MgO component follows Henry's law and the ZnO component Raoult's law in the range of compositions accessed experimentally. However, there is an inconsistency between the partial molar volume of wurtzite-structured MgO deduced from this model and that inferred from lattice parameter measurements by X-ray diffraction in the quenched samples. This discrepancy may be caused by unquenchable disordering of some significant fraction of the substituting Mg2+ into normally vacant octahedral interstices of the wurtzite structure.

Thermo-mechano-electrochemical performance of the composition-graded electrode particles

Higher capacity and longer life of the electrode is highly desired for the performance of lithium-ion batteries, which is subject to challenge of thermal and mechanical stability under working conditions. We proposed a thermo-mechano-electrochemical model for the diffusion-induced stresses of the electrodes with graded Young's modulus, partial molar volume, and thermal expansivity gained by modulating distribution of the compositions, i.e., concertation-grated substance. Modeling analysis revealed that lowering Young's modulus or raising thermal expansivity of the particle surface could protect the electrode from being cracked during lithiation or delithiation while reducing the partial molar volume could prevent fracture during lithiation. The ratio gain induced by graded Young's modulus is effective at any stress level. However, the value gains of graded partial molar volume and thermal expansivity enable the electrode to be prestressed.

Lithiation-induced swelling of electrodes

Understanding the correlation between the deformation field and the diffusion of lithium in electrode materials can provide the knowledge of the effects of materials constants on the deformation behavior of lithiated electrode materials. In this work, we study the lithiation-induced swelling and deformation of three different electrodes of plate-like, cylindrical-shell and spherical-shell shapes under the framework of linear elasticity. Linear relationships between the lithium concentration and hydrostatic stress are derived for all the three different electrodes. For the diffusion of lithium with negligible contribution of stress-limited diffusion to the diffusion of lithium in electrode materials, the lithium-induced swelling is proportional to the partial molal volume of lithium and the state of charge for the three different electrodes of plate-like, cylindrical- and spherical shapes. The increase rate of the normalized surface displacement (the swelling) is the largest at the onset of lithiation and decreases gradually to a plateau with the increase of the lithiation time. Both the constraint and geometrical shape of an electrode play important roles in controlling the swelling and stress evolution in the electrode.