Behavior of powder under weak perturbations: the concept of the effective surface. I.

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Behavior of powder under weak perturbations: the concept of the effective surface. I.

E. Gurovich, S. Maruk

To cite this version:

E. Gurovich, S. Maruk. Behavior of powder under weak perturbations: the concept of the effective surface. I.. Journal de Physique II, EDP Sciences, 1991, 1 (10), pp.1147-1166. �10.1051/jp2:1991125�.

�jpa-00247581�

J. Phys. II France 1(1991) l147-l166 OCTOBRE1991, PAGE l147

Classificafion Physics Abstracts

05.40 05.60

Proofs _not corrected by ^the authors

Behavior of powder under weak perturbations

_:

^the concept ^of

the effective surface. 1.

E. V. Gurovich

(*)

and S. V. Maruk

Departrrent of Theoretical Physics, Insfitute for lligh Pressure Physics, Moscow Region, Troisk, 142092, U-S-S-R-

(Received J3 May J99J, accepted in final form J7 June J99J)

Abswad. On the base of the semiphenomenological model one-panicle approach describing

the behavior of powder under weak perturbation is developed. The notion of the effective surface which restricts the possible shift of _a grain due to the presence of surrounding ones is put forward.

The dynamics of swelling and packing down is considered _{as a} synergetic self-organizing _process through Poincare's _maps. The equilibrium density for _a given intensity of perturbations is

described. The origin of the convective motion in vibration and its threshold character are

investigated.

1. Inwoduction.

The

investigation

of

powders (so-called granular

_or

free-flowing materials)

is traced back to the studies of

Faraday (1831)

and

Reynold's (1885). They II,

2] described

unique

static and

dynamic properties

^of a medium

consisting

of _a great many of

grains,

in interaction of which friction is of great

importance.

^Behavior of

powder

was revealed to

depend non-trivially

_upon the

density

and the _manner of

grains packing.

This leads to the existence of _a number of

original phenomena breaking

_{up, as} it seems, the fundamental laws of classical mechanics.

So, under _a rather

powerful

vibration in

granular

material _a convective stream of _mass

occurs in the

opposite

direction to force of

gravity [I].

Another well known

phenomenon

is the

residing

of

powder

at

perturbations.

Under

jostle, shaking

_or vibration _a

change

of initial volume _can reach the

magnitude

of the order of 10 fb.

This effect is

customarily

considered _{as a} trivial _one and

explained by

the

tendency

of the

dissipative

mechanical system to minimize

potential

_energy. However, the

stirring

_can lead not

only

to

residing

but also to

swelling

and

expanding.

Finally,

an

interesting phenomenon

such _as

segregation

of

polydisperse powder

should be noted. Under chaotic disturbance _a mixture of

grains

^of different sizes and _{masses can} be

separated.

The

larger (sometimes

_even

denser) panicles

rise to the top,

coming

to the

surface,

and smaller _ones

(though

less

dense)

filter down

sinking

in

spite

of _» Archimed's law.

l148 JOURNAL DE PHYSIQUE II N 10

Due to apparent industrial

applications

these

phenomena

have been the

subject

of

considerable

study.

At present ^there is _a number of

technological

and

applied

_{papers as a} rule of _a

descriptive

character which can't be reviewed _even

briefly.

Some references to this type ^of papers can be found in references [3-27~ mentioned below. It is to be noted

only

that

these

phenomena

have been used

long

_ago in

powder

industries.

Nevertheless, _as

paradoxical

_as it is, the

microscopic grounds

^and

quantitative description

of these effects, in

spite

of their apparent

simplicity,

_are absent until _now

(').

The _reason for this is that the consistent

theory

of

powder

_stops at the very

complicated problem

of classical

mechanics,

namely,

the

many-body problem.

In

spite

of the number of

grains

in the unit of volume

being

in 106-108 _{times less than in}

typical problems

of the molecular

dynamics

this

problem

can't be

directly

solved. As is

usually

for this type ^of

problems,

the

following questions

arise _: how to connect the _movement of _any selected

particle

with the others, how to

connect the local geometry of

packing

with the

properties

of medium as a whole ?

There exist different _ways to solve the

problem

concemed. The main part ^{of the work is} devoted to the

analysis

of random

packing [3, 17-24] usually

_a

packing

of

spheres

in the three-dimensional _{space or} disks _on the

plane.

The _common

geometrical probabilistic [23, 24]

methods,

data obtained

by

natural

II

8,

21]

and numerical

[17,

19,

20] experiments

_are used _as the base of this work.

The

regular packing

of

spheres

and disks have

long

_ago been studied and classified in

crystallography (Fig, I).

The share of the volume

occupied by particles

in

hexagonal

and cubic

spatial packings

_are

equal

~

3~i~~~~

^~

^~~~~~

^~~~~

The densities of

corresponding regular plane packings

_are well known too.

They

_are

respectively,

The _nature of random

packings

_are to a great extent _more

complicated. Simulating

the

strucrue of monoatomic

liquid

Bemal

[17]

discovered the

density jump

between the

hexagonal

and the random closed

packings.

^While

experimenting

with the metallic balls [18]

Scott showed the

density

of

poured powder depending

_on ^{friction forces} between

grains.

The maximum

density

that _can be achieved

by stirring

and

pressing

is

f 0.637

(1.3)

Finney following

Bemal and Skott

reproduced [19]

their results and carried out the

deep

statistic

processing

^of three-dimensional random closed

packing by

_computer simulation.

Computer ^{research of}

plane

random

packing

of disks

[20]

shows the maximum

density

of it to be

equal

to

f 0.821

(1.4)

(~) There is _a lack of studies and the origin of the convective motion is still unexplained _» (Douady,

Fauve and Laroche, 1989), _a basic phenomenological understanding of segregation mechanism has remained elusive (Rosato, Strandburg, Prinz, Swendsen, 1987), _« the most elementary problems have not been fully addressed (Edwards, Oakenshott, 1989).

N 10 BEHAVIOR OF POWDER UNDER PERTURBATIONS l149

a) _~~,,

,

b)

,

dn

~2'~3

,_j , , ,_,

~Z,X3

Iii _d)

^~i

_c)

,,

, , , ,,,

~fi

,

~

xi ^,

~

/

,

~

~~d~

,

, ,

Xz>X3 ^{~ '~} i

W

Fig. I. a) Regular plane packing characterized by the angle p, b) the closed random plane and c) _a

loose random plane packings, d) _a local geometrical configuration in the regular three-dimensional packing characterized by the angle p, here loose two-dimensional layers have hexagonal synlmetry, e)

random packing in polydispersive powder. The particles having occupied _{one row} (layer

4

_in the _more compact packings ~b) and occupiying after swelling (c) a number of _rows (more thick layer d~) areshaded.

Nowadays

there exist _{numerous papers}

concerning

computer simulations of both

poly-

and

mono-disperse powders (see

references in

[22-23]).

However, the values of maximum densities

(1.3-4)

_are still

intriguing

as is the notion of random closed

packing

itself.

The random

packings

simulate arrangement ^of

powder particles.

Nevertheless, in the papers devoted to the mathematical

analysis

of

packings

the

physical

results _are

practically

absent. In these _papers

particles

_are

regarded

_as

purely geometric objects,

_mass and

gravity

field

being

absent while

grain

arrangement fixed. To go over from the

packing

_geometry to

properties

of

granular

medium in this

approach

is

impossible.

The second,

phenomenological approach,

tumed out to be _more fruitful.

Postulating

_some

properties

of

powder

as a continuous medium the authors of references

[4,

5, 8,

13-16] explain

some or other effects. In this

approach

the basic notion

going

back to

Reynolds

[2] is concept of compact and friable

(loose) packings

which reveal the solid and

liquid phase properties

respectively.

So, when

describing

the convective motion

[I,

I1,

26]

under the vibration De Gennes

[13]

has modernized this concept

using

the terms of active and

passive phases. During

the active

phase

of vibration the

particle

acceleration is directed

against

of

gravity force,

the

powder being

swollen.

During

the

passive

_one

particles

fall while

powder density increases,

and arrangement ^of

grains being

fixed. It results in vibration

instability

of the surface and

arising

of the convective stream directed

upward.

II 50 JOURNAL DE PHYSIQUE II N 10'

The concept ^of fluidization of

powder

under

perturbation

was earlier used

[15,

16] to

account for

shrinkage.

Outer disturbance

destroys

_a metastable state, and

powder

shrinks

trying

to minimize [3,

4]

its

potential gravity

_energy. This

explanation,

however, is _open to criticism.

Really,

if before

stirring

the

powder

has closed random

packing, practically

_any

stirring

will lead to the

swelling

of the

powder.

The

phenomenological

ideas of fluidization _» and minimization of

potential

_energy cannot describe the effect of

swelling.

The

shrinkage

and

swelling

of

powder

_were described

by

_a unified _way in the framework of another

phenomenological

model [5], where the

equilibrium

value of

density

reflects _a balance between the

amplitude

of the

perturbations

and

restoring

force such _as

gravity.

In order to

explain

the effect of size

segregation [3,

12, 14,

25]

it is maintained in

[15]

that the resistance to the motion

upward

of _a

large particle

is less then _one to motion down. This results in «

coming

to the surface

large grains

under the accidental

perturbation.

However the

question

_: how does the resistance

depend

_on the relative size of

grains

and their _mass ?

has _{no answer} in the

phenomenological approach.

A brief _survey of basic studies shows that the

phenomenological

_way

gives

_a

fragmentary picture

in which _numerous

hypotheses

_are introduced _to

explain

various effects. It _seems that this way doesn't allow the

macroscopic properties

of free

flowing

medium to be united with the

geometric

characteristics of

packing.

A third

microscopic approach

to the

powder

is

nowadays developing.

The authors

[10]

computer-simulated

the

phenomenon

of

segregation

in _a mixture of

particles

of two different

(large

and

small)

^sizes in

shaking

and

explained

it _as follows _: the

large particle

is moved

upward

_as smaller

particles

fill voids beneath it. In order for _a

large particle

to _move back down _a

large

void must open beneath it. Because several small

particle

must move

simultaneously

in order to create such

large

void _appearance of _a such

large

void is

relatively unlikely,

and thus the

larger particle experience

a net

upward

motion

[10].

This

explanation

_assumes the

perturbation

to be

enough powerful. Really,

for the

large particle

to _move

upward (to

the

right,

to the

left)

under _a weak

perturbation

it is _necessary

that several smaller

particles

should create _a

large

void _over it

(to

the

right,

to the

left),

that is

unlikely

either. Moreover, the process of

segregation depends

_on the relative _masses of

grains [3,

12, 14,

25].

For

example

the Brazil nut »

(in

the term of

[10])

will be _on the top ^{but the} steel ball of the _same size will sink. In the

procedure [10]

_«mass is not

important

for

segregation

process »,

The

original microscopic theory

_was

developed by

Edwards and coworkers

[6-9],

who used the ideas and methods of statistical mechanics _as

applied

to

powder physics.

The

hypothesis

about

equal probability

^of realization of the _same

density

states in the ensemble of

powders

formed

by

the _same extensive

manipulation

is taken _{as a} basis (2) ^{of their}

theory [6,

7] so to say the

powder analogy

of the microcanonical distribution

[28].

This _«

powder ergodic»

hypothesis

is not trivial and,

generally speaking,

demands _some

argumentation.

Numerous

questions

do arise _: what constant

plays

the role of the Boltzmann constant, what is it determined

by

and

dependent

_on ? How _{can one}

integrate

the states of

granular

_{system over}

phase

_space of

configurations

of

impenetrable particles

_to calculate the real

physical

parameters

through

distribution function ? The last

problem

for

powder

_as contrasted to the system of

penetrable particles (ideal gas),

which _are the main

approach

for all

problems

of statistical

physics

is

unlikely

solved

directly.

The

only problem

solved

directly

in framework of the method [6] ^{is the} one-dimensional

powder.

In the _more

complicated

two- _or three-

dimensional _cases, obtained results [6] ^became «intractable _».

(2) ^{The similar} hypothesis was put ^{forth earlier in the work} [24] devoted to mathematical analysis of random packings.

M 10 BEHAVIOR OF POWDER UNDER PERTURBATIONS _list

In the

following splendid

_papers

[8-9]

Edwards and coworkers

investigating

the

binary

mixture of

large

and small size

grains

used the

analogy

between the geometry of local

packing

and the well known

eight-vertex

model. This

approach

set forth _some

questions

too. lvhat

vertex model must

poly-granular

medium

given

in

figure

I be described

by

?

The

difficulty

of method of statistical

physics

to describe the

powder

is due to the

nonergodicity

[9] ^of the latter. In fact _any state of

powder

is metastable. In the lack of disturbance the distribution function of

grain configurations

has _no universal character but is determined

exclusively by

initial conditions. To put ^{it in other} way, it is not the initial _state

(that analysis

of random

packing)

but

only

behavior of

granular

^medium with certain initial conditions under

perturbations

^of certain

intensity obeys physical law-govemed

nature. It is

by

this _way that the

question

is put ^{forward in the} present paper.

The first part of _our work is built _{up as} follows. In section 2 the basic for _our

procedure

concept ^{of the effective surface}

(one-particle approach)

^is

introduced, equations

of

dynamics

are put ⁱⁿ

paragraph

3 the mean-field

approach

for effective surface is stated and the

simple

models of effective surface _are discussed _; sections 4, 5 and 6 _are devoted to

description

of

swelling shrinkage

^{effects and} to

origin

of convective motion under vibration. The size and

density segregation

and

density

of free

poured powder

will be discussed in second part.

2. The effective surface

(one-particle approach).

Any

model of

powder

^is to reflect two obvious

properties

the dominant roles,

first,

of the

impenetrability

of

grains (the

effect of excluded volume which prevents from both

analysis

^of

random

packing

and direct realization of Edward's

approach [6])

and,

secondly,

the role of friction forces. The form of

grains spheres, ellipsoids, polyhedrons

_as well _{as concrete} type of

perturbation, jostle, stirring, shaking,

vibration, _as it _was shown

by experiments,

don't

change

the

qualitative picture

of the effects discussed in

paragraph

I. We'll consider

spheres

and disks _as

powder particles,

their

regular (I.1-2)

and random

(1.3-4) packings being

commented _on above.

From the

microscopic point

^{of view} an outside

perturbation (for example, banding

_on the container walls in which

powder is)

reveals in transmission of momentum from

underlaying layers

to upper ones, which _are free to move.

Figure

2 shows

(certain, schematic)

_an

instantaneous

configuration

of

grains

at

stirring.

Because of mutual

impenetrability only

Zf

'~,~

,

~~$_

,

~___

,

~

'

~2 >~3

Fig. 2. An instantaneous state of powder under stirring. The _« active _» (or_« liquid-like ») phase is the

particles 3, 4, 5, 6, 12 which _are capable to move at the _moment. The _« passive _» (or _« solid-like ») phase is the grains1, 2, 7, 8-11, which haven't at the present moment any free _space for moving. Local

geometric configurations, inside which _a shift of shaded grain in respect with the surrounding ones can occur, are shown too. The tendency of shaded particle to emerge depends _on the local geometry of packing _: the more density (the less distance I between grains in _a layer), the more tendency to swell (the _more vertical component ^{of the} momentum obtained under stirring).

II 52 JOURNAL DE PHYSIQUE II N 10

particles

to be

capable

to

change

their

position

in respect ^with surrounding _ones at this instant

moment are those, which have vacant voids for

moving

at the moment.

So, having

obtained _a

knock from

particles

I _or 2

particle

3 is

capable

to

change

its

position. Having

obtained _a bit from

particles

3 _or 4

grain

5

changed

its

position. Impulse

obtained

by particle

7 from

particle

6

(or by particle

it from

particle 12)

Mill _not result in _any

change

of

position

of the last but will pass ^to

particles

8 _or 9

(or

to

particle13).

The

suggested picture

is similar to

Reynolds' [I]

and De Gennes's

[13]

ideas

already

mentioned in the introduction. In _our

description

active _or

«liquidlike» phase ~particles3-6, 12)

and

passive

«solidlike» _one

(particles

1,

2,

7-9,

11)

coexist in the bulk

simultaneously

but _are

arranged

in different

points

of the volume.

Let _us introduce _a coordinate system x;, the axis _xi

being

^directed

vertically upward.

Let _us consider _an

arbitrary

selected

particle (for example

one number

3)

and follow its

positions

in

the coordinate system ^{connected with} the container after the each

n-th,

n+lst,

n + 2d

perturbations. Figure

3 and comments to it show all the

possible

_types of behavior _: when test

particle

_goes

down,

lifts _or remains

approximately

at its

place.

In

principle

the _same

procedure

can be fulfilled for any

grain.

If while

averaging

the

procedure

_over all

grains

of medium _one of these situations take

place

it will _mean that the

powder

settles and its

density rises,

swells _or is in its

equilibrium

state under the

given perturbations.

The

displacement

vector of _some

grain

_at the n-th disturbance is

3u)~~

= r,l~~ r,l~ ^'

(2.1)

The full vector of

displacement

of the

grain

after _n disturbances is

equal

~(n)(~(0))

= ~in)

~lo)

~

( _~~(k)

(~

~)

' ' ' '

k =1 '

~

~+

^l,

^{n+/~~ ~+6~~j~+4}

~~~4 ~$/

_n+2

^{n+3~~ ~~~ ~~7}

n+6

~

~~ n+I

a)

^~

b) c)

Fig. 3. Behavior of _an arbitrary selected grain a) sinking down, b) going to the surface, c) remaining

near its place under the n-th, _{n +} lst, _{n +} 2d knocks in container walls.

and

depends

_upon the initial

position

of the

grain

^r(°). In _{contrast to} the continuous medium the

dependence

_{u,l~) upon} r1°~'

strictly speaking,

isn't smooth and differentiable. It is due _to

grains

aren't

clamped together

and _can intermix under chaotic

perturbations. So,

there exists

a random component of

displacement

_vector caused

by

the occasional _{momentum trans-}

mission in the random

packing

at the _same time with the

regular

component caused

by

the

gravity

force. Because of the

regular

_component of the

displacement

vector directed

perpendicular

to the

gravity

force

(along

the _{axes x~,} x~)

equals identically

_{to zero,} the components 3u)~~,

3u)~)

have _a

merely

accidental nature. Let the

typical

distance _on which

grains

are shifted at _a

single

weak

perturbation along

_{x~, x~}

equal

to 2

ix (.

i _3"l~~i i

=

i 3"'~~

= 2

lx1

(2.3)

M 10 BEHAVIOR OF POWDER UNDER PERTURBATIONS II 53

The value

ix depends

_on the

density

_p~ of

packing

and the

intensity

of n-th

perturbation.

It increases,

certainly,

with

diminishing

of the

density

and with

rising

of

intensity.

To smooth the

dependence

of

displacement

vector

3u)~)

_upon the

prior

coordinate

r;(~ ^' of _a

particle

let _us average it _{over a}

volume,

which contains _many

grains

and is much

more then the

typical

occasional

displacement

_x. For smoothed

by

a such _way

displacement

vector

(3u,l~~)

one can introduce the strain _{tensor at} the n-th

perturbation

If at the n-th disturbance

powder density

doesn't

change

_p~

= p~ _j and convective _streams

are absent _one obtains

j~~jn)j

=

j~~jn)~

₌

j~~jn)j

= o

11

3"i~~ii

=

ii3Ui~~ii

= 2

(x(

and all the components of the deformation tensor _are

equal

to zero. If the n-th

perturbation

leads to the

swelling ~packing down)

of

powder

_p~ » p~_ _j one has

(3u)~~)

^w0;

(3u)~~)

=

(3u)~~)

= 0

i13"l~~ii

=

i18"l~~ii

=

2

(x(

and the

only

nonzero component ^of strain tensor is

(3u)[~)

what

corresponds

to the uniform

tension

~pressing) along

axis _xi.

After the n-th perturbation which results in

swelling

_or

packing

down of

powder,

_a distance

dx)~~

^'~ between _two

arbitrary

selected

particles

will

equal

(dx)~~)

= l ₊ 2

3u)/~ (dx)*'~)

=

ij

+ 2

3u))~ (dx)°))

k =1

Substituting

in the

expression

the definition of the deformation tensor

(2.4)

_{one can}

easily

make _sure that

n a @~

lk)j

^~ ~(n)j

(dx)"~)

= l ₊

~

^(~

(dx)°~)

=

l ₊ ~

(dx)°))

,

(2.5)

~ j 3Xj 3Xj

here u,(") ^{is the full} vector of

displacement (2.I).

An infinitesimal volume

(dV(°))

after _n

perturbations

will turn into the volume a

j~)n)j

(dV~~~)

₌ + ~

(dV~~~) (2.6)

3Xj The volume

dV(~)

_{contains not the}

same

particles

which _were contained in

dV(°)

_before

swelling (as

it _were in continuous

medium)

but

only

the same number of

particles.

The

applicability

of formulas

(2.5-6)

is restricted

by

the finite size D of grains. Let us

consider _a horizontal

layer

of the closed random

packing ~paragraph I)

the

grains,

which

are crossed

by arbitrary

drowned

plane

_xi

= const. Over

(or under)

these

grains

the next

layer

is

disposecd-

the

grains,

which

directly

touch the

grain

of the first

layer.

The distance

IN JOURNAL DE PHYSIQUE II M 10

d

_between

_(mass center)

of the two

layers

is

naturally

to

identify

with the

layer

^thickness of the random closed

packing.

Let k

perturbations

lead in _a uniform

loosing

of closed

packing (Fig, I). According

to the formula

(2.5)

the thickness of _a

swelling layer

is

3(u)~~) -3(u)~))

~~ ~

fi

^{~ ~ ~ ~k' ~k ~}

_~~jo) ^~~'~)

value u~ = u~_j characterizes

by

what average

magnitude

the thickness of _a

layer

have increased after the

swelling, by

what distance have raised _{on average a}

grain

with respect to the

grains

of

underlaying

layer under the k-th

shaking.

Thus,

under the k-th disturbance _average

grain

_» free to move at the moment receives _a

momentum from _one of the

surrounding grains

of the

underlaying layer (Fig. 3) along

the

normal to the tangent plane of

grains

and shifts in respect with the

grains

_on tile distance u~ = u~ _j

along

_xi and _on the distance 2 x~ in the

perpendicular

^direction

(see (2.3)),

the

powder tuming

into _{a new} metastable

packing.

It is worth _{to note} that full

displacement

of _a

grain

_at the k-th

shaking

is

given by (2,I).

But most part ^{of this}

displacement 3u)~~

_a

grain

_{moves as a} whole with

surrounding

_ones. Therefore,

only

_{a part u~}

= u~_ _j,

x~-x~_j of the

displacement 3u,l~)

connected with _a relative movement in respect to

neighboring grains

leads to _a local

repacking and,

hence, to a

change

of the

density.

Let _us introduce

(Fig. 4)

the effective surface

u(x)

inside which the _« average ^motion » of the average

grain

in

regard

to the

surrounding neighbors

_occurs. The

turning

of

powder

u(x)

~n-

f

~n-

X~+

_i

U~+ U~

~kif"k+

U~

Xk-f"

Fig. 4. The effective surface inside which _« the _average motion _» of _« the average grain _» in respect to its neighbors occurs under perturbations. The turning of powder from _one rnetastable state pi _{_k} into another state pk ^at the k-th perturbation corresponds to the vaulting _over from _one point of the effective surface to another. Compare with figure 3.

from _one metastable state pk- _i into another _{p~ at the} k-th

perturbation corresponds

to the

vaulting

_over from _one

point (u~

j, x~ _j

)

of the effective surface _to another

(u~, xk) Point.

It is

impossible

for _average

particle

to come out of the

boundary

of effective

surface,

it

being

formed

by

the

surrounding impenetrable particles.

M lo BEHAVIOR OF POWDER UNDER PERTURBATIONS l155

Under _a weak

stirring

or

jostle

_a

grain

_{gets an} momentum from _one of the

neighboring grains along

the normal to the

plane

of their tangency. ^{Because of the effective surface is} created

by

the

surrounding grains

this momentum is directed

along

normal to it.

Having

received _a momentum _a

particle

_moves in the

gravity

field. In the first collision with the

surrounding having repacked grains (with

_{a new}

point

of the effective

surface)

the

grain

adheres _to them

(to

the effective

surface) losing

its kinetic energy. The last fact shows the dominant role of friction forces in the

properties

^of

powders.

The volume restricted

by

the effective surface is much less then

grain

volume

(see paragraphs

3,

4), and,

hence, under _a weak _one

perturbation

_a shift of _a

grain

in respect to

neighboring

_ones is much less then

grain

size. In order to prevent

misunderstanding

it should be noted that _our concept ^does not

assume any localization of real

grains

after _numerous disturbances. The effective surface is

situated in _a

configuration (but

not the

real)

_space. It determines the correlation of the

positions

of

neighboring grains

before and after _a

perturbation.

The form of the effective surface

being given,

it won't be difficult _to

investigate

the

behavior of the «average

grain».

Let the

particle,

^which had been in the

position

x~, u~ before n-th

perturbation,

_get the

velocity

_v~

along

the normal to the effective surface

u(x).

Then it _moves free in the

gravity

field

d~r/d~

= g

up ^to collision with the surface in the

point

_x~~

j, un~ _j. Under the next n + lst

jolting

the

grain

gets the

velocity

_vn~

j and _moves from the

position

_{x~~j, un~j} to the

position

x~~~, u~~~ and adheres there and _{so on,} and _{so on.} The

points

_{x~, u~} and x~~i,

u~~ j, x~~ _i, u~

~ j and x~~~, u~~~ are connected due to the

folloming elementary

formula of motion in the

gravity

field

U(Xn+I)~U(Xn)

"

(Xn~Xn+1) (~(Xn)) ~) _(i

₊

~

(Xn)) ~) (Xn~Xn+1)~

~

(2.8)

Here

..., n, n + I, _{n +}

2,..,

are the numbers of disturbances

fioltings, stirrings).

In formula

(2.8)

_we moved _on from the

particles

of finite size _to the

description

of their center of

gravity.

In accordance with

(2.6-7)

_a

position

of the

particle

on the effective surface determines the

density

^of

powder

_:

~

=

~~'~~(~

= l

+u~/d. _(2.9)

Pn

(dV )

Here

d

_{is the}

_layer

_{thickness of random closed}

packing,

f is the

density

of the closed

packing (1.3-4).

The transition of _a

point

to the bottom of effective surface

corresponds

to a

consolidation of

powder

dp d~

dlxl

^~°'

S<°. ^(2,io)

The minimum

(the bottom)

of the surface

corresponds

to the closed random

packing.

As to maxhnum value of _u it is defined rather

by

_{a type} of _structures stable in

gravity

^{field than}

by geometric

considerations.

The chain of

equations (2.8-9)

relates the

position

of «average

grain»

_{x~, u~ on} the effective surface and the

density p~(x~)

of

powder

^before n-th

jolting

to tile _one of

l156 JOURNAL DE PHYSIQUE II M 10

x~~ j, u~

~ j and the

density

_p~

~ j(x~

~ j

)

after it. The

equations (2.8-9) formally

solve the

problem

of the weak

perturbation

in

one-particle approach.

3, The _mean field

approach

for the effective surface.

In this

paragraph

_we

develop

the _mean

(or self-consistent)

field

theory

for the effective surface.

Any

state of

homogeneous free-floving

medium is described

by

the

position

of the

average

grain

» on the effective surface. A certain

position

_u is

corresponded by

the

given density

_p of _a

powder

in accordance ~vith formula

(2.9).

On the other

hand,

_every

density

is

corresponded by

the

packing

^which _possesses the

perfectly

certain local

geometric

character- istics,

specifically,

the certain _average

angle

of momentum which is transmitted from

underlayer

to a

grain

free to move

(Fig. 3).

For

example,

^this

angle

is marked

by

p and

equals

60° and 45° for the

plane hexagonal

and cubic

packings (Fig. I) respectively.

It means, that every

given density

_p and,

consequently,

_every given

position u(p )

of the

particle

on the effective surface _are

corresponded by

certain value of the

angle

of _momentum transmission, and, hence, the certain tangent

du/dx

of the effective surface in the

point x(p ). (Remind

that due to the definition of the effective surface the direction of _momentum

transmission is

perpendicular

to

it.) Putting

it another way, the

dependence u(x)

= u

fl _{(x) (3.1)}

is not

arbitrary

but

strictly

self-consistent.

To obtain this

dependence

let _us consider two random

packings

of different densities shown

by figure

I and put ^the

regular

_ones

(Fig. I)

of the _same

density

in accordance to these random

packings.

^It is

quite

natural to expect in the

spirit

of meanfield

theory,

that the local

geometrical

characteristics of the random

packings

and of the

corresponding regular

_ones will coincide. In the framework of this

supposition

the form of the effective surface _can be obtained _as follows.

In the

plane hexagonal packing (p

= 6

fl

^{the thickness}

^d

^of a

layer

_or of _{a row}

(for example

dashed _on in

Fig. I)

is

equal

to D

3/2,

where D is the

grain

diameter. In

decreasing density

the

particles

earlier

being

in the _{same row}

(layer)

will occupy several _rows (see

Fig.

I

and comments to

it).

The thickness of _a

layer

of _a

regular plane packing

characterized

by

the

angle

fl is D sin fl, and the number of _rows which _are

occupied by

the

particles

earlier

being

in the _same

layer

of

hexagonal packing

is

equal

to 2 _cos fl. So _a

layer

^which in the

hexagonal packing

had the thickness D

l12

_{has the thickness 2D sin fl}

cos fl after

swelling

of the

powder.

The difference between these thicknesses is

naturally

to be identified with the ordinate _u of the _« average »

grain

_on the effective surface, the ordinate

being

counted from the bottom of the effective surface which

corresponds

to the most closed

packing. According

to the concept ^{of the effective surface the}

angle

fl

equals

3u/3x.

Expressing

the

trigonometrical

functions fl

through

this derivative and

substituting

them in the difference of the thicknesses of the closed and unclosed

packings layers

_{one can} obtain the

following

differential

equation

describing

the effective surface for the

powder

of disks _:

an unessential constant

coming

to the

displacement

of the coordinate

origin being

omitted in the

right

hand of differential

equation (3.2).

The solution of the

equation

is the

following

x/D

=

~

+ ln

(1 ~)

+ 0.053

(3.3)

M 10 BEHAVIOR OF POWDER UNDER PERTURBATIONS _l157

0.83 0.637 ~

O.60

1 ₁₁

O.50

o88

O.OO 2.0O

1,oo 1.12

O.97 1.02

1 1

O.94 0.92

O.

a) b)

Fig. 5. The effective surface and dependence of the density _on the point on it in the mean-field approach for the a) plane powder of disks, b) three-dimensional _case, powder ^of spheres.

The

arbitrary

constant in the

right

hand _was chosen in such _{a way} that the bottom of the effective surface should

correspond

to the closed random two-dimensional

packing (0.821).

The form of the effective surface for

powder

of _a disks is shown in the

figure

5.

In the

regular plane packing

characterized

by

the

angle

fl the share of volume

occupied by grains equals w/4

sin fl. Therefore coordinate _x of the

particle

in the effective surface is

corresponded by

the

following density

of the

packing

p(x)=( ₍₁₊ ()

)~) (((

)~~ ^(3.4)

The values of these densities _range

(Fig. 4)

within interval 0.79-0.821.

The _same considerations _can be

applied

to the three-dimensional _case, where

swelling

of

powder

results in the

thickening

of two-dimensional

layers

^of

hexagonal

syrnrretry

(Fig.

I).

So in the closed

hexagonal

three-dimensional

packing

(p ₌ II

/)

_{the thickness}

d

_of

a

layer

is

equal

to D

~fijj.

_In

an unclosed

regular packing

the

particles,

which before

swelling occupied

a one two-dimensional

layer

of the closed

packing (row

of the

hexagonal packing

with thickness D

~fijj),

_{after the}

_swelling

_{will take}

up another _more loose and thicker

layer (number 3Dcos~fl

_of

rows

having

thickness D sin fl

everyone).

The overall thickness d~ ^{of the} new layer

consisting

of the

particles

which _were

disposed

in _one

layer

of the

hexagonal packing

_may be

easily

shown to

equal

3 D _cos fl ^~ sin

fl. Identifying

the

thickening

d~

d

_{of the}

_layer

of

swelling packing

_as

compared

with the

layer

of the closed

(hexagonal)

one with the ordinate of the «average

particle

_{» on} the effective surface _u

(see (2.7))

_we

obtain

(having expressed

the

angle

fl

through

the tangent

3u/3x)

the

following equation

for the form of the three-dimensional effective surface

~

=

31~ _l~li

₊

Ill

l~l~~~ ^(3.5)

l158 JOURNAL DE PHYSIQUE II N 10

The

arbitrary

constant connected Pith the

option

of the coordinate

origin

is omitted. The

parametric

solution of the

equation (3.5)

is

I

=

~

~

~ + 0.949 D

(I+t)/~

2 1/2 2

~=3(fi) (2-(fi)j _(3.6)

D I+t I+t

and is shown in

figure

5. The numerical constant in the

right

hand side of the first

equation (3.6)

_was chosen _so that the minimum of the effective surface should

correspond

to the random closed

packing

of

spheres (0.637).

One _can make _sure that volume share

occupied by grains

in the

regular

three-dimensional

packing

characterized

by

the

angle

fl is

w/9 /cos~

_{fl sin fl}

~particularly,

for the closed

hexagonal

_one

fl

=

I/ /

_{and the share of volume is}

_equal

_to

_w/3 /).

_{So, the}

_position

x(p

of the _« average

particle

» on the effective surface

corresponds

to the

density

of

packing

~ d~ ² 3/2 d~ -2

~ ~~~

9i~

^~ _dX dX ^{' ~~ ~~}

the

density ranging

vithin 0.52-0.637.

Formulas

(3.3-4)

_or

(3.6-7)

obtained above present ^the

examples

of _a constructive realization of the idea of the effective surface _set forth in

paragraph

2.

Comparing (3.5,

_7~

(or (3.2.4))

_one finds that formulas

(3.4, 7)

obtained from _a

purely geometric

considerations

satisfy

to and _are

only particular

case of the

general

^forrnula

^(2.9).

Let _us consider the

following simple

models of the effective surfaces

@

=a

(j(, _(3.8)

~ =(j~j~ _(3.9)

The

expression (3.8) approximates

the effective surface

(3.3)

for the

plane powder

^of disks in the whole ange of densities, _so does

expression (3.9)

for the effective surface of

spheres (3.6)

except ^of the very narrow

vicinity

of the surface bottom

corresponding

to the closed random

packing.

Note, that

according

to results

(3.3, 6)

under _a weak

perturbation grains

of

powder

_move in respect with

neighbor

_{ones on a}

magnitude

of order of

10~'

_{-1 of their diameter. That}

coincides with the results of natural

experiments.

Thus the parameter a introduced in

(3.8, 9)

have to be assumed

a l 10~ ^'

(3.10)

and

physically

allowable values of the

position

x range within 0.lD-lD.

4. Swelling ^and subsiding ~packing down) of powder under perbwbation.

The _nature of

subsiding (or swelling)

of

powders

is illustrated

by figure

3. There local

geometrical configurations (of

the type ^{of the}

configuration

of the

grains1,

2, ³ and 11, 12, 14) inside which _a relative rearrangement ^of a

particle

_can occur are shown. The direction of

an momentum transmission is _seen to

depend

_upon the distance

I,

between

grains

ⁱⁿ

packing

and,

hence,

_upon the

density

^{of the}

packing.

The

higher

^the

density

is the _more the vertical component ^of momentum obtained in

stirring

does and the _more is _{« a}

tendency

_» for

powder

N lo BEHAVIOR OF POWDER UNDER PERTURBATIONS 1159

to swell. It _means that the less

density (the higher

_a

point

on the effective

surface)

is the _more vertical the walls of the effective surface _are. It is this situation that takes

place

for the effective surfaces

(3.9)

and

(3.3, 6)

obtained in the _mean field

approach.

That is

why

under the _same outward disturbance

(under

the _same average

magnitude

of the

handing velocity

~ the behavior of

powder depends

_on its state, a friable

powder subsiding,

a compact one

swelling.

However both the last and the former _{are to} arrive at the

equilibrium

for the certain

regime

of

perturbation density

_p,

regardless

of their initial _state.

The behavior of

free-flowing

medium is described

by

the behavior of the average »

grain

on the effective surface. The behavior of the

grain

is determined

by

the chain of

equation (2.8),

which

gives

_a scheme of _one dimensional Puankare's _maps

Un+I "

Un+I(Un),

_Xn+I

"Xn+I(Xn),

_Pn+I

"

Pn+I(Pn).

Here x~, u~ is _a coordinate of the «average»

grain

on the effective surface and

p~ is the

density

of

powder corresponding

to the

point

_{x~, u~.}

The method of Poincark's _maps makes it

possible

to

investigate complicated

_many

dimensional

dynamic

_systems to which the

powders undoubtedly belong.

To

study qualitat- ively powder dynamics

we

apply

the _very illustrative

technique

of _a construction of Poincark's

map

diagrams [29].

^{For this} _purpose ^{scheme of} _maps

given

in _a

implicit

form should be rewritten in the

explicit

form.

Analytically

it _can be done for the

simple

forms of the effective surfaces

(3.8, 9).

Having

substituted the

expressions (3.8, 9)

in the recursive correlations

(2.8)

and

solving

the last in respect to +~~ _j one can obtain for the first effective surface the

following explicit

forms of the _maps

respectively

jx~~j>0, ^x~>0

x~~j =x~±

(a +a~')M~'

^~ _{x~~j <0,}

_x~<0

x~~j =x~±

(a -a~')(2M)~'± ₍₄₁₎

±~/(a-a~')~±8ax~M(2M)~'

jx~~j>0, ^x~<0

~~

x~~j <0,

x~>0,

where

A=

~~~),

^M=

^~(l+a~~),

2V ²

and for the effective surface

(3.9)

_as follows

(A-a)x(+ (A-2a)a~~x~

x~~ j ⁼ ~ ~ ,

(4.2) (A+a)x~+Aa~

where

A=

~~~),

^amA.

2V

Here V is the average

velocity

obtained

by grains during repacking

which characterize the

intensity

of

perturbation.

Typical diagrams corresponding

to the _maps

(4.1, 2)

_are shown in

figure

6. The sequence of the

points

_{n, n +} I, _{n +} 2,

(...,

k, k ₊ I, k ₊ 2, describes the

packing

down

l160 JOURNAL DE PHYSIQUE II N lo

x

~i

x

Fig. 6. The typical diagrams of _a packing down and _a swelling of powder calculated for the effective surfaces a) 3.8, b) 3.9. The _sequences of points _{n, n +} I, _{n +} 2, (... k, k ₊ I, k ₊ 2, describe a

packing down (swelling) of powder after n-th, _{n +} lst, _{n +} 2d, (k-th, k ₊ lst, k ₊ 2d, ...) stirring.

Both sequences converge ^to the fixed 2-cycle, which is corresponded by the density equilibrium for the certain intensity of _a shaking.

p P

P

~

/~

n n n+I

I n-I n n+i I n-I _n n+[

P P

~

/~ ~k+I

p ~k+[

~

~ ^k-I

~ ~

l. k-I k k+i i k-I k k+I

Fig. 7. A possible behavior of _a density under perturbations of a given intensity which is described by the maps (4.1, 2).