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Gauss's Law: Spherical Symmetry

Gauss's Law: Spherical Symmetry

A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a uniform...

Electric Field of a Non Uniformly Charged Sphere

Electric Field of a Non Uniformly Charged Sphere

Gauss's law states that the electric flux through any closed surface equals the net charge enclosed within the surface. This law is beneficial for determining the expressions for the electric field for a particular charge distribution if the electric flux is known.
Consider a non-uniformly charged sphere, for which the density of charge depends only on the distance from a point in space and not on the direction. Such a sphere has a spherically symmetrical charge distribution. Here, the electric...

Spherical and Cylindrical Capacitor

Spherical and Cylindrical Capacitor

A spherical capacitor consists of two concentric conducting spherical shells of radii R1 (inner shell) and R2 (outer shell). The shells have equal and opposite charges of +Q and −Q, respectively. For an isolated conducting spherical capacitor, the radius of the outer shell can be considered to be infinite.
Conventionally, considering the symmetry, the electric field between the concentric shells of a spherical capacitor is directed radially outward. The magnitude of the field, calculated by...

Gauss's Law: Cylindrical Symmetry

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...

Gauss's Law

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.

Cylinders in Three-Dimensional Space

Cylinders in Three-Dimensional Space

A cylindrical surface is generated when a two-dimensional profile curve is translated along a straight line in three-dimensional space. The translated copies of the curve form a surface composed of parallel rulings, each oriented in the same fixed direction. This construction allows many three-dimensional forms to be described using relatively simple planar equations.In Cartesian coordinates, a cylindrical surface is often recognized by an equation that omits one of the three variables. For...

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Updated: Jun 19, 2026

Mechanical Mapping of Spheroids Using Brillouin Spectroscopy

Mechanical Mapping of Spheroids Using Brillouin Spectroscopy

Published on: December 12, 2025

Spherical brushes within spherical cavities: a self-consistent field and Monte Carlo study.

Juan J Cerdà¹, Tomás Sintes, Raúl Toral

¹Institute for Computational Physics, Universität Stuttgart, 70569 Stuttgart, Germany. jcerda@icp.uni-stuttgart.de

The Journal of Chemical Physics

|October 10, 2009

Summary

This study numerically investigates spherical brushes in spherical cavities. Self-consistent field and Monte Carlo methods reveal distinct pressure behaviors under compression, with Flory theory excelling at high compressions.

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Mechanical Mapping of Spheroids Using Brillouin Spectroscopy

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Preparation and Friction Force Microscopy Measurements of Immiscible, Opposing Polymer Brushes

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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

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Area of Science:

Polymer Physics
Soft Matter Physics
Computational Chemistry

Background:

Spherical brushes are crucial in various applications, including colloid stabilization and drug delivery.
Understanding their behavior under confinement is essential for designing advanced materials.
Previous theoretical models often simplify complex interactions within confined polymer systems.

Purpose of the Study:

To numerically investigate the behavior of spherical brushes confined within a spherical cavity.
To determine monomer and end-chain density profiles and cavity pressure.
To compare theoretical and simulation methods for accuracy.

Main Methods:

Extensive numerical study using self-consistent field (SCF) theory.
Off-lattice Monte Carlo (MC) simulations.
Comparison with Flory theory for polymer solutions.

Main Results:

SCF and MC methods accurately predict density profiles and pressure.
Flory theory provides excellent agreement for pressure in strongly compressed regimes.
A scaling relationship P ~ v(alpha) was observed at high compressions, with SCF yielding alpha=2.15±0.05 and MC yielding alpha=2.73±0.04.

Conclusions:

SCF is a viable alternative to MC for free and softly compressed brushes.
Flory theory accurately models pressure in highly compressed systems.
SCF's mean-field approach underestimates pressure scaling due to limitations in accounting for monomer density correlations.