Brownian motion question [Archive]

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Alison Chaiken

Nov3-04, 10:06 AM

PanSynthesis@netscape.net

Dec15-04, 12:39 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Alison Chaiken wrote:\n> I\'m interested in the question of Brownian motion for non-spherical\n> particles. Assuredly this problem was worked out the \'20s if not the\n> \'10s. I see that Rev. Mod. Phys. has had 4 reviews with Brownian in\n> the title, the most recent in 1952. Rep. Prog. Phys. has no articles\n> with Brownian in the title. Who can recommend a review article that\n> treats the non-spherical case and much, much more? I am an\n> experimentalist so a pedagogical article is best for me. My\n> application is in microfluidics. Thanks in advance.\n>\n> --\n> Alison Chaiken "From:" address above is valid.\n> (650) 236-2231 [daytime] http://www.wsrcc.com/alison/\n> What was written as gossip can soon be read as History, like many a\n> trifle before it. -- Freya Stark\n\nI don\'t know of a good review article, but since no one responded\nwith exactly what you need, here are a few remarks that are somewhat\nrelated to your request, and are oriented to experiments.\n\nAn aspherical particle will execute a random walk described by a\ndiffusion coefficient D (mean squared displacement increases linearly\nin time). For a spherical particle of radius R we have D = kT / 6 pi\neta R where eta is the viscosity of the fluid. 1 / 6 pi eta R is the\ndrag force per velocity. This is the ratio of two vectors and in\ngeneral for an aspherical particle one will have to properly recognize\nit as a tensor and the mathematics are sure to be complicated. (Besides\ndrag, there will be lift, a fluid-exerted force component that isn\'t\nparallel to the velocity.) Physica A might have articles where people\nhave worked out such tensors. In general, Physica A is a good source\nfor physics-oriented articles on similar topics.\n\nIf all you can measure is the translational mean squared displacement,\nI think you would have a hard time seeing effects caused by the\naspherical shape. It might be, though, that at early times there is a\nballistic regime (where the mean square displacement increases like\nt^2, or, at any rate, non-linearly) where one might see consequences of\nit being easier for the particle to move in certain directions than\nothers. An out-of-date discussion of the ballistic regime of a\ndiffusing spherical particle is in the book of reprints, edited by\nNelson Wax, published by Dover. (It is in one of the papers by\nUhlenbeck.) In that discussion, account is taken of the particle\'s\nmass, which causes it to tend to keep moving in the same direction\nduring an exponentially short pre-diffusional regime. The treatment is\nout of date because in fact the pre-diffusional regime lingers on with\nan algebraic, not exponential, decay. This is the celebrated long-time\ntail (algebraic, rather than exponential, decay of the velocity\nautocorrelation function at long times) first discovered in molecular\ndynamics computer simulations by Alder and Wainwright. (The velocity\nautocorrelation function of a particle moving purely diffusively is a\ndelta function: instant decay.) It is called the long-time tail, but it\naffects the early part of the motion of a diffusing particle. It is\ncaused by the fact that 1 / 6 pi eta R is the drag force per velocity\nonly if the velocity is constant. In general, the drag depends on the\nhistory of the velocity, because of eddies in the fluid, created by the\nparticle, that can come back and influence the particle. This is a big\nsubject. You can find experimental articles on it by Weitz in Physics\nReview Letters in the early 90\'s - spherical particles of course.\nI\'m suggesting that you might observe modifications of the long time\ntail that are due to the aspheric shape.\n\nParticles diffuse in orientation as well as in position. Rotational\ndiffusion is discussed in the book "Dynamic Light Scattering" by\nBerne and Pecora. Spherical particles execute rotational diffusion, of\ncourse, but that ordinarily does not produce an observable effect. If\nthe particles are aspherical, their changing orientation with respect\nto the incident light causes detectable fluctuations in the scattered\nlight. In this way, Pusey observed rotational diffusion of rod-shaped\nparticles as written up in PRL in the 80\'s. Piazza and/or DiGiorgio\nlooked at particles spherical in shape but optically anisotropic. They\nclaimed to have seen a surprising dependence of the rotational\ndiffusion coefficient on particle concentration, but others suggested\nthat they had a multiple scattering problem.\n\nYou mention microfluidics, so your interest may be in the rheological\nbehavior of suspensions of aspherical particles. Check the Journal of\nRheology for an entry into a big literature. Many non-Newtonian effects\ncan occur, such as shear-thinning. Aspherical particles can tumble or\norient in a shear flow, as discussed by Jefferies in the 20\'s in\nProceedings of the Royal Society. Liquid particles (i.e., droplets in\nan emulsion) will deform in a shear flow, as discussed by Taylor some\nyears before Jefferies, also in Proc. Roy. Soc. Even if there is no\ndeformation, which will be the case if the surface tension is high,\nthere will be shear flows induced inside the droplets which cause the\nviscosity of the emulsion to have a larger concentration coefficient\nthan does a suspension of solid particles. This is shown in Taylor\'s\npaper, which references the earlier paper by Einstein treating solid\nspherical particles. (Einstein worked out the theory in order to\nprovide experimentalists with a practical method to determine particle\nconcentration, and so, ultimately, particle radius, in order to test\nhis prediction for the diffusion coefficient.) For a liquid drop in a\ntime-dependent shear flow, there will be a lag between the time\ndependence of the flow and the time dependence of the drop deformation.\nThis can lead to viscoelasticity, as shown in experiments by Sengers\nwritten up in Physical Review E in the early 90\'s. Pine has described\nmany scattering experiments on how structure is deformed by shear in\npolymer systems. These can be found in PRA and PRE at about the same\ntime.\n\nHope some of this helps.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Alison Chaiken wrote:
> I'm interested in the question of Brownian motion for non-spherical
> particles. Assuredly this problem was worked out the '20s if not the
> '10s. I see that Rev. Mod. Phys. has had 4 reviews with Brownian in
> the title, the most recent in 1952. Rep. Prog. Phys. has no articles
> with Brownian in the title. Who can recommend a review article that
> treats the non-spherical case and much, much more? I am an
> experimentalist so a pedagogical article is best for me. My
> application is in microfluidics. Thanks in advance.
>
> --
> Alison Chaiken "From:" address above is valid.
> (650) 236-2231 [daytime] http://www.wsrcc.com/alison/
> What was written as gossip can soon be read as History, like many a
> trifle before it. -- Freya Stark

I don't know of a good review article, but since no one responded
with exactly what you need, here are a few remarks that are somewhat
related to your request, and are oriented to experiments.

An aspherical particle will execute a random walk described by a
diffusion coefficient D (mean squared displacement increases linearly
in time). For a spherical particle of radius R we have D = kT / 6 \pi\eta R where \eta is the viscosity of the fluid. 1 / 6 \pi \eta R is the
drag force per velocity. This is the ratio of two vectors and in
general for an aspherical particle one will have to properly recognize
it as a tensor and the mathematics are sure to be complicated. (Besides
drag, there will be lift, a fluid-exerted force component that isn't
parallel to the velocity.) Physica A might have articles where people
have worked out such tensors. In general, Physica A is a good source
for physics-oriented articles on similar topics.

If all you can measure is the translational mean squared displacement,
I think you would have a hard time seeing effects caused by the
aspherical shape. It might be, though, that at early times there is a
ballistic regime (where the mean square displacement increases like
t^2, or, at any rate, non-linearly) where one might see consequences of
it being easier for the particle to move in certain directions than
others. An out-of-date discussion of the ballistic regime of a
diffusing spherical particle is in the book of reprints, edited by
Nelson Wax, published by Dover. (It is in one of the papers by
Uhlenbeck.) In that discussion, account is taken of the particle's
mass, which causes it to tend to keep moving in the same direction
during an exponentially short pre-diffusional regime. The treatment is
out of date because in fact the pre-diffusional regime lingers on with
an algebraic, not exponential, decay. This is the celebrated long-time
tail (algebraic, rather than exponential, decay of the velocity
autocorrelation function at long times) first discovered in molecular
dynamics computer simulations by Alder and Wainwright. (The velocity
autocorrelation function of a particle moving purely diffusively is a
\delta function: instant decay.) It is called the long-time tail, but it
affects the early part of the motion of a diffusing particle. It is
caused by the fact that 1 / 6 \pi \eta R is the drag force per velocity
only if the velocity is constant. In general, the drag depends on the
history of the velocity, because of eddies in the fluid, created by the
particle, that can come back and influence the particle. This is a big
subject. You can find experimental articles on it by Weitz in Physics
Review Letters in the early 90's - spherical particles of course.
I'm suggesting that you might observe modifications of the long time
tail that are due to the aspheric shape.

Particles diffuse in orientation as well as in position. Rotational
diffusion is discussed in the book "Dynamic Light Scattering" by
Berne and Pecora. Spherical particles execute rotational diffusion, of
course, but that ordinarily does not produce an observable effect. If
the particles are aspherical, their changing orientation with respect
to the incident light causes detectable fluctuations in the scattered
light. In this way, Pusey observed rotational diffusion of rod-shaped
particles as written up in PRL in the 80's. Piazza and/or DiGiorgio
looked at particles spherical in shape but optically anisotropic. They
claimed to have seen a surprising dependence of the rotational
diffusion coefficient on particle concentration, but others suggested
that they had a multiple scattering problem.

You mention microfluidics, so your interest may be in the rheological
behavior of suspensions of aspherical particles. Check the Journal of
Rheology for an entry into a big literature. Many non-Newtonian effects
can occur, such as shear-thinning. Aspherical particles can tumble or
orient in a shear flow, as discussed by Jefferies in the 20's in
Proceedings of the Royal Society. Liquid particles (i.e., droplets in
an emulsion) will deform in a shear flow, as discussed by Taylor some
years before Jefferies, also in Proc. Roy. Soc. Even if there is no
deformation, which will be the case if the surface tension is high,
there will be shear flows induced inside the droplets which cause the
viscosity of the emulsion to have a larger concentration coefficient
than does a suspension of solid particles. This is shown in Taylor's
paper, which references the earlier paper by Einstein treating solid
spherical particles. (Einstein worked out the theory in order to
provide experimentalists with a practical method to determine particle
concentration, and so, ultimately, particle radius, in order to test
his prediction for the diffusion coefficient.) For a liquid drop in a
time-dependent shear flow, there will be a lag between the time
dependence of the flow and the time dependence of the drop deformation.
This can lead to viscoelasticity, as shown in experiments by Sengers
written up in Physical Review E in the early 90's. Pine has described
many scattering experiments on how structure is deformed by shear in
polymer systems. These can be found in PRA and PRE at about the same
time.

Hope some of this helps.

Robert M. Mazo

Dec17-04, 07:53 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Wed, 15 Dec 2004 18:39:11 +0000 (UTC), PanSynthesis@netscape.net\nwrote:\n\n>Alison Chaiken wrote:\n>> I\'m interested in the question of Brownian motion for non-spherical\n>> particles. Assuredly this problem was worked out the \'20s if not the\n>> \'10s. I see that Rev. Mod. Phys. has had 4 reviews with Brownian in\n>> the title, the most recent in 1952. Rep. Prog. Phys. has no articles\n>> with Brownian in the title. Who can recommend a review article that\n>> treats the non-spherical case and much, much more? I am an\n>> experimentalist so a pedagogical article is best for me. My\n>> application is in microfluidics. Thanks in advance.\n>>\n>> --\n>> Alison Chaiken "From:" address above is valid.\n>> (650) 236-2231 [daytime] http://www.wsrcc.com/alison/\n>> What was written as gossip can soon be read as History, like many a\n>> trifle before it. -- Freya Stark\n>\n\nSee my book, Brownian Motion, Oxford, 2002, Chapter 15.. For a more\nthourough treatment of the rotational case, see Coffey et al. The\nlangevin Equation, World Scientific, 2004.\n\nRobert Mazo\n\n\n>I don\'t know of a good review article, but since no one responded\n>with exactly what you need, here are a few remarks that are somewhat\n>related to your request, and are oriented to experiments.\n>\n>An aspherical particle will execute a random walk described by a\n>diffusion coefficient D (mean squared displacement increases linearly\n>in time). For a spherical particle of radius R we have D = kT / 6 pi\n>eta R where eta is the viscosity of the fluid. 1 / 6 pi eta R is the\n>drag force per velocity. This is the ratio of two vectors and in\n>general for an aspherical particle one will have to properly recognize\n>it as a tensor and the mathematics are sure to be complicated. (Besides\n>drag, there will be lift, a fluid-exerted force component that isn\'t\n>parallel to the velocity.) Physica A might have articles where people\n>have worked out such tensors. In general, Physica A is a good source\n>for physics-oriented articles on similar topics.\n>\n>If all you can measure is the translational mean squared displacement,\n>I think you would have a hard time seeing effects caused by the\n>aspherical shape. It might be, though, that at early times there is a\n>ballistic regime (where the mean square displacement increases like\n>t^2, or, at any rate, non-linearly) where one might see consequences of\n>it being easier for the particle to move in certain directions than\n>others. An out-of-date discussion of the ballistic regime of a\n>diffusing spherical particle is in the book of reprints, edited by\n>Nelson Wax, published by Dover. (It is in one of the papers by\n>Uhlenbeck.) In that discussion, account is taken of the particle\'s\n>mass, which causes it to tend to keep moving in the same direction\n>during an exponentially short pre-diffusional regime. The treatment is\n>out of date because in fact the pre-diffusional regime lingers on with\n>an algebraic, not exponential, decay. This is the celebrated long-time\n>tail (algebraic, rather than exponential, decay of the velocity\n>autocorrelation function at long times) first discovered in molecular\n>dynamics computer simulations by Alder and Wainwright. (The velocity\n>autocorrelation function of a particle moving purely diffusively is a\n>delta function: instant decay.) It is called the long-time tail, but it\n>affects the early part of the motion of a diffusing particle. It is\n>caused by the fact that 1 / 6 pi eta R is the drag force per velocity\n>only if the velocity is constant. In general, the drag depends on the\n>history of the velocity, because of eddies in the fluid, created by the\n>particle, that can come back and influence the particle. This is a big\n>subject. You can find experimental articles on it by Weitz in Physics\n>Review Letters in the early 90\'s - spherical particles of course.\n>I\'m suggesting that you might observe modifications of the long time\n>tail that are due to the aspheric shape.\n>\n>Particles diffuse in orientation as well as in position. Rotational\n>diffusion is discussed in the book "Dynamic Light Scattering" by\n>Berne and Pecora. Spherical particles execute rotational diffusion, of\n>course, but that ordinarily does not produce an observable effect. If\n>the particles are aspherical, their changing orientation with respect\n>to the incident light causes detectable fluctuations in the scattered\n>light. In this way, Pusey observed rotational diffusion of rod-shaped\n>particles as written up in PRL in the 80\'s. Piazza and/or DiGiorgio\n>looked at particles spherical in shape but optically anisotropic. They\n>claimed to have seen a surprising dependence of the rotational\n>diffusion coefficient on particle concentration, but others suggested\n>that they had a multiple scattering problem.\n>\n>You mention microfluidics, so your interest may be in the rheological\n>behavior of suspensions of aspherical particles. Check the Journal of\n>Rheology for an entry into a big literature. Many non-Newtonian effects\n>can occur, such as shear-thinning. Aspherical particles can tumble or\n>orient in a shear flow, as discussed by Jefferies in the 20\'s in\n>Proceedings of the Royal Society. Liquid particles (i.e., droplets in\n>an emulsion) will deform in a shear flow, as discussed by Taylor some\n>years before Jefferies, also in Proc. Roy. Soc. Even if there is no\n>deformation, which will be the case if the surface tension is high,\n>there will be shear flows induced inside the droplets which cause the\n>viscosity of the emulsion to have a larger concentration coefficient\n>than does a suspension of solid particles. This is shown in Taylor\'s\n>paper, which references the earlier paper by Einstein treating solid\n>spherical particles. (Einstein worked out the theory in order to\n>provide experimentalists with a practical method to determine particle\n>concentration, and so, ultimately, particle radius, in order to test\n>his prediction for the diffusion coefficient.) For a liquid drop in a\n>time-dependent shear flow, there will be a lag between the time\n>dependence of the flow and the time dependence of the drop deformation.\n>This can lead to viscoelasticity, as shown in experiments by Sengers\n>written up in Physical Review E in the early 90\'s. Pine has described\n>many scattering experiments on how structure is deformed by shear in\n>polymer systems. These can be found in PRA and PRE at about the same\n>time.\n>\n>Hope some of this helps.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 15 Dec 2004 18:39:11 +0000 (UTC), PanSynthesis@netscape.net
wrote:

>Alison Chaiken wrote:
>> I'm interested in the question of Brownian motion for non-spherical
>> particles. Assuredly this problem was worked out the '20s if not the
>> '10s. I see that Rev. Mod. Phys. has had 4 reviews with Brownian in
>> the title, the most recent in 1952. Rep. Prog. Phys. has no articles
>> with Brownian in the title. Who can recommend a review article that
>> treats the non-spherical case and much, much more? I am an
>> experimentalist so a pedagogical article is best for me. My
>> application is in microfluidics. Thanks in advance.
>>
>> --
>> Alison Chaiken "From:" address above is valid.
>> (650) 236-2231 [daytime] http://www.wsrcc.com/alison/
>> What was written as gossip can soon be read as History, like many a
>> trifle before it. -- Freya Stark
>

See my book, Brownian Motion, Oxford, 2002, Chapter 15.. For a more
thourough treatment of the rotational case, see Coffey et al. The
langevin Equation, World Scientific, 2004.

Robert Mazo

>I don't know of a good review article, but since no one responded
>with exactly what you need, here are a few remarks that are somewhat
>related to your request, and are oriented to experiments.
>
>An aspherical particle will execute a random walk described by a
>diffusion coefficient D (mean squared displacement increases linearly
>in time). For a spherical particle of radius R we have D = kT / 6 \pi>\eta R where \eta is the viscosity of the fluid. 1 / 6 \pi \eta R is the
>drag force per velocity. This is the ratio of two vectors and in
>general for an aspherical particle one will have to properly recognize
>it as a tensor and the mathematics are sure to be complicated. (Besides
>drag, there will be lift, a fluid-exerted force component that isn't
>parallel to the velocity.) Physica A might have articles where people
>have worked out such tensors. In general, Physica A is a good source
>for physics-oriented articles on similar topics.
>
>If all you can measure is the translational mean squared displacement,
>I think you would have a hard time seeing effects caused by the
>aspherical shape. It might be, though, that at early times there is a
>ballistic regime (where the mean square displacement increases like
>t^2, or, at any rate, non-linearly) where one might see consequences of
>it being easier for the particle to move in certain directions than
>others. An out-of-date discussion of the ballistic regime of a
>diffusing spherical particle is in the book of reprints, edited by
>Nelson Wax, published by Dover. (It is in one of the papers by
>Uhlenbeck.) In that discussion, account is taken of the particle's
>mass, which causes it to tend to keep moving in the same direction
>during an exponentially short pre-diffusional regime. The treatment is
>out of date because in fact the pre-diffusional regime lingers on with
>an algebraic, not exponential, decay. This is the celebrated long-time
>tail (algebraic, rather than exponential, decay of the velocity
>autocorrelation function at long times) first discovered in molecular
>dynamics computer simulations by Alder and Wainwright. (The velocity
>autocorrelation function of a particle moving purely diffusively is a
>\delta function: instant decay.) It is called the long-time tail, but it
>affects the early part of the motion of a diffusing particle. It is
>caused by the fact that 1 / 6 \pi \eta R is the drag force per velocity
>only if the velocity is constant. In general, the drag depends on the
>history of the velocity, because of eddies in the fluid, created by the
>particle, that can come back and influence the particle. This is a big
>subject. You can find experimental articles on it by Weitz in Physics
>Review Letters in the early 90's - spherical particles of course.
>I'm suggesting that you might observe modifications of the long time
>tail that are due to the aspheric shape.
>
>Particles diffuse in orientation as well as in position. Rotational
>diffusion is discussed in the book "Dynamic Light Scattering" by
>Berne and Pecora. Spherical particles execute rotational diffusion, of
>course, but that ordinarily does not produce an observable effect. If
>the particles are aspherical, their changing orientation with respect
>to the incident light causes detectable fluctuations in the scattered
>light. In this way, Pusey observed rotational diffusion of rod-shaped
>particles as written up in PRL in the 80's. Piazza and/or DiGiorgio
>looked at particles spherical in shape but optically anisotropic. They
>claimed to have seen a surprising dependence of the rotational
>diffusion coefficient on particle concentration, but others suggested
>that they had a multiple scattering problem.
>
>You mention microfluidics, so your interest may be in the rheological
>behavior of suspensions of aspherical particles. Check the Journal of
>Rheology for an entry into a big literature. Many non-Newtonian effects
>can occur, such as shear-thinning. Aspherical particles can tumble or
>orient in a shear flow, as discussed by Jefferies in the 20's in
>Proceedings of the Royal Society. Liquid particles (i.e., droplets in
>an emulsion) will deform in a shear flow, as discussed by Taylor some
>years before Jefferies, also in Proc. Roy. Soc. Even if there is no
>deformation, which will be the case if the surface tension is high,
>there will be shear flows induced inside the droplets which cause the
>viscosity of the emulsion to have a larger concentration coefficient
>than does a suspension of solid particles. This is shown in Taylor's
>paper, which references the earlier paper by Einstein treating solid
>spherical particles. (Einstein worked out the theory in order to
>provide experimentalists with a practical method to determine particle
>concentration, and so, ultimately, particle radius, in order to test
>his prediction for the diffusion coefficient.) For a liquid drop in a
>time-dependent shear flow, there will be a lag between the time
>dependence of the flow and the time dependence of the drop deformation.
>This can lead to viscoelasticity, as shown in experiments by Sengers
>written up in Physical Review E in the early 90's. Pine has described
>many scattering experiments on how structure is deformed by shear in
>polymer systems. These can be found in PRA and PRE at about the same
>time.
>
>Hope some of this helps.