Brownian force

Brownian forces can disrupt the particle chains that produce a yield stress, but can be made negligible if the electrostatic forces holding particle structures together are strong enough. From Eq. (8-3), we can compute the characteristic depth of the potential well holding 

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The structure formation in an ER fluid was simulated [99]. The characteristic parameter is the ratio of the Brownian force to the dipolar force. Over a wide range of this ratio there is rapid chain formation followed by aggregation of chains into thick columns with a body-centered tetragonal structure observed. Above a threshold of the intensity of an external ahgn-ing field, condensation of the particles happens [100]. This effect has also been studied for MR fluids [101]. The rheological behavior of ER fluids [102] depends on the structure formed chainlike, shear-string, or liquid. Coexistence in dipolar fluids in a field [103], for a Stockmayer fluid in an applied field [104], and the structure of soft-sphere dipolar fluids were investigated [105], and ferroelectric phases were found [106]. An island of vapor-liquid coexistence was found for dipolar hard spherocylinders [107]. It exists between a phase where the particles form chains of dipoles in a nose-to-tail... 

The behavior of a bead-spring chain immersed in a flowing solvent could be envisioned as the following under the influence of hydrodynamic drag forces (fH), each bead tends to move differently and to distort the equilibrium distance. It is pulled back, however, by the entropic need of the molecule to retain its coiled shape, represented by the restoring forces (fs) and materialized by the spring in the model. The random bombardment of the solvent molecules on the polymer beads is taken into account by time smoothed Brownian forces (fB). Finally inertial forces (f1) are introduced into the forces balance equation by the bead mass (m) times the acceleration ( ) of one bead relative to the others ... 

The dumbbell relaxation time (t) in the preceding model is coil deformation dependent. Neglecting Brownian forces, the dumbbell relaxation time is given by t ssf H/fs. Equation (45) is then tantamount to saying that t increases approximately in proportion to the root mean square end-to-end separation distance R [52] ... 

J. T. Padding and A. A. Louis, Hydrodynamic interactions and Brownian forces in colloidal suspensions coarse-graining over time and length scales, Phys. Rev. E 74, 031402 (2006). 

Following FerrelK, the second term in Equation 2 can be expressed as a Green-Kubo integral over a flux-flux correlation function. The transport is due to a velocity perturbation caused by two driving forces, the Brownian force and frictional force. The transport coefficient due to the segment-segment interaction can be calculated from the Kubo formula(9 ... 

The probability flux is determined by a balance of generalized mechanical and Brownian forces, of the form... 

The elastic force is given by the sum of a mechanical force —dU/dq and a corresponding Brownian force. The form of the Brownian force may be inferred by requiring that Fa vanish when /( ) = v /eq( ). in order to guarantee that the probability flux 7 vanishes in thermal equilibrium. This requirement yields an elastic force... 

The local equilibrium value of the kinetic force thus yields the Brownian force of the phenomelogical diffusion equation. Equation (2.94) also allows us to take D pa) /Dt = 0 on the left-hand side (LHS) of Eq. (2.91), to obtain a force balance... 

Note that the averaging with respect to momentum fluctuations has the effect of replacing the system s potential energy U by an effective potential U — kT In /m, exactly as in the equilibrium distribution for a rigid system, which is of the form /eq( ) oc exp —(C/ — kTln )/kT. Substituting Eq. (2.98) for the average mechanical force and Eq. (2.95) for the Brownian force into Eq. (2.96) yields an effective force balance... 

In general, the motion of a polymer chain in solution is governed by intermolecular interaction, hydrodynamic interaction, Brownian random force, and external field. The hydrodynamic interaction consists of the intra- and intermolecular ones. The intramolecular hydrodynamic interaction and Brownian force play dominant roles in dilute solution, while the intermolecular interaction and the intermolecular hydrodynamic interaction become important as the concentration increases. 

The external electric field is in the direction of the pore axis. The particle is driven to move by the imposed electric field, the electroosmotic flow, and the Brownian force due to thermal fluctuation of the solvent molecules. Unlike the usual electroosmotic flow in an open slit, the fluid velocity profile is no longer uniform because a pressure gradient is built up due to the presence of the closed end. The probability of the particle position is obtained by solving the Fokker-Planck equation. The penetration depth is found to be dependent upon the Peclet number, which is a measure of significance of the convective electroosmotic flow relative to the Brownian diffusion, and the Damkohler number, which is a ratio of the characteristic diffusion-to-deposition times. 

Effects of colloidal forces (i.e. van der Waals, electrostatic, and Brownian forces). 

Brownian motion must be taken into account for suspensions of small (submicron-sized) particles. By their very nature, such stochastic Brownian forces favor the ergodicity of any configurational state. Although no completely general framework for the inclusion of Brownian motion will be presented here, its effects will be incorporated within specific contexts. Especially relevant, in the present rheological context, is the recent review by Felderhof (1988) of the contribution of Brownian motion to the viscosity of suspensions of spherical particles. 

Fig. 9.3. Acid mist removal candle filter being installed atop a stainless steel H2SO4 making tower. It is one of many. Exiting gas passes inward through the candle fabric and out the top of the candle - then out of the tower. The acid mist is caught in the candle fabric by impact, diffusion and Brownian forces (Brink, 2005 Friedman and Friedman, 2004 Lee and Byszewski, 2005 Ziebold and Azwell, 2005). The large total area of the candles gives a low gas velocity through the fabric, which allows 99+% capture of the mist. The captured mist trickles down the fabric and drips back into the tower or into collection pipes (Outokumpu 2005).

Because of Brownian forces, each molecule in the molten state continually changes its configuration, but for long polymer molecules composed of hundreds or more monomers, the time-averaged mean-square distance R )o separating one end of the molecule from the other obeys the random-walk formula... 

The third term is the Brownian force, F = kfiTS In /9R. It represents the average force exerted by bead 2 as a result of random bombardments by the surrounding (mostly solvent) molecules. 

The forces one must include in such a simulation include electrostatic, hydrodynamic, and steric forces. For small particles, Brownian forces might also be present, but since these break up particle structures, it is desirable to use particles big enough (> 1 /xm) to suppress Brownian motion. Ordinarily, particle inertia can be neglected. Simulations can be greatly simplified by making drastic approximations, including the point-dipole approximation, and the Stokes -drag approximation. Both of these approximations are only really valid for widely separated particles. 

When Brownian forces are not negligible—that is, when A. defined in Eg. (8-7) is less... 

In the model of Agarwal and Khakhar [57] the polymer molecules are taken to be bead-rod chains with the hydrodynamic forces concentrated at the beads. The chains may bend about a bead, and a spring force acts to restore the chain to is equilibrium conformation, which is a straight chain. The connecting rods are inextensible. The system is confined to a plane, and the chains diffuse due to Brownian forces resisted by hydrodynamic forces. Hydrodynamic forces resulting from an imposed shear flow deform and orient the molecules. Two chains may react and combine to form a longer chain if the chain ends approach to within the capture radius (a) and if the angle between the chains is less than the critical value (0 ). The reaction is assumed to be very fast (kfj k j ) so that every collision that satisfies the above criteria results in... 

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