Dislocations in amorphous materials

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S.F. Edwards, Anita Mehta

To cite this version:

S.F. Edwards, Anita Mehta. Dislocations in amorphous materials. Journal de Physique, 1989, 50 (18), pp.2489-2503. �10.1051/jphys:0198900500180248900�. �jpa-00211076�

Dislocations in amorphous ^materials

S. F. Edwards and Anita Mehta

Cavendish Laboratory, University ^of Cambridge, Madingley Road, Cambridge ^CB3 OHE, ^G.B.

(Reçu ^{le 9 mars 1989,} accepté le 25 avril 1989)

Résumé. ²⁰¹⁴ Le but principal ^{de cet article est} de fournir une meilleure compréhension ^des

matériaux amorphes, plus particulièrement ^des systèmes ^{vitreux et} granulaires, ^en indiquant ^les

domaines où le comportement ^{de ces derniers} peut être semblable. Une attention spéciale ^est portée ^aux systèmes granulaires ^{en tant} que ^nouveau sujet ^d’étude, d’importance croissante, ^en physique ^{et nous} présentons ^{une revue des} progrès ^récents accomplis ^{dans la} compréhension ^de

leurs propriétés statiques ^et dynamiques. ^{La liaison} sous-jacente ^{entre ces} dernières, ^{en terme de} transmission de force conduisant à des flux, ^{est discutée est} des spéculations ^{sont faites sur un}

mécanisme possible de transmission.

Abstract. ²⁰¹⁴ The general aim of this paper is ^to provide ^{a better} understanding ^of amorphous materials, with special ^{reference to} glassy ^and granular systems, indicating ^{areas where the}

behaviour of these two may be similar. We lay special emphasis ^on granular systems, ^{as befits} their importance ^{as a} newly emerging ^{field of} study ⁱⁿ physics, ^and present ^{a review of recent} progress in understanding their statics and dynamics. ^The underlying connection between the latter in the form of transmission of stress leading ^to flow, ^is discussed, ^and speculations ^{are made}

as to the possible mechanism of this transmission.

Classification

Physics ^Abstracts

05.20 ^- 05.40 ^- 05.60 ^- 05.90

Introduction.

In 1964 Jacques ^Friedel

[1]

published ^{a treatise on} dislocations, which covered that subject magisterially, showing how the mechanical properties ^of crystalline solids could be explained

and tracing back the way in which the electronic ^structure of metals was related to their mechanical properties using ^such concepts. ^In writing in honour of Professor Friedel, ^we thought it would be interesting ^to look forward to new fields where the concepts required ^are

still being ^{worked out and one} day may reach the elegance achieved in Friedel’s book.

One is often struck in physics by the fact that a particular ^{class of} problems is resolved by

certain concepts ^which clearly ^{do not} apply ^{in a related} problem. ^For example, ^{one can solve}

the problem of electrons in ^a metal

(a

subject with massive and

illuminating

contributions from

Friedel)

^{and hence see} why ^a metal is cohesive. But there are other crystalline ^solids

which are manifestly ^{not metals, but which are} cohesive and with remarkably ^similar binding energies ^to metals. The general principle ^{is that} something ^{must stick the atoms} together, ^but

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180248900

it can be the metallic, ionic, ^{or covalent} bonds, ^or indeed the Van der Waals or hydrogen

bonds at a weaker level.

Now in metals, ^for example, ^the strength ^{of the} material, ^{and its} plastic flow, depend ^{on the}

dislocations in the metal. It depends ^{on how} easily they ^are created and destroyed, ^what

energy threshold is

required

^{to move} them, ^{and so on.}

Strength

and flow arise however in

non-crystalline materials such as glasses ^{as well as in} granular systems. ^{There must be a} way of

sustaining ^{stress in these} materials, ^{and a} way in which they flow. The manifestations of small deformation elasticity ^{such as} compressibility ^can obviously be tackled by the behaviour of forces due directly ^{to the atoms or molecules.}

But the general ^{class of} amorphous materials is much wider than that of crystals, ^and

dislocation theory ^{must be a} particular instance of a more general way of looking ^{at stress}

transmission and stress crisis in the substance. Sometimes it is clear from the nature of the material how stress is transmitted and the physical circumstances are seen to be very ^remote from crystalline solids. For example margarine ^is typical ^{of a mixed} system ^which consists of

crystals ^of lipid surrounded by oil. The fat crystals ^{tend to stick to one} another, ^{so one can see}

two limiting regimes. ^{The first}

(Fig. 1)

^{is when one has a low} population density ^{of fat} crystals,

which tend to link up ^to form chains ; the second is when one has a well connected network

(Fig. 2)

^of crystals permeated by ^oil.

Fig. 1. Fig. 2.

Fig. ^{1. -} Configuration showing ^low population density ^of lipid crystals ^{in oil.}

Fig. ^{2. -} Configuration showing high population density ^of lipid crystals ^{in oil.}

It is obvious that the second of these cases represents margarine in its bulk solid state, whereas the first would behave like a highly ^viscous liquid. Clearly ^{one can} arrange the behaviour of the material empirically ^{to suit} the needs of the consumer who wishes to apply ^so

much force to his knife to

spread

^the margarine ^{on his bread ; but one} also should be able to calculate the strength directly. ^This example ^{is remote from} crystalline systems. ^An apparently ^{more similar} system would be that of amorphous ^{silica, common} glass, which looks like crystalline material and has ^a very similar strength ^{to covalent or ionic} crystals. ^However

when one starts to consider this problem ^{in detail,} it has very different features, ^{and shares} difficulties with the mathematical description ^of granular systems. ^{We shall} study ^these problems ^here by reviewing progress that has already ^{been made, and} by offering ^{some new}

ideas and speculations ^{of our own.}

1. The description ^of amorphous ^materials.

Although ^our primary interest is in glassy systems, ^the problem ^{of a} powder ^{is a} good ^{one to}

consider for it removes any temptation ^{to use} quasi-crystalline descriptions. ^The ergodic

theorem is normally thought ^{of as a} consequence of the dynamics ^{of atoms, but it is} important

to realise that any operation ^{which acts} macroscopically

(i.e.

^{not on} individual

entities),

^will

result in conditions which can be called ergodic. ^Thus given ^a powder, operations ^{such as} shaking, stirring ^or rolling, ^{do not} discriminate between individual grains and result in a

material which has a minimal specification. ^For example, close-packed spheres ^{have an}

average coordination of twelve, but this value can almost reach thirteen. Thus a demon can construct a solid whose density ^{is, at least}

initially,

higher ^than close-packed. Maybe ^a patient

demon could do this on a large ^{scale also, but if a} volume of hard spheres is treated by shaking

or stirring ^or rolling, ^{one achieves a} density which is below face-centred cubic, and which lies between two well defined limits, ^a maximum random packing, ^{and a} minimum. For a powder

of spheres ^{i.e. a} system ^where temperature ^is unimportant, ^{but one} for which the Van der Waals forces are minor compared ^{to the stresses which are} easily transmitted from outside - the kind of powder ^one knows well in everyday ^{life - the} system clearly ^{has a} total number of

particles N, ^{a volume} V, ^{and an} entropy S ^{but no} temperature T, total energy E, ^{and free}

energy F. The question then arises as to how one describes it. The same problem appears in

principle ⁱⁿ glasses ^{when one} is well below

Tg,

^{the glass} transition temperature.

We can argue

[2-4]

^{that the matter} is resolved by ^{a table of} analogies. ^For ordinary

statistical thermodynamics, ^the system is conservative and has a Hamiltonian H such that all states with E ⁼ H are equally probable. Thus in this the microcanonical ensemble, ^the probability distirbution is

where the entropy S is the normalization got ^from realising that because

we must have

The transition to the more useful canonical ensemble comes from defining temperature ^{T and} free energy F via

In a granular system where the conventional definition of E does not constitute a key thermodynamic quantity, ^we suggest

[2,

3,

4]

^{that the} key quantity ^must be the volume V and that there must be a function W which gives the volume in terms of the positions ^and

orientations of the constituent grains ^{of the} powder, ^or molecules of the glass. ^{Thus we can}

define

where À, ^the analogue of Boltzmann’s constant, is such that S has the dimension of volume.

From this one can define a compactivity

and an effective volume Y by

Now at this point the reader might complain that this is a very elaborate way of writing

down the well-known concept ^{of free} volume, ^{but to our mind the current} literature lacks the

precision of these formulae which are direct analogues ^of statistical mechanics. Having ^thus

set out the basic formalism of our statistical mechanical approach ^to powders, ^{we can now} proceed ^to apply ^{it to somewhat more} realistic situations.

We reproduce here the model calculation of Edwards

[2]

^to illustrate the spirit of the above

approach ^{- the} problem of the mixture of two grains ^is mapped ^{onto the} Bragg-Williams problem of the A-B alloy, ^{and we seek a} solution for the variation of the domains of A, B and A-B with respect ^{to the} compactivity X ^{and the} degree ^to which the grains aggregate preferentially among themselves. We model the W function so as to reproduce the crudest

possible assumption ^- less volume is ^« wasted ^» on average if grains ^{of the same} size cluster

together ^{than if} grains of different sizes do :

with

niA(B) being 1(0)

depending ^{on whether an}

A(B)

grain ^is present

(absent).

^{This is} mapped

as usual onto an Ising ^model

with J, ^the exchange given by

and ai = ^± 1

depending

^on whether site i is occupied by ^{an A or a B atom. The mean field}

solution of Bragg and Williams gives

[2]

^{as usual}

with z as the coordination number of the lattice. Thus for

J/AX

^{1, the two} powders ^are totally ^{miscible, but as}

J/AX >

1, ^the

powders

^{tend to have} unequal mixed domains until at

X - 0, the material separates into domains of pure A and pure B.

We see from the above that even this relatively ^crude modelling reproduces ^a qualitative

feature of powder mixtures that one observes in nature : however a more detailed analysis

[3]

gives ^{one the two ordered states} observed in nature, where in addition to the so-called

« ferromagnetic ^{» one} gets ^{a «} stacking ^» solution, i.e., ^{one where a «} layered ^» arrangement of the grains ^is preferred ^so that each A rests on a base of B and vice versa. This can be done

[3]

by mapping ^the binary ^{mixture onto the} eight-vertex ^{model of} spins, and the results thus obtained show good agreement with results of computer simulation experiments

[5].

Thus despite ^{the crude nature of some of our} assumptions, e.g. those involved in neglecting

the non-lattice-based aspects ^{of such} granular systems, the above shows in our view the

underlying strengths ^{of our} statistical mechanical approach ^to the statics of these systems. ^In the next section, ^{we will} attempt ^to relate this framework to the transmission of stress in such systems, ^{and to the} consequent flow, ^once resistance to motion

(e.g.

in the form of

friction)

has been overcome.

2. Statics and dynamics.

We have argued that there are two limits to the natural density ^of packed amorphous

material. Clearly ^{if the} material is at its maximum density in the presence of boundaries, ^{it can}

transmit stress but it cannot flow. Thus if the compactivity parameter X ⁼ 0 there is no flow and the « viscosity » of the material appears ^to be infinite. The resistance to flow then decreases until X = 00 which is the lowest solid density. After that the material has the attributes of ^a liquid ^{and is} outside this discussion. The compactivity ^{has an} average value

across the materials, but will fluctuate as will the effective volume Y and it will be these fluctuations which permit ^{flow. Of course} all that is being said is that some configurations permit rearrangement, ^{and some do not. It can} be shown that the coordination of a particle ^is

a good criterion for discussing the condition of an amorphous ^material and the minimum free volume comes with the maximum coordination which has X ⁼ 0, whereas the lowést density

comes when all

(meta)stable

coordinations are equally likely and X = 00. This gives ^{a useful}

way ^to think of the cooling ^{of a} glass, ^{for one can} summarize well known facts about cooling ⁱⁿ

the symbolic diagrams that constitute figures ^{3 and 4.}

Fig. 3. Fig. 4.

Fig. 3. - Diffusion as a function of temperature for various cooling ^rates.

Fig. ^{4. -} Density ^{as a} function of temperature for different cooling ^rates.

The time taken to reach

Tg

^{of the glass}

^depends

^{on the cooling rate.} The fastest rate brings

one to the highest

Tg, Tg max,

the slowest to the

Tg, Tg

^{min. For a thermal system such as a glass,}

there will now be slow processes that will take the system down from

Tg max

^to

Tg

^{min - for a powder,} there exist analogues ^to these slow relaxation processes ^on which work is currently in progress

[6].

In general ^{these are} complex ^and highly cooperative, ^but when, ^{as in the case of a} glass, they ^are thermally driven it is well known that they ^are governed by ^a simple law, variously

known as the law of Vogel and Fulcher or Doolittle or Williams, Landel and Ferry

[7].

^{The lax}

is that there is an essential singularity in the run-up ^to

Tg,

^{i.e. a} relaxation time r such that

A simple derivation of this law has been given by Edwards and Vilgis

[8]

who argue, in the current language, ^{that an excess at one}

point

waits until a reduction occurs at an adjacent point. This reduction itself depends ^on another reduction. It is

illuminating

^to discuss this in the language of barriers ^- if ^a is the probability ^{that a} barrier obstructs the motion of the

« excess », i.e. the particles ^under consideration, ^then clearly ^{the time taken} by the latter to move will be increased by ^{a factor} involving ^{a. To be more} precise : ^if Do is the free diffusion

coefficient, and D that in the presence of barriers, ^then

Do

^{1 and}

^D-1,

^the corresponding

diffusion times, ^must be related by ^{a series in a -} this is because if there is only ^{one barrier,}

the time taken is increased by a ; ^{if there are two} barriers, ^{one of which can} only ^{move if the}

other one does so first, then the time will be increased by

a 2 ;

^{and so on.} Thus in the mean

field approximation ^{we can} write,

or

and ^{a =} 1 corresponds ^{to the} glassified state, i.e. where the barriers ^are frozen in so that ^no further diffusion occurs. This is the so-called glass transition. The above was derived under the assumption that each barrier moves independently of the others ^- however when a

cooperative ^motion of the barriers _occurs, so that e.g. three barriers ^move around in ^a loop ^at

the same time, the motion is further modified, and the series is, with this correction

[8]

where _{a 1} is the weight ^{of a closed} loop. ^{This can} be evaluated to give

with B a constant ^- on recognising ^that

D-1/r,

^this gives ^{us back}

(18),

^the Vogel-Fulcher

(V-F)

^form.

This calculation becomes rigorous ^for systems ^{where mean field} analysis ^is rigorous e.g. for

long rod molecules moving ^{in an} entanglement ^{tube, but the} universality of the V-F form suggests, ^as is often the case, that mean field arguments ^are much better than one might expect. Indeed if the V-F is a mean field formula, ^more subtle versions can be expected ^{to fit} experiment ^{a little} better, and formulae like

exp - A / (T - Tg) (1 + f3)

^are found in the literature. If the relaxation time has the V-F form, it allows us to consider the set of curves in

figures 3 and 4 for then one can argue that if the cooling ^rate exceeds the relaxation rate of the

cooling

liquid,

^the liquid glassifies. ^{Thus of}

Tgo

^{is the glass} temperature

(i.e.

the minimum

Tg

^{of our}

^diagrams)

^and

Tg

^{the glass} temperature found with a cooling ^rate

d1J/dt ==

^R, say, then

so that for R ⁼ 0,

Tg

⁼

T gO.

The formula has no validity for R >

8 -

^{1 for that} gives

Tg

^{= oo} and in fact there is a maximum

7g

corresponding ^{to X =} oo, after which point ^a glass

cannot occur as it would not be connected. Clearly ^{a more elaborate} argument ^is required ^but

formula

(23)

does fit much data quite well. Thus ^a

quite

reasonable first approximation

emerges for which ^a detailed argument

[9]

^{will be} published ^later.

The picture ^{so far has} only ^{talked of «} fluctuations » and not attempted ^to quantify ^their

size. Clearly there will be a spread ^of sizes, ^{and the} relaxation will depend ^{on this} spread. ^For example if the relaxation time T is really

7 ( A )

^{such that}

our crudest analysis above will relate

top 7- > .

^But the actual formulae should use

T (À )

^{and be} averaged ^{in some correct} way. In particular ^a correlation function will decay ^like

not like e-’/’. A simple example, ^{when T - e À’, and P is} like p

exp ( v À b),

gives ^on using steepest ^{descents to solve}

(25)

^for large ^times,

the stretched exponential. The forms of P and T require ^{models of} course, which must be

sufficiently complicated ^to contain the indices a and b.

Much of the above discussion has concentrated on glasses ; ^the dynamics ^of granular systems ^{is much more} complicated although ^{it is our view}

[6]

^{that some of the} underlying principles ^here might ^have analogies with those that hold in glassy systems. ^{In order to} indicate where these resemblances may lie, ^we will review in the next section what is known about granular dynamics, and include some brief remarks about our work in this area at

present. Finally ^{in section 4, we} will include some speculations about the transmission of stress in these systems ^and try ^{to see whether or not the} analogue ^{of a} dislocation can be the

operative mechanism for stress propagation ⁱⁿ amorphous systems.

3. Dynamics ^of granular systems.

The dynamics ^of granular systems, ^or powder ^{mechanics as it was}

traditionally

known, ^has been extensively ^studied by ^chemical engineers

[10, 11] ;

^however physicists have turned their attention to it only recently, ^so that while many ^current experimental configurations

[12, 13]

(when

shorn of their digitised

sophistication)

^are remarkably ^{similar to} those studied earlier

[10, 11],

^{the nature of the}

questions

addressed reflects the different approaches ^{of the two} disciplines. It will be our purpose in this section first ^to sketch the development of this difficult and fascinating field, ^{and then to} pose the questions ^{that are} currently being addressed, giving

indications of possible ^answers where available.

To start with, ^we lay ^{out three} principles ^of powder ^mechanics

[10]

^{- as} early ^{as 1885,}

Reynolds [14]

observed that « a

tightly

packed ^{mass of} granules inclosed within a flexible

envelope invariably ^increases in volume when the envelope is deformed : if the envelope ^is

inextensible but not inflexible, ^no deformation is possible ^{until the} applied forces-rupture ^the

bag

^or fracture the granules » - this expresses the fundamental principle of dilatancy ^of powders. ^The application ^{of shear stress then causes erstwhile} granular contacts to slacken : and surfaces of sliding, permitting ^relative displacements ^of granules, ^are thereby ^formed.

The

principle of

mobilisation of friction ^{then states} that the frictional force between any ^two

grains ^{in a} powder ^{at rest can} take any value between ^zero and ^some threshold value for the onset of relative motion - thus the stress distribution in a powder ^{at rest is} indeterminate. If the frictional force due to shear of a powder reaches its limiting value, ^a surface of sliding ^is

formed. Another consequence of this principle ^{is the} occurrence of a range of equilibrium

states, hence of bulk densities and angles of repose. As the powder begins ^to flow, however,

the grains ^are constantly rearranged - for conditions of steady flow e.g. through apertures, ^it

seems that there must be a restriction on the previous condition. This then leads to the

principle of minimisation of energy, which ^states that under such conditions, energy is minimised in a well-defined manner thus enabling ^{one to} calculate flow rates at apertures, ^for instance.

It was realised before

[10]

^{that to cause a} powder ^{to flow, it had to be} brought ^{into a state of} ready sliding, ^{or to a} so-called « critical voids ratio » ^- although ^the identity ^{of these} descriptions ^has yet ^{to be} rigorously proven, it ^seems reasonable to conjecture ^{that this must}

be so, and this is indeed the hypothesis ^{that we} have made in an earlier section. One of the chief questions ^{then at this} stage, ^{concerns the onset of flow - in a} system with open boundaries, ^{is there a critical} compactivity

(or

^more precisely, ^{when one} recognises ^the possible first-order nature

[13]

^{of this} transition and takes into account the principle ^of

mobilisation of friction, ^{is there a} small range of critical

compactivities)

^at which flow will

begin ? Also, ^{once flow has} begun, ^{is there some} simple way in which it ^can be characterised ? We have made some preliminary attempts ^{to answer these} questions ^{based on the} theory ^of generalised avalanche processes, in work which is ^to be published shortly

[15].

^An alternative and closely related way is ^{to state} that what one really ^{needs is a} good microscopic theory ^of

surfaces of sliding, ^which separate rapidly moving ^from nearly stationary material - this underlines the need for ^an ab-initio approach ^{based on} dynamical equations ^{of motion, which}

are the subject ^{of some of our current work}

[16].

A thorough ^and comprehensive ^account of many experiments dealing ^{with the} dynamics ^of granular piles ^{is to} be found in the excellent book by ^Brown and Richards

[10].

^{We choose}

however to focus on an experiment ^{which has been the} motivation for several experiments ⁱⁿ

the recent past, ^{of which a} great variety ^of questions ^has been asked - this will illustrate the

point made in the opening paragraph ^{of this section, as well as serve to link us with the state of}

the art, ^{as it} were, ^at present.

The so-called rotating cylinder ^method of Franklin and Johanson

[17]

^{was used} by ^{them to}

measure the instantaneous surface angle ^{of a} granular pile ^that partially ^{filled a} rotating ^drum.

We quote, for historical interest, and for purposes of comparison ^{with current} experiments

the account given ^{in reference}

[10] :

« At very low rotation speeds the material surged ^{as it}

was raised beyond its maximum angle of repose, subsequently collapsing ^{to a low} angle.

Increasing ^the speed of rotation increased the frequency ^of surging ^{until, at} about 2.5- 3 r.p.m., the surface became substantially steady,

exhibiting

only ^minor ripples

(Fig. 5).

^At higher speeds

(but

well below the critical speed ^at which material becomes centrifuged ^{to the}

curved

surface)

the free surface assumed an elongated S-shape... ^»

Fig. ^{5. - Free} surface of granular pile ⁱⁿ rotating cylinder experiment ^{above a threshold} frequency ^of

rotation.

This served as a clear signal ^{to modern} experimenters of the appearance of instabilities in the free surface of granular piles when these were submitted to enforced motion ; equally ^the

« surges »

(or

avalanches, ^{in more modem}

parlance)

^{seemed to be} clearly worthy ^of study.

Before proceeding ^{to describe recent} studies of these matters, ^we _{pose the} questions ^{that even}

this sketchy ^{account would arouse : how,} for instance would the free surface of a granular pile

look as a function of different frequencies ^and amplitudes of rotation and/or vibration ? What, if any, of the experimental parameters, e.g. the aspect ^{ratio and} shape ^{of the} cell, size, shape ^{and texture} of the beads ^or the nature of the surrounding medium, might influence this transition ? Could one explain theoretically ^{the nature} of the instabilities ? Can one

understand theoretically ^{the onset of an} avalanche ? Could one measure the frequency spectrum ^{of these} avalanches, and understand them, ^{at least at some} crude level ?

To answer the first of these

questions,

Rajchenbach ^and Evesque

[12]

^{submitted a}

parallelepipedic

^cell partially filled with monodisperse glass spheres ^{to vertical} vibrations of

varying amplitude. Beyond ^{a threshold} amplitude, they observed that the horizontal free surface became unstable, ^and the surface spontaneously ^{assumed an} angle 0 ^{to the} horizontal ; ^a permanent ^current of avalanches was observed to flow in this regime. ^The instability appeared ^{to have a} convective nature in that the transport ^of particles ^{down the} slope ^was compensated

(Fig. 6)

^for by ^{a reverse} transport ^of particles in the bulk from the bottom to the top. ^{This state was seen to} persist ^until yet another threshold was reached,

which heralded the onset of chaotic behaviour, ^with temporal ^and spatial intermittency ^of particle ^{flow at} the free surface.

Fig. ^{6. -} Appearance ^of convective instability ^above threshold of amplitude ^of vibration ; the downward flow along ^the slope ^is compensated ^for by ^an upward flow in the bulk.