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Qualitative theory of the solid-liquid transition
Physica
1llA
(1982) 217-239
QUALITATIVE
North-Holland
THEORY
Publishing
Co.
OF THE SOLID-LIQUID
TRANSITION’
Arno HOLZ* Fachrichtung
Theoretische
Physik,
Uniuersitiit Received
des Saarlandes,
26 August
66 Saarbriicken,
W.-Germany
1981
Some qualitative attempts to the elucidation of the melting transition in solids are presented. The screening of the long range elastic interactions in the liquid phase is considered to originate from the presence of freely mobile infinitely extended dislocation loops. In contrast to other similar theories it is, however, assumed that cooperative diffusive motion (or “convective” motion) plays an equally important role in releasing any trapped in dislocation configurations which may lead to immobility of the system. The apparent universal nature of the discontinuity of the melting transition is studied on the basis of these concepts via possible nucleation mechanisms. The technical means used to treat dislocation configurations are combinatorics and the path integral formalism. The conclusions reached are that the universal nature of the discontinuity of the melting transition is a consequence of the strongly coupled dynamics of dislocations and cooperative diffusive motion.
1. Introduction The study
of phase
transitions
in solids in particular
structural
transitions
of
martensitic type if they occur discontinuously require the knowledge of the operating nucleation mechanism. This nucleation mechanism may usually be described as a highly exceptional but still probable path in phase space leading through a tiny pin hole into the new state. It seems intuitively obvious that an understanding of the nucleation mechanism must give in many cases an explanation of the structural and dynamic properties of the new phase. In his work
on martensitic
transformations
U. Dehlinger’)
recognized
the im-
portance of centers of internal strain for the formation of the new phase in particular dislocation configurations which allow the embedding of the new phase into the host matrix and the rearrangement of atoms into their new positions.
The problem
of nucleation
can therefore
be formulated
in terms
of
two strongly coupled questions, how is the nucleus formed and how is it embedded into the host matrix. In order to avoid any misunderstanding it should be pointed out that in phenomenological nucleation theories the embedding energy is taken into account over a surface free energy and the bulk energy of the nucleus over the chemical potential of the new phase. This approach presupposes ad hoc the bulk phase in the nucleus whereas it is clear ’ Supported * To Professor
in part by Deutsche Ulrich Dehlinger
0378-4371/82/0000-0000/$02.75
Forschungsgemeinschaft under on the occasion of his eightieth
@ 1982 North-Holland
SFB 130. birthday.
A. HOI>Z
?I8
that also that part of the nucleus further surface.
growth
of the nucleus
It is emphasized
properties
that
of this activation
has to climb occurs
over
we want
barrier
over an activation structural
to study
the structural
or say free energy
barrier
transformations barrier
until at its
and dynamic for the melting
transition. In order
to put the problem
of melting
into the appropriate
context
;I short
review of the development of ideas in this field will be given. The idea that a crystal melts because it will loose its inherent resistance to shear goes back to a suggestion of Boltzmann’). Essentially it is proposed that the shear modulus C11-C,2 vanishes at the melting temperature. Because the melting transition is strictly observed to be of first order and such a hypothesis implies a continuous transition this point of view has been abandoned by, most people working in this field. The idea that melting
originates
to Frank’) and attempts to describe a liquid state model were initiated
through
the generation
of defects
goes back
how such defects may be incorporated by Mott and Gurney4) in their work
in on
microcrystalline arrays. Early attempts to describe melting as dislocation originated and the liquid state as dislocation saturated have been critisized by specialists working in this field and have not been included in the review on dislocation theory by Seeger’). The reason for that is very simple. First up to now no thermal dislocation loop formation as a premelting phenomena has been observed experimentally, is that strong that the Burgers
and second the damage of good area in liquids circui?) is only badly defined if it is defined at
all. Work on dislocation like structure of liquids was initiated by MacKenzie and Mott’) and Mott’), and led over the works of Rothstein’), Mizushima’), Ookawa”), Siol”) to the theory of melting by Kuhlmann-Wilsdorf”). In the latter work it was conjectured that melting occurs when the free energy of dislocations gets negative leading over spontaneous formation of dislocations discontinuously into a dislocation saturated liquid. The essential point of her derivation is that she calculates the free energy of the dislocation state over the vibrational
entropy
of localized
and extended
lattice
modes
in contrast
to
later work where the conformational entropy of dislocations is calculated similar as in polymer physics. The nature of the dislocation like structure of liquids has been elucidated by Cotterill”). The most recent study of the solid-liquid transition and the liquid state has been done by Edwards and Warner14). These authors treat the problem according to the principles of polymer statistics but take also long range interactions into account. Their treatment approximates the rather awkward Biot-Savart type interactions between dislocations by averaging over the Burgers vectors. In this way the problem is mapped onto the simpler problem of vortex loops characterized by current strengths. It is then stated that the
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
219
discontinuous transition is a consequence of the exponential screening of long range interactions in the liquid similar as it is the case in metal insulator phase transitions. In an earlier paper by the author”) on the phase transition in the planar rotator model and where vortex loops are operative in driving the phase transition it has already been pointed out that the screening mechanism advocated by Edwards and Warner14), does not necessarily lead to discontinuous melting. The crucial point is very simple. The planar rotator model is known to undergo a continuous phase transition and Edwards and Warner14) have mapped their system onto that model modulo some irrelevant features. The latter are related to the range of the short range interaction and which may influence the dynamics of the loop motion and in addition to packing effects. Volume expansion during the transition, however, is ignored by these authors. Let us point out that volume expansion is very often observed during melting transitions but not always. For instance it is observed that for alkali metals under strong hydrostatic pressure a discontinuous entropy change is accompanied by a smooth volume changei6). Apparently the electronic system in such metals is restructured in such a manner that a liquid state without net volume change is possible. Because the electronic system in Li and Na does not undergo a phase transition during the melting transition (it is metallic and remains metallic although correlations may change as it is the case in K, Rb and Cs) its response to the reordering during melting must be smooth. Under these conditions, however, it cannot be explained why the discontinuous melting transition without volume change does not take place between two temperatures Tr and T2 required to drive the system through the coexistence region if discontinuous volume change really would be prerequisite for discontinuous melting. A consequence of volume expansion in most systems under ordinary conditions is, however, that melting starts at the free surface of the system as is well known and as will be discussed later. As a matter of fact volume expansion at the solid-liquid transition is a consequence of the asymmetry of the atomistic interaction forces and which favors energetically the formation of vacancies with respect to interstitials. Because, as will be explained in the sequel, a liquid state is not possible without the operation of nonconservative processes and which can be considered as mainly vacancy mediated volume expansion under ordinary conditions is a general rule. Under strong hydrostatic pressure successive delocalization of the electron shells results until eventually a Wigner crystal is formed. In this state the interaction forces between the nuclei are essentially symmetric leading essentially to no volume expansion for vacancy interstitial pairs. The importance of the experiments reported in ref. 16 is therefore that they show that with increased delocalization of the shell electrons the volume dis-
A. HOLZ
220
continuity
flattens
out but still a first order
sion can therefore
safely
As an introduction following reach
section
a state
In section reminiscent
be considered
to the problem the
problem
extension
via energetically
of infinite the
of melting
combinatorical
3 a path integral of
transition
formulation
approach
polymers. however,
The constraints the conservation
observed expression
by just
giving
of
occurs.
as a secondary
we will first
how
favorable
imposed of matter
to
be satisfied.
Here
in the
loop
may
conformations.
will be given
constrained
which
systems
is of
on the system in the present case are, in the system which is not automatically
an arbitrary
set of dislocation
loops
because
. r” X dr”) = AN must
expan-
study
a dislocation
of the problem Edwards”)
Volume
effect.
the
(1)
b” is a Burgersvector
and the integral
and sum ran
over the whole dislocation system and A is the volume of the unit cell. The right-hand side of eq. (1) represents the number of unpaired interstitials or vacancies depending on AN being positive or negative respectively. In section 4 the consequences of this constraint will be studied in two simple models. Another constraint imposed on the system of dislocations is related to the concept of curvature and torsion’8) of the dislocated state and reflects metrical properties of the path integrals. In section 5 we draw the conclusions from the results reached
itself in the so far and
propose the idea that melting must be discontinuous as a consequence of the strong interaction between cooperative conservative and nonconservative motion in the liquid. An excellent introduction to functional integration methods can be found in refs 19. 20 and 21.
2. Dipolar conformations Thermal
generation
of dislocations
of dislocation
loops as a premelting
phenomena
have so
far not been observed experimentally. At first glance therefore the attempt to develop a statistical theory of dislocation loops should be doomed to fail. There are at least two reasons why this will not be the case. The first is that one wants to know why it does not work but would perhaps become operative at higher temperature if energetically more favorable nucleation mechanisms would be inactivated, for instance by depriving the system of free surfaces etc. The second reason is that a nucleation mechanism may be a rather complex evolutionary type process leading over a steep activation barrier into the new phase such that its presence is only reflected in the short wave length
QUALITATIVE
THEORY OF SOLID-LIQUID
TRANSITION
221
and high frequency domain of the structure factor, where it is difficult to detect. It seems therefore not completely unrealistic to devote ones attention to the following problem. Given a dislocation loop of N links, where one link corresponds roughly to one lattice unit, a core energy of the order of yc per link and the usual Biot-Savart type interaction between different loop parts; find the free energy of such a loop as a function of temperature disregarding its translational degrees of freedom. If one assumes that the loop is generated over a random walk process having at each step the choice of about z possibilities to continue (in three dimensions one gets roughly .z = 5) then the free energy of the loop is bounded from below by F rooP2 N(-yc - kT In 2).
(2)
On the right-hand side of eq. (2) long range interactions have been neglected thus leading to the lower bound. From eq. (2) one obtains an estimate of the melting temperature in the form kT, 2 y&n z.
(3)
Choosing appropriate values for yc and z and making a reasonable estimate on the packing of dislocation cores in the liquid allows to estimate the melting temperature and latent heat of simple solids in a manner similar to Kuhlmann-Wilsdorf”). As a matter of fact this approach is completely equivalent to hers. It is obvious that the estimate represented by eq. (2) is good only for N 9 1. For finite N, eq. (2) should be substituted by L,P 2 N(y(N,
T) - kT In z(N, T)),
(2’)
where y(N, T) 2 yc and z(N, T) c z should hold. The first inequality is a consequence of the fact that the long range Biot-Savart type interaction between loop parts on the average will increase the energy per link of the loop. The second inequality originates from the condition that the formation process of the loop starts from small size. Consequently it cannot avoid crossing itself when trying to reach its possible conformations, and many possible conformations within the random walk model are not reached due to unsurpassable obstacles or due to trapped in conformations. Unsurpassable obstacles must be considered as topological constraints as represented e.g. by eq. (1) and trapped in conformations such as considered in the theory of crystal plasticity**). Observing therefore for finite N that y(N, T)/ln z(N, T) * r&r z
(4)
possibly holds it is very unlikely that thermal formation of dislocation loops in
222
A. HOLZ
solids will occurs.
be observed
and
necessarily
a discontinuous
melting
transition
Unfortunately, however, eq. (4) is not easily justified as far as its dependence on y(N, T) is concerned. In ref. 15 the author has studied this problem in the context of the order-disorder transition in the 3-dimensional planar rotator model where vortex loops interacting via the Biot-Savart law drive the phase transition. Because the long range interactions &(y(N. holds
for
T)IkT the
this phase transition is continuous the screening with increasing N must be such that
- In z(N, T)) 2 0,
planar
rotator
model
of
for all N, and T < T,, and
also
for
the
(5)
dislocation
loop
ap-
proximation if all special solid state effects on z(N, T) as stated above are ignored and only y(N, T) is considered. Although eq. (5) may follow from the knowledge of the exact result it is not easily derived as shown by the discussion presented in ref. IS. In order to demonstrate the successive build up of screening of long range interaction with increasing N one may develop the following rather crude model. For small N one uses dipolar conformations of loops, i.e., double folded loops as depicted in fig. 1. If the energy per double link is assumed to be Yd 2 2-y, then according to F%,, b F(2rC
the free
energy
of this
loop
may
be bounded
from
below
- kT In z).
(6)
Because eqs. (2) and (6) are bounds for the asymptotic behavior of single loops in the small and large N limit the free energy of the loop should be bounded from below as depicted in fig. 2. The dashed part of the estimated free energy studied
bound
is an interpolation
in the appendix
over
between
the tree
the asymptotic
like structures
behaviors
of dipolar
and is
conformed
b
Fig.
1. Dipolar
structure
conformation
is the reference
of a dislocation
loop with
graph used in the appendix.
Burgers
vector
b. The dashed tree like
QUALITATIVE
THEORY OF SOLID-LIQUID
TRANSITION
>> 1 saddle
’
’ Loop
extension
223
point
N
Fig. 2. The curve bordering the hatched area is the lower bound of the free energy Fluupof a loop of N links. I marks the inflection point and AE represents the activation barrier for a transition from the large N- to the small N-saddle point. The dashed part of the curve is interpolated. The dash pointed curve with slope r’E- kT In z applies to high density renormalization of core energy yC. This schematic drawing applies to T < T,.
loops as depicted in fig. 1. Although it follows from these calculations that the magnitude of the tree like structures readily increases it seems that the magnitude of stochastically conformed loops is approached only by a logarithmic law if it approaches it at all, a question which cannot be answered on account of the approximative treatment given in the appendix. A quantitative estimate of the height of the activation barrier AE has also not been achieved. In section 4 this problem will be attacked from a different point of view. Let us point out that if 7: = y,(N * ~0,T) > y,(N = 6(l), T) is assumed then the dash pointed free energy curve shown in fig. 2 arises and no activation barrier and no inflection point is necessary. As this assumption is unfounded within the present simplified approach we will only come back to that possibility in section 4. It follows from fig. 2 and the curve bordering the hatched area that eq. (5) cannot hold for all T below T, because the free energy of the loop should display an inflection point I. Accordingly one may be inclined to postulate two saddle points of the free energy for temperatures in the interval T, < T s T,, where the large N saddle point is metastable below T,. In a similar fashion one gets two saddle points above T, with the stability properties interchanged. On account of this reasoning a first order transition should arise as has been postulated by Edwards and Warner”). The many reasons why this is not necessarily the case have been extensively discussed in ref. 15 and will not be repeated here but only two important observations will be made. The first point is that the occurrence of two saddle points will depend on
224
the
A. HOLZ
height
regimes,
of the
energy
barrier
dE
seperating
the
two
if it is smaller than kT,, it can be overcome The interpolated dashed drawn curve in fig. 2 should
fluctuations. calculated point
free
e.g.,
as can be done
using the theory
is that the asymptotic
far as the internal
energy
estimate
given
given
in the appendix.
by eq. (2) is certainly
term of the free energy
is concerned.
asymptotic by thermal therefore be The second sensible
as
This has been
demonstrated by Banks et al.“) using renormalization group methods. Intuitively it is also obvious that the orientation dependence of the Biot-Savart interactions for large and stochastically conformed loops must average out. Uncertain to a certain degree is the precise value of yC to be used in the small and larger N limits due to interactionand anharmonic effects. The entropy part of the estimate, however, may be completely wrong. The reason for that is that using for the entropy the quantity S = Nk In z presupposes that the loop is really able to move through all is certainly not necessarily the case glasses and quenched in disorder. The can therefore only be guaranteed if it exponentially the case then litude phonon
stochastically conformed shapes. This as one knows from the existence of existence of the large N saddle point can be shown that this state leads to
screened elastic interaction, i.e.. that it is liquid. If this is not it still may be a saddle point if it is stable against small ampexcitations. But in this case it is a defect state of the type ob-
tained by means of work hardening of solids or by means of the production of entangled vortex loops in the planar rotator model at low temperatures. The existence of such states does naturally give no clues whatsoever to the nature of phase transitions between thermodynamic equilibrium states in such systems. In concluding
this section
we point
out that the reasoning
presented
with
respect to the properties of the bounds of the free energies of loops in connection with fig. 2 are still not completely waterproof due to the various uncertainties related to the calculation of y(N, T) and z(N, T). More sophisticated applied
methods to deal in section 4.
with
this
problem
are developed
in section
3 and
3. Path integral formulation The
path
integral
formulation
of a dislocation
calculating the grand partition function configurations into account. The interaction represented in the form
V = f 3
r’ dS,,i
rr':i'0
7” dS,,.i,V,,,,,(r,,iL,,i; 0
problem
is equivalent
to
taking all possible dislocation of the dislocation system can be
r,,,iv; i,,,,ir),
(7)
QUALITATIVE
THEORY
OF SOLID-LIQUID
225
TRANSITION
potential between dislocations’) of Burgers V,,,# is the interaction and rO,i = r(S,,i), L,,i = dr,,i/dS,,i. Under the vector b” and b”‘, respectively condition that the following potential inversion can be solved with respect to where
V’. &R”V,+(r,
where
i; r”, S”)V,$mt(r”, i”; r’, i’) = &&‘(r
d6R” = d3r”d3r”, it can be shown
exp{-
- !$g, where
that the following
VILT} = N-’ n / S[@,] exp{-i Cr
the normalization
3
I d6R 1 d6R’@,(r, factor
N is defined
Eqs. (9) and (10) are functional integrals. problem can now be written in the form
r’)6’(i
-
identity
7’ dS,@,(r,,i, 0 L)Vib(r,
- i’),
(8)
holds:
i0.i)
$; r’i’)@Jr’,
if)],
(9)
by
The grand
partition
function
of the
z=1+$
x c Cr,u’
x~dS,idSuidS,r
d6R’oC(r,
i)V$(r,
i; r’, +‘)@,,(r’, i’)
@r’+allpossibleirred.graphsj-11. c(
(11)
where the sums extend over all possible Burgers vectors but excluding the inverse Burgers vectors. The oriented lines drawn in eq. (11) are the propagators of the theory
A. HO12
226
;, = G(r,
;G
S,,; r’, 0; @,,,)
r..s
=
1 d[r(S)Iexp( r’..S=O
-jdSl~i~~ixh”.(rxi)+i~~,(r.i)_i./~Ti}, 0
In eq. (12), f(, is the free energy Burgers
vector
(12)
per link of the core
of the dislocation
with
b”, i.e.. (13)
f(, = y,, - kT In z,,.
where y(, and z,, are defined as in eq. (2). The “kinetic” term in eq. (I?) enters over the Gaussian approximation to path integrals and mJ2 = 3/2 (see refs. 19 and 20). The K-integration in eq. (11) enters over the integral representation of eq. (1) where the right-hand side is set equal to zero for the sake of simplicity. The factors gcrBvappearing in eq. (I 1) are related to branching processes of dislocations.
If the internal
energy
of a node
is t,+
then
%JBY= exp(-•,,,/kT).
(14)
The first loop integral in eq. (I I) carries a factor l/S,, in the integrand to guarantee correct counting of graphs and E = ( (1). Let us point out that in eq. (I 1) only the constraint eq. (1) has been taken into account and besides AN = 0 has been assumed for the sake of simplicity. Constraints entering over torsicn and curvature have not been taken into account either. In order to work with eq. (1 I), eq. (8) should be solved. One observes first that V&r, r; i’, f’) is bilinear in i- and L’. For an isotropic material one can deduce from the expressions given in ref. 5 V,,,+(r,,, i,,; r,,,, r,,,) = K[-(
1 + v)(b” . fi”)(b”
+ Cc,,,(W,d)(R” where
u and /J are Lame
constants,
. a<‘)
. f2”)]i,,f,,,/lr,,
- r,,J.
(15)
K = ~/47~( 1 - U) and
c,,r,(n,,,) = b” * b”’ + (b” . n,,,,‘)(b”’ . n<,<,.), nWd = (r, - r,,,)/ir,, - rrr,/; Because integrals
the “kinetic” in the interval
rr-+3jr(rr),
k = 0 ;.
term in eq. (12) favors r-values in the functional (0, 1) it is sensible in eq. (IS) to make the substitution (16)
where j,(x) is the spherical Besselfunction of order 1. (Notice that 3j,(x < I) = x and that the firs&zero of jr(x) appears for x > rr.) Only in that case eq. (8) is sensible and can be solved. Confining oneself to an iterative solution, i.e., inverting the diagonal terms only one obtains
QUALITATIVE
THEORY OF SOLID-LIQUID
TRANSITION
d3qV;;,(q; r; rJ) eiq.(r-r’),
V,$(r, i; r’, i’) = &
227
(17)
I
where
v;$(q; i,
i’) = $[V;‘(q@j(b”
- d2)(b”’ - 0’) + V;‘(qW
{(2b”‘- (b” - q?')((l V;‘(q’?=
c4T;3K
/
- i2’)1W~Mr+), (174
- v)b”‘- (b” * q’?‘)),
(2bu2 - (b” * q”>‘>,
(17c)
where q” = q/q. The general inversion formula corresponding by means of the ansatz V$(q, r, +‘) = (2/3T)q’j,(+) + T;;3;){2bA
(17b)
to eq. (17a) can be obtained
[ hfliZr’(qO)L?j * b” -(b*
- qq(b”
. 49);;.],
(18)
which represents an orthogonal decomposition of V;$. Here the second term takes care of the right-hand side of eq. (8) and the matrix function Z(q? is obtained via an inhomogeneous system of equations easily obtained from eq. (8) Zr”‘(q? = 9,(,‘~3~)beV~‘(qqb~V6,l(qq x (-(1 + r)b:b;‘V::;(q’))+
I)$‘-‘,
(18a)
where V&(q@)= 2(b” - b@)- (ba . q@)(b’ . qy and I is the unit matrix. From the second term of eq. (18) it can be seen that a linearly independent set of Burgers vectors has to be used in order that Vi: is well behaved. If linearly dependent sets of Burgers vectors are used eq. (8) is not anymore sensible. An extension of the theory to cover this case which certainly gets important at high temperatures is easily accomplished but will not be presented here. For the functional potential Oo(r, i) in eqs. (9) to (12) the following ansatz can be made: @,(r, i’) = CL(r)
- i,
(19)
where d&(r) is an arbitrary vector function. Then the integrals appearing in eqs. (10) and (11) can be evaluated by means of the substitution given by eq. (16) which is necessary in order that these integrals are defined. Let us point out that so far we have taken into account only long range direction dependent interactions. If short range direction independent interactions are also
22x
A. HO12
included
in V,,, then it can be shown
@
short
range
that eq. (19) is substituted
by
* i.
interaction
(19’)
are a consequence
of nonlinear
volume
expan-
sion effects related to the core of dislocations. They are discussed in the non linear elasticity theory of dislocations’) and contribute to the discontinuous volume
expansion
during
the melting
the asymmetry of the atomistic themselves for the discontinuous
transition.
They
interaction forces melting transition
are a consequence
of
but are not responsible by as explained in section 1.
at the end of Section 1 they will not be taken into However, if a melting theory of ionic crystals is
Due to the reasons given account in the following.
formulated then due to the alzpearance of charged dislocations long range Coulomb forces have to be considered and then x(,(r) must be included. The propagators defined by eq. (12) can be represented in the form G(r, S,,; r’; 4,) = where the Schrodinger
ff” =
F
$k(r.
&)q!!;(r’,
&(f
A(r) = n,(r)
H(r)
The
7 - f&,(r) -i
K(b” X
=
to the
r))‘.
of a spinless
(21) charged
particle
moving
in the
+ !j rc(b” X r)
corresponds rot n,(r)
constraint
therefore
(20)
+ f,,/kTSL
set {I&} represents an orthonormal set of eigensolutions like eigenvalue problem with Hamiltonoperator
This is the Hamiltonoperator vector potential
to which
&) expt-(EL
on
a constant
a magnetic
(22)
field
+ Kb”. the
(22’)
conservation
magnetic
of
matter
field in the direction
given
by
eq.
(1) imposes
of the Burgers
vector
b”
and being of variable strength K. In addition one has a space dependent magnetic field which originates from the direction dependent interaction between dislocations. Having n linearly independent Burgers vectors one gets n propagators of the type given by eq. (20), which are coupled among each other through the strength K of the constant magnetic field. Further coupling between them is provided by the V&term in eq. (11). It should be pointed out that eq. (11) is not in the usual form to be treated by the methods of field theory. In that case it should be brought into Siegert’sz4) representation. It is, however, in a form which is useful for the following purposes because all diagrams appearing in a perturbative treatment of the partition function can be identified easily with the physical processes they correspond to.
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
229
Finally it should be pointed out that eq. (11) does not include phonon excitation. As far as the harmonic phonon excitations are concerned this is no problem because the free energy of those is just an additive term in the total free energy of the problem. Anharmonic phonon effects are more severe and lead to T-dependent renormalization of all coupling constants appearing in eq. (11) and in addition to many body interactions which have to be added to V,,,. Such effects have been studied in a similar two-dimensional problemz5). 4. Single loop approximation The evaluation of eq. (11) will be confined to a single loop approximation, i.e., an ideal gas of noninteracting loops. This is not necessarily a good approximation but it is sufficient for the present purposes. Four cases listed below in increasing order of difficulty will be considered. a) The free loop. b) The “magnetically” confined loop. c) The conservative loop. d) The magnetically embottled loop. In the present approximation it is sufficient to calculate the single loop integral in eq. (11) using the eigenfunction spectrum of eq. (21). a) The free loop In this case the eigenfunction spectrum of eq. (21) is continuous and the eigenfunctions can be normalized in a big sphere of radius R. One obtains easily for r+O G(r, S; 0; 0) = y,f 7T
dk k2jo(kr)jo(0) e-v’kT+k2’2m)s,
(23)
0
where jo(x) is a spherical Besselfunction of zeroth order, and 0(x) is the Heaviside function. Further integration occuring in eq. (11) is done using the identity r
I $!
m
emeSg(E) =
J5
e-(‘+x)‘Og(E)
(24)
0
‘0
and letting at the end of the calculation lo+ 0 in such a manner that only the finite part of the integral survives. Then E in the exponential of eq. (24) can be neglected and one obtains for the single loop partition function in the large loop approximation m
=dS
@O, 0) = 1 s
10
G(0, S; 00) = $1
277
dk k* o f/kT + x + k2/2m e-xb’
dx 1 0
(25a)
A.HOLZ
230 Or
(25b)
valid
for f 2 0. The
elementary
lattice
extra
integration
cell. This procedure
in eq. takes
(25b)
is performed
over
care of the probabilistic
one
property
of the path integral where a “closed” loop has to be interpreted as one coming back into the same cell where its initial point is located. In eq. (25a) this fact is taken care of by means of a cut-off in the eigenfunction lattice units are taken as 1. The quantity G-(0,0) increases function of decreasing f/kT and diverges for f + 0. Using
spectrum, monotonically for f + 0
f/kT = IT - T,J a representative heat Cr easily Cr=
where as a
(26)
quantity to describe the divergence of G(O, 0) is the specific obtained over the free energy F = -kT log Z
l/IT - T-,1”?.
(27)
b) The magneticully confined loop In this case only the constant magnetic term described by the vectorpotential eq. (22) is taken into account. For that case the propagator given by eq. (12) has been first calculated by Feynman (see Feynman and Hibb?). However, in order to make the subsequent integrations needed here tractable it is more convenient to start from eq. (20) and use a cut off in momentum space as done in eq. (25a). One obtains
G(O, Kb)
=
!$ j-dx i
(28)
0 where L represents the linear dimension of the normalization eigenfunctions of H are unbounded in b-direction). Furthermore and n and n, ran over all positive
integers
where
the prime
volume (the k; = (2~r/L)n
on the summation
sign confines them to the momentum cut off k, = 2~. The further integrations are straightforward but lead to awkward expressions. For consistency the final K-integrdoII is also confined by the momentum cut off via IK,,,,bI = kf. In case that eq. (25b) is used this is not necessary. The fact that we still talk about a single loop approximation is due to the cumulant expansion used in evaluating the exponential in eq. (11). The divergence when the system approaches T, is best represented by the specific heat singularity corresponding to eq. (27), i.e., c,,
=
1T - T,/“’
(29)
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
231
c) The conservative loop Although the magnetically confined loop does satisfy conservation of matter globally as required by eq. (1) it does not do that locally. This as a matter of fact is also not necessary but if one wants to distinguish between cases where transport of matter is easy or not then also the case where matter is conserved locally is of interest. The motions of the loop are then confined by its possible glide planes which can be reached by cross slipz2). Because the two-dimensional surface on which the loop is located after purely conservative glide motion has no intrinsic curvature it can be laid out on a plane surface as shown in fig. 3. The Greensfunction of the closed loop can then be calculated by first evaluating it in the (x, y)-plane between two walls being a distance 1 apart. In the second step the straight segment 1 is allowed to display its conformations in the (y, z)-plane. This implies crumbling the straightened out surface, or say leaf, back into its physical shape where selfintersections are allowed. Allowing all values of I between 0 and infinity gives all possible conservative loop conformations. The quantity corresponding to G(O,O) eq. (25a) can now be represented in the form G,(O,0) = j-dl 0
z: j- y 42
j- dSz 1’ dS, SI
I
m c
X
dx Gz((x, 1- E, Sz; E, O)Gz(x, 1 - E, S,; E, O), J OY
(30)
where Gz is the two-dimensional Greensfunction corresponding to eq. (12), i.e., fz = y - kT In ~2. Here t2 is the number of possibilities that exist at each step for the loop generating random walk process, i.e., z2 = 3 and m2 appearing in eq. (12) instead of m, is for the two-dimensional problem given by
Fig. 3. Defolded glide plane of a dislocation loop with Burgersvector b. Hatched areas mark inpenetrable walls for reference particle described by Schradinger like boundary value problem. I represents maximal linear extension perpendicular to b of the set of loops covered by this drawing. x,y-coordinates are in plane and z-coordinate points out of plane.
A. HOLZ
232
rnz = 1 (see refs. It cannot the
19 and 20). The quantity
be set to zero in the present
Hamiltonian
ditions, mations
of the
i.e., they
present
E in eq. (30) is of the order scheme
problem
vanish
for y = 0 and to be used in eq. (20) are
are
because subject
y = 1. The
of unity.
the eigenfunctions to the boundary
eigenfunctions
and
sin k,y,
to consum-
(31a)
c -$ j- dk, j- dk,,
(31b)
respectively, where L % 1 is a normalization length for the eigenfunctions. The subsequent calculations are straightforward but lengthy. The essential point of the result is that the specific heat singularity is the same as in case a) (32)
(‘CL= I/IT - T;(“‘, but T:/ T, = Evaluating from
In z In 2~ + (l/S)ln’
eq. (33) for z = 5 and z2 = 3 one obtains
eq. (33) that due to the reduced
loop the problem,
(33)
z2’
number
of freedom
of the
system is stabilized. Observe that for the purely two-dimensional i.e., without cross slip a specific heat singularity c = l/IT ~ T,,,:I is
obtained where T&T,,, = In 513. d) The magnetically embottled loop So far the interaction between loop parts represented f&,(r) in eq. (21) has not been taken into account. function
Tf/T,,, = 1.29. It follows
of degrees
which
appears
in the functional
integration,
by the vector Because n,,(r) the eigenfunctions
potential is some to eq.
(21) are not easily determined. One may. however, choose such functions C&,(r) for which the integrand of the functional integral eq. (1 I) displays a saddle point. Such methods are described in refs. 19 and 20. Because the term in eq. (1 I) involving the inverted interaction V,,,:, is multiplied by kT one expects that such vector potentials which maximise the loop integral may give the main contribution at low temperatures. What is required in that case is a magnetic field which confines the charged particle described by the Hamiltonian eq. (21) to the origin or say a magnetic bottle. From this hypothesis the title of this paragraph derives. The detailed calculations for the many magnetic bottle fields which have been done by the author will be published elsewhere*‘). The main result obtained so far is negative. Setting K = 0, i.e., ignoring conservation of matter no saddle point is found except for the origin
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
233
C&(r) = 0. This result is preliminary because so far only real and purely imaginary vector potentials have been studied and no complex vectorpotentials. If the present result turns out to be true then the small and large N limits studied in section 2 and explained by fig. 2 are not seperated by an activation barrier, but the dash pointed free energy curve applies. This then would also provide a simple explanation why in the planar rotator model a continuous phase transition is observed. An interesting question is if a saddle point takes off the origin once the homogeneous magnetic field representing the mass constraint is turned on, because once two saddle points are present the possibility of a discontinuous phase transition arises. Unfortunately so far no results with respect to that problem have been reached.
5. Discussion
and conclusion
We have studied in this paper various mechanisms defined over dislocation loop conformations which we hope will give some insight into the problem of the nucleation mechanism of the solid liquid transition. In section 2 we have first considered the problem of screening of long range interactions. At first sight it appears obvious that a small loop in order to become a large loop or say an infinitely extended loop has to pass over a free energy barrier. This would allow to explain the discontinuity of the transition as observed empirically. Unfortunately vortex loop systems interacting also via a long range Biot-Savart law as in the three-dimensional planar rotator model do exhibit a continuous transition. This puts the argument into considerable doubt and in fact there exist already on an elementary level reasons that a free energy barrier does not have to exist necessarily. One reason is that the small loop may grow over dipolar configurations of a tree like structure into a large loop. We have studied this problem in the appendix over an integral equation for the number of tree like structures. Because the restriction to tree like structures implies a monotoneous increase of trees with increasing size the possibility to see the development of an activation barrier is not inherent in this approach. The results obtained suggest to a certain degree that the cardinality of dipolar loop conformations approaches steadily the cardinality of the stochastically conformed loops if the constrained degrees of freedom are successively liberated. Because no mechanism is built into this model which allows a change-over from dipolar conformed loops to stochastically conformed loops a more powerful approach to this problem is needed. In section 4 we have studied this problem again under case d, using a functional integral formulation of the problem. The preliminary result obtained is that there exists no activation barrier (see fig. 2).
234
A. HOLZ
The answer
to the problem
of the discontinuity of freedom
which
are operative
at least three additional instance
in the
stated
law of melting
servation
of matter.
Second contrast
the dislocation to the current
rotator
may be simply
that the universality
of the additional
in solids but not in vortex
properties
planar
above
is a consequence of solids not found
model.
This implies
First
transport
that loop coformations
loops in solids are characterized strengths characterizing vortex
systems.
in vortex
degrees There
systems
of matter must
satisfy
are
as for
and
con-
eq. (I).
by a Burgers vector in systems. This allows a
three-dimensional manifold of defect states in solids in contrast to the one-dimensional manifold of defect states described by vortex systems. The third property is that the motion of dislocations changes the metrical properties of solids over torsion and curvature. The consequences
of the first distinguishing
property
stated
above
has been
studied in section 4 under cases b. and c. Case b. showed that taking matter conservation on the single loop level into account disregarding long range interaction weakens the singularity connected with infinite loop extension considerably. As can be shown and as one expects it does not imply a discontinuous transition. Because transport of matter in solids is a thermally activated process and because conservative and nonconservative processes do not have to take place with the same speed it is reasonable to study also the conformations of a loop where already locally matter is conserved, i.e., which is formed by a pure glide process. This has been done in section 4 listed under case c. Clearly such a constraint stabilizes the loop against expanding, and it could be shown that the transition temperature is increased in that case. The essential point of this example is demonstrated by studying the following fictitious process. Consider a loop formed by conservative and non conservative elementary random walk steps each showing y< and qnc possibilities of orientation respectively formations 2 of a loop of N steps
at each step. The total is then given by
number
of con-
(33) Because each step contributing to eq. (33) takes place over an activation barrier it is reasonable to attach the probabilities pc and pnc to the occurrence of conservative and nonconservative steps per time unit respectively, where pc + pnc = 1. Instead of eq. (33) one obtains now
z+
Z’ = (pcq, + pncqnc)N.
In the authors thermodynamic ses contribute
(33’)
opinion eq. (33’) should be prefered to eq. (33) in calculating properties in a system where strongly rate dependent procesto the thermal formation of structures like dislocation loops
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
235
requiring extensive matter transport. This is in fact obvious because if the partition function is calculated only with sets of final states like loop conformations then all processes the system has to activate in order to get from one state to another are disregarded. Although on a very large time scale the number of conformations of the loop of N steps is given by eq. (33), in a smaller time interval At the number of realizations counted starting from some arbitrary conformation is given by eq. (33’). This phenomena is also known from the theory of quenched disorder or the theory of glasses where the free energy has to be calculated for one possible sector of the phase space of the system. Applying this idea to the nucleation process of melting we may assume that at some favorable place at the surface or in the bulk an energy fluctuation produces a small loop over a shear process. Because things happen on a small time scale the arguments given above imply that only such loop conformations are reached where the connecting processes to the initial state are the quickest. Hence many possible loop conformations reached only over complicated intermediate processes (e.g. trapped in configurations) will not be activated. In particular we think that the diffusive motion of matter in the solid state occurs on time scales too large to guarantee the occurrence of all possible loop conformations with equal probability per time unit. However, once the loop has reached by chance a critical size the stiffness may be reduced in the area affected by the loop’s polarizability that much that diffusive motion of matter may assume a collective character. Then suddenly many more degrees of freedom are in action per time unit and the statistical description of matters must shift over from eq. (33’) to eq. (33). The example developed under case c, in section 4 demonstrates then easily that such a phenomena may be connected with a discontinuous transition. We would like to emphasize that the present discussion does not rely on any specific properties of the initial state on top of which the nucleus forms but is of purely generic character. Eventually we propose that melting is discontinuous because the activation of conservative and nonconservative processes is such interdependent that none of them can activate its degrees of freedom without the other having them activated as well. Because the polarization potential of a loop depends on the loop having a critical size where it is able to screen long range interactions, and the collective diffusive motion requires just that polarization to get into motion, whereas the polarizability of the loop sets only in once the collective diffusive motion releases trapped in configurations, the strongly coupled qualities of both processes are at once obvious. It follows also that the transition must be discontinuous. That melting usually starts at free surfaces can also easily be expiained within that scheme. Polarization and transport properties are facilitated by the surface
A.HOLZ
236
due to mirror-forces, facilitating conditions requires
additional
Because
volume
consequence
free space and heat transport, whereas in the bulk such do not exist and besides the usual volume expansion
work to be dissipated expansion
of the increased
easy formation
of vacancies
once a nucleus
in the liquid density at free
state
under
of vacancies surfaces
is formed
in the bulk.
ordinary
conditions
is a
as stated
in section
I the
may be considered
as the main
driving force for the nucleation of the liquid phase at such surfaces. Finally we like to mention that the branching processes contributing partition function given by eq. (11) have not been studied in this Because these will lead to a interaction are where the third a simple model
to the paper.
branching processes are entropy generating in all orders they destabilization of the ground state for T > 0 if the elastic not properly taken into account. A free propagator model term of eq. (12) is neglected will therefore make no sense and taking such processes into account is not easily conceived.
Acknowledgements The author likes initiated melting, amorphous state melting in polymer
Appendix
to thank Professor H. Gleiter for discussions on surface Prof. H. Kronmtiller for stimulating discussions on the and Prof. J. Naghizadeh for interesting discussions on systems.
A
The counting problem for dipolur configurations In order
to count
the tree like structures
of a loop having
n links as the one
e.g. shown in fig. 1 we have to add up all possible .trees with the origin fixed and total trunk and branch length N = n/2. To generate the tree we use the building blocks formally depicted in fig. 4a. A simple unit of the tree requires two links of the original loop, a branching element of order (Y3 2 requires 2(a! + 1) links of the original loop. The total number of tree configurations W(N) obeys an integral equation which is graphically depicted in fig. 4b where for the sake of simplicity we have confined ourselves to branching elements of order three. Assume that the simple unit of the tree has q. possibilities and a branching element of order (x, qz” possibilities of orientation respectively. If rno is the number of simple units along the trunk of the trees in fig. 4b and mk the number of branched off trees of total branch length
QUALITATIVE
THEORY
-
Tr
’
TRANSITION
237
&l
+--,---I
c--_(
-
OF SOLID-LlQUID
’
+-$++
a=2
’
a=3
Fig. 4. a) Elementary units used in constructing dipolar structures as depicted in fig. 1. From left to right are drawn a simple tree unit, and branching units of order 2 and 3 respectively. Full drawn lines refer to original oriented links of loop where segments are indicated by vertical dashes. Dashed lines mark tree skeleton. b) Graphical representation of eq. (A. I). The hedgehogs attached to the root or trunk of the graph on the left- and right-hand side of the equation respectively refer to the complete set of possible trees with N and k links respectively.
k 2 1 then fig. 4b is translated
into the equation
(A.11
where each term corresponds to one graph of fig. 4b. The prime on the summation sign implies that such sets {mk}N are used which satisfy N=mo+g
mk(k+3). k=l
(A.21
Next we define the generating function
G(Z) =
N$o ZNWW
C-4.3)
A. H0I.Z
238
where
2
M =
mr, k=O
Zk = &,oq,Z + (I - &.“)W(k)qZZk+3
(A.4)
and Z is a complex number. Eq. (A.3) has been derived using the multinomial theorem and analytical continuation. Writing eq. (A.3) in the form G(z) = (I - [qoz + q;z’(i: observing
h-0
W(k)zk
- W(O)!)i
W(0) = 0 and the definition
G(Z) = (1 - qoz - (q?z)‘G(z)) This equation
can be solved
‘,
of G(z) one obtains (A.5)
‘.
for G(z)
yielding
(A.61
. Because this is an analytical one obtains using the Cauchy W(N)
where
=
the contour
oneself
determined
’ - (1 -
Because
is finite
at the origin
(G(0) = I)
is around
(A.7) the origin
avoiding
the square
for N + I using the saddle
that the leading
contribution
point
comes
root branch method.
from
cuts of
One easily
the saddle
point
by 4(qzz)’
i
which
&f d$y,
G(z). Eq. (A.7) is best solved convinces
function theorem
there
“?
qozY~ = l/N.
N % 1.
is no need to get more than an estimate
just for two reasonable W(N)
= (2qo)““;
W(N)
= ($4”7qo)n”;
numbers
we will give the results
of q:.
qz = q,,/4”‘,
(A.9)
q? = qo.
(A.10)
Here we have used N = n/2 and n is the number of original ing that qo= 5 one notices that the tree like structures cardinality of the stochastic loop structures because in that front of qo in eqs. (A.9) and (A.10) should be of the order however, be pointed out that we have not taken all degrees
links. Rememberdo not have the case the factor in of qo. It should, of freedom of the
QUALITATIVE
THEORY OF SOLID-LIQUID
TRANSITION
239
dipolar structure into account. The two foldedness of the trees allows to a certain degree rotational degrees of freedom to be activated. In addition to each tree characterized by the dashed trunk as in fig. 1 many more dipolar structures (enveloping oriented line in fig. 1) correspond than just the one drawn in fig. 1. These dipolar structures are characterized by crossing points of the oriented lines at the trees branching points in a planar representation. These effects will increase qo_
References
2) 3) 4) 5) 6) 7) 8) 9) 10) ll) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27)
U. Dehlinger, Encyclopaedia of Physics Vol. VII, 2, S. Fliigge, ed. (Springer, Berlin, 1958) p. 211. L. Boltzmann, Vorlesungen iiber Gastheorie, Vol. I (Verlag von Johann Ambrosius Bart, Leipzig, 1896) p. 7. F.C. Frank, Proc. Roy. Sot. A170 (1939) 182. N.F. Mott and R.W. Gurney, Trans. Faraday Sot. 35 (1939) 364. A. Seeger, In Encyclopaedia of Physics Vol. VII, 1, S. Fhigge, ed. (Springer, Berlin, 1955) p. 383; Phys. Status Solidi 1 (1961) 669. J.K. MacKenzie and N.F. Mott, Proc. Phys. Sot. London 63 (1950) 411. N.F. Mott, Proc. Roy. Sot. A215 (1952) I. J. Rothstein, J. Chem. Phys. 23 (1955) 218. S. Mizushima, J. Phys. Sot. Japan 15 (1960) 70. A. OoKawa, J. Phys. Sot. Japan 15 (1960) 2191. M. Siol, Z. Physik 164 (1961) 93. D. Kuhlmann-Wilsdorf, Phys. Rev. A140 (1965) 1599. R.M.J. Cotterill, Phys. Rev. Lett. 42 (1979) 1541. S.F. Edwards and M. Warner, Phil. Mag. 40 (1979) 257. A. Holz, Physica 97A (1979) 75. A.M. Nikolaenko, I.N. MaKarenko and S.M. Stishov, Solid State Comm. 27 (1978) 475. S.F. Edwards, Proc. Roy. Sot. 91 (1967) 513. E. Kroner, Kontinuumstheorie der Versetzungen (Springer, Berlin, 1958). F.W. Wiegel, Phys. Reports Cl6 (1975) 59. K.F. Freed, Adv. Chem. Phys. 22 (1972) I. Functional Integration and its Applications, A.M. Arthurs, ed. (Clarendon, Oxford, 1975). A. Seeger, in Encyclopaedia of Physics Vol. VII, 2, S. Fliigge, ed. (Springer, Berlin, 1958) p. I. T. Banks, R. Myerson and J. Kogut, Nucl. Phys. B129 (1977) 493. A.J.F. Siegert, Physica 26 (1960) S50. A. Holz, Physica 1WA (1981) 58. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). A. Holz, to be published.
1llA
(1982) 217-239
QUALITATIVE
North-Holland
THEORY
Publishing
Co.
OF THE SOLID-LIQUID
TRANSITION’
Arno HOLZ* Fachrichtung
Theoretische
Physik,
Uniuersitiit Received
des Saarlandes,
26 August
66 Saarbriicken,
W.-Germany
1981
Some qualitative attempts to the elucidation of the melting transition in solids are presented. The screening of the long range elastic interactions in the liquid phase is considered to originate from the presence of freely mobile infinitely extended dislocation loops. In contrast to other similar theories it is, however, assumed that cooperative diffusive motion (or “convective” motion) plays an equally important role in releasing any trapped in dislocation configurations which may lead to immobility of the system. The apparent universal nature of the discontinuity of the melting transition is studied on the basis of these concepts via possible nucleation mechanisms. The technical means used to treat dislocation configurations are combinatorics and the path integral formalism. The conclusions reached are that the universal nature of the discontinuity of the melting transition is a consequence of the strongly coupled dynamics of dislocations and cooperative diffusive motion.
1. Introduction The study
of phase
transitions
in solids in particular
structural
transitions
of
martensitic type if they occur discontinuously require the knowledge of the operating nucleation mechanism. This nucleation mechanism may usually be described as a highly exceptional but still probable path in phase space leading through a tiny pin hole into the new state. It seems intuitively obvious that an understanding of the nucleation mechanism must give in many cases an explanation of the structural and dynamic properties of the new phase. In his work
on martensitic
transformations
U. Dehlinger’)
recognized
the im-
portance of centers of internal strain for the formation of the new phase in particular dislocation configurations which allow the embedding of the new phase into the host matrix and the rearrangement of atoms into their new positions.
The problem
of nucleation
can therefore
be formulated
in terms
of
two strongly coupled questions, how is the nucleus formed and how is it embedded into the host matrix. In order to avoid any misunderstanding it should be pointed out that in phenomenological nucleation theories the embedding energy is taken into account over a surface free energy and the bulk energy of the nucleus over the chemical potential of the new phase. This approach presupposes ad hoc the bulk phase in the nucleus whereas it is clear ’ Supported * To Professor
in part by Deutsche Ulrich Dehlinger
0378-4371/82/0000-0000/$02.75
Forschungsgemeinschaft under on the occasion of his eightieth
@ 1982 North-Holland
SFB 130. birthday.
A. HOI>Z
?I8
that also that part of the nucleus further surface.
growth
of the nucleus
It is emphasized
properties
that
of this activation
has to climb occurs
over
we want
barrier
over an activation structural
to study
the structural
or say free energy
barrier
transformations barrier
until at its
and dynamic for the melting
transition. In order
to put the problem
of melting
into the appropriate
context
;I short
review of the development of ideas in this field will be given. The idea that a crystal melts because it will loose its inherent resistance to shear goes back to a suggestion of Boltzmann’). Essentially it is proposed that the shear modulus C11-C,2 vanishes at the melting temperature. Because the melting transition is strictly observed to be of first order and such a hypothesis implies a continuous transition this point of view has been abandoned by, most people working in this field. The idea that melting
originates
to Frank’) and attempts to describe a liquid state model were initiated
through
the generation
of defects
goes back
how such defects may be incorporated by Mott and Gurney4) in their work
in on
microcrystalline arrays. Early attempts to describe melting as dislocation originated and the liquid state as dislocation saturated have been critisized by specialists working in this field and have not been included in the review on dislocation theory by Seeger’). The reason for that is very simple. First up to now no thermal dislocation loop formation as a premelting phenomena has been observed experimentally, is that strong that the Burgers
and second the damage of good area in liquids circui?) is only badly defined if it is defined at
all. Work on dislocation like structure of liquids was initiated by MacKenzie and Mott’) and Mott’), and led over the works of Rothstein’), Mizushima’), Ookawa”), Siol”) to the theory of melting by Kuhlmann-Wilsdorf”). In the latter work it was conjectured that melting occurs when the free energy of dislocations gets negative leading over spontaneous formation of dislocations discontinuously into a dislocation saturated liquid. The essential point of her derivation is that she calculates the free energy of the dislocation state over the vibrational
entropy
of localized
and extended
lattice
modes
in contrast
to
later work where the conformational entropy of dislocations is calculated similar as in polymer physics. The nature of the dislocation like structure of liquids has been elucidated by Cotterill”). The most recent study of the solid-liquid transition and the liquid state has been done by Edwards and Warner14). These authors treat the problem according to the principles of polymer statistics but take also long range interactions into account. Their treatment approximates the rather awkward Biot-Savart type interactions between dislocations by averaging over the Burgers vectors. In this way the problem is mapped onto the simpler problem of vortex loops characterized by current strengths. It is then stated that the
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
219
discontinuous transition is a consequence of the exponential screening of long range interactions in the liquid similar as it is the case in metal insulator phase transitions. In an earlier paper by the author”) on the phase transition in the planar rotator model and where vortex loops are operative in driving the phase transition it has already been pointed out that the screening mechanism advocated by Edwards and Warner14), does not necessarily lead to discontinuous melting. The crucial point is very simple. The planar rotator model is known to undergo a continuous phase transition and Edwards and Warner14) have mapped their system onto that model modulo some irrelevant features. The latter are related to the range of the short range interaction and which may influence the dynamics of the loop motion and in addition to packing effects. Volume expansion during the transition, however, is ignored by these authors. Let us point out that volume expansion is very often observed during melting transitions but not always. For instance it is observed that for alkali metals under strong hydrostatic pressure a discontinuous entropy change is accompanied by a smooth volume changei6). Apparently the electronic system in such metals is restructured in such a manner that a liquid state without net volume change is possible. Because the electronic system in Li and Na does not undergo a phase transition during the melting transition (it is metallic and remains metallic although correlations may change as it is the case in K, Rb and Cs) its response to the reordering during melting must be smooth. Under these conditions, however, it cannot be explained why the discontinuous melting transition without volume change does not take place between two temperatures Tr and T2 required to drive the system through the coexistence region if discontinuous volume change really would be prerequisite for discontinuous melting. A consequence of volume expansion in most systems under ordinary conditions is, however, that melting starts at the free surface of the system as is well known and as will be discussed later. As a matter of fact volume expansion at the solid-liquid transition is a consequence of the asymmetry of the atomistic interaction forces and which favors energetically the formation of vacancies with respect to interstitials. Because, as will be explained in the sequel, a liquid state is not possible without the operation of nonconservative processes and which can be considered as mainly vacancy mediated volume expansion under ordinary conditions is a general rule. Under strong hydrostatic pressure successive delocalization of the electron shells results until eventually a Wigner crystal is formed. In this state the interaction forces between the nuclei are essentially symmetric leading essentially to no volume expansion for vacancy interstitial pairs. The importance of the experiments reported in ref. 16 is therefore that they show that with increased delocalization of the shell electrons the volume dis-
A. HOLZ
220
continuity
flattens
out but still a first order
sion can therefore
safely
As an introduction following reach
section
a state
In section reminiscent
be considered
to the problem the
problem
extension
via energetically
of infinite the
of melting
combinatorical
3 a path integral of
transition
formulation
approach
polymers. however,
The constraints the conservation
observed expression
by just
giving
of
occurs.
as a secondary
we will first
how
favorable
imposed of matter
to
be satisfied.
Here
in the
loop
may
conformations.
will be given
constrained
which
systems
is of
on the system in the present case are, in the system which is not automatically
an arbitrary
set of dislocation
loops
because
. r” X dr”) = AN must
expan-
study
a dislocation
of the problem Edwards”)
Volume
effect.
the
(1)
b” is a Burgersvector
and the integral
and sum ran
over the whole dislocation system and A is the volume of the unit cell. The right-hand side of eq. (1) represents the number of unpaired interstitials or vacancies depending on AN being positive or negative respectively. In section 4 the consequences of this constraint will be studied in two simple models. Another constraint imposed on the system of dislocations is related to the concept of curvature and torsion’8) of the dislocated state and reflects metrical properties of the path integrals. In section 5 we draw the conclusions from the results reached
itself in the so far and
propose the idea that melting must be discontinuous as a consequence of the strong interaction between cooperative conservative and nonconservative motion in the liquid. An excellent introduction to functional integration methods can be found in refs 19. 20 and 21.
2. Dipolar conformations Thermal
generation
of dislocations
of dislocation
loops as a premelting
phenomena
have so
far not been observed experimentally. At first glance therefore the attempt to develop a statistical theory of dislocation loops should be doomed to fail. There are at least two reasons why this will not be the case. The first is that one wants to know why it does not work but would perhaps become operative at higher temperature if energetically more favorable nucleation mechanisms would be inactivated, for instance by depriving the system of free surfaces etc. The second reason is that a nucleation mechanism may be a rather complex evolutionary type process leading over a steep activation barrier into the new phase such that its presence is only reflected in the short wave length
QUALITATIVE
THEORY OF SOLID-LIQUID
TRANSITION
221
and high frequency domain of the structure factor, where it is difficult to detect. It seems therefore not completely unrealistic to devote ones attention to the following problem. Given a dislocation loop of N links, where one link corresponds roughly to one lattice unit, a core energy of the order of yc per link and the usual Biot-Savart type interaction between different loop parts; find the free energy of such a loop as a function of temperature disregarding its translational degrees of freedom. If one assumes that the loop is generated over a random walk process having at each step the choice of about z possibilities to continue (in three dimensions one gets roughly .z = 5) then the free energy of the loop is bounded from below by F rooP2 N(-yc - kT In 2).
(2)
On the right-hand side of eq. (2) long range interactions have been neglected thus leading to the lower bound. From eq. (2) one obtains an estimate of the melting temperature in the form kT, 2 y&n z.
(3)
Choosing appropriate values for yc and z and making a reasonable estimate on the packing of dislocation cores in the liquid allows to estimate the melting temperature and latent heat of simple solids in a manner similar to Kuhlmann-Wilsdorf”). As a matter of fact this approach is completely equivalent to hers. It is obvious that the estimate represented by eq. (2) is good only for N 9 1. For finite N, eq. (2) should be substituted by L,P 2 N(y(N,
T) - kT In z(N, T)),
(2’)
where y(N, T) 2 yc and z(N, T) c z should hold. The first inequality is a consequence of the fact that the long range Biot-Savart type interaction between loop parts on the average will increase the energy per link of the loop. The second inequality originates from the condition that the formation process of the loop starts from small size. Consequently it cannot avoid crossing itself when trying to reach its possible conformations, and many possible conformations within the random walk model are not reached due to unsurpassable obstacles or due to trapped in conformations. Unsurpassable obstacles must be considered as topological constraints as represented e.g. by eq. (1) and trapped in conformations such as considered in the theory of crystal plasticity**). Observing therefore for finite N that y(N, T)/ln z(N, T) * r&r z
(4)
possibly holds it is very unlikely that thermal formation of dislocation loops in
222
A. HOLZ
solids will occurs.
be observed
and
necessarily
a discontinuous
melting
transition
Unfortunately, however, eq. (4) is not easily justified as far as its dependence on y(N, T) is concerned. In ref. 15 the author has studied this problem in the context of the order-disorder transition in the 3-dimensional planar rotator model where vortex loops interacting via the Biot-Savart law drive the phase transition. Because the long range interactions &(y(N. holds
for
T)IkT the
this phase transition is continuous the screening with increasing N must be such that
- In z(N, T)) 2 0,
planar
rotator
model
of
for all N, and T < T,, and
also
for
the
(5)
dislocation
loop
ap-
proximation if all special solid state effects on z(N, T) as stated above are ignored and only y(N, T) is considered. Although eq. (5) may follow from the knowledge of the exact result it is not easily derived as shown by the discussion presented in ref. IS. In order to demonstrate the successive build up of screening of long range interaction with increasing N one may develop the following rather crude model. For small N one uses dipolar conformations of loops, i.e., double folded loops as depicted in fig. 1. If the energy per double link is assumed to be Yd 2 2-y, then according to F%,, b F(2rC
the free
energy
of this
loop
may
be bounded
from
below
- kT In z).
(6)
Because eqs. (2) and (6) are bounds for the asymptotic behavior of single loops in the small and large N limit the free energy of the loop should be bounded from below as depicted in fig. 2. The dashed part of the estimated free energy studied
bound
is an interpolation
in the appendix
over
between
the tree
the asymptotic
like structures
behaviors
of dipolar
and is
conformed
b
Fig.
1. Dipolar
structure
conformation
is the reference
of a dislocation
loop with
graph used in the appendix.
Burgers
vector
b. The dashed tree like
QUALITATIVE
THEORY OF SOLID-LIQUID
TRANSITION
>> 1 saddle
’
’ Loop
extension
223
point
N
Fig. 2. The curve bordering the hatched area is the lower bound of the free energy Fluupof a loop of N links. I marks the inflection point and AE represents the activation barrier for a transition from the large N- to the small N-saddle point. The dashed part of the curve is interpolated. The dash pointed curve with slope r’E- kT In z applies to high density renormalization of core energy yC. This schematic drawing applies to T < T,.
loops as depicted in fig. 1. Although it follows from these calculations that the magnitude of the tree like structures readily increases it seems that the magnitude of stochastically conformed loops is approached only by a logarithmic law if it approaches it at all, a question which cannot be answered on account of the approximative treatment given in the appendix. A quantitative estimate of the height of the activation barrier AE has also not been achieved. In section 4 this problem will be attacked from a different point of view. Let us point out that if 7: = y,(N * ~0,T) > y,(N = 6(l), T) is assumed then the dash pointed free energy curve shown in fig. 2 arises and no activation barrier and no inflection point is necessary. As this assumption is unfounded within the present simplified approach we will only come back to that possibility in section 4. It follows from fig. 2 and the curve bordering the hatched area that eq. (5) cannot hold for all T below T, because the free energy of the loop should display an inflection point I. Accordingly one may be inclined to postulate two saddle points of the free energy for temperatures in the interval T, < T s T,, where the large N saddle point is metastable below T,. In a similar fashion one gets two saddle points above T, with the stability properties interchanged. On account of this reasoning a first order transition should arise as has been postulated by Edwards and Warner”). The many reasons why this is not necessarily the case have been extensively discussed in ref. 15 and will not be repeated here but only two important observations will be made. The first point is that the occurrence of two saddle points will depend on
224
the
A. HOLZ
height
regimes,
of the
energy
barrier
dE
seperating
the
two
if it is smaller than kT,, it can be overcome The interpolated dashed drawn curve in fig. 2 should
fluctuations. calculated point
free
e.g.,
as can be done
using the theory
is that the asymptotic
far as the internal
energy
estimate
given
given
in the appendix.
by eq. (2) is certainly
term of the free energy
is concerned.
asymptotic by thermal therefore be The second sensible
as
This has been
demonstrated by Banks et al.“) using renormalization group methods. Intuitively it is also obvious that the orientation dependence of the Biot-Savart interactions for large and stochastically conformed loops must average out. Uncertain to a certain degree is the precise value of yC to be used in the small and larger N limits due to interactionand anharmonic effects. The entropy part of the estimate, however, may be completely wrong. The reason for that is that using for the entropy the quantity S = Nk In z presupposes that the loop is really able to move through all is certainly not necessarily the case glasses and quenched in disorder. The can therefore only be guaranteed if it exponentially the case then litude phonon
stochastically conformed shapes. This as one knows from the existence of existence of the large N saddle point can be shown that this state leads to
screened elastic interaction, i.e.. that it is liquid. If this is not it still may be a saddle point if it is stable against small ampexcitations. But in this case it is a defect state of the type ob-
tained by means of work hardening of solids or by means of the production of entangled vortex loops in the planar rotator model at low temperatures. The existence of such states does naturally give no clues whatsoever to the nature of phase transitions between thermodynamic equilibrium states in such systems. In concluding
this section
we point
out that the reasoning
presented
with
respect to the properties of the bounds of the free energies of loops in connection with fig. 2 are still not completely waterproof due to the various uncertainties related to the calculation of y(N, T) and z(N, T). More sophisticated applied
methods to deal in section 4.
with
this
problem
are developed
in section
3 and
3. Path integral formulation The
path
integral
formulation
of a dislocation
calculating the grand partition function configurations into account. The interaction represented in the form
V = f 3
r’ dS,,i
rr':i'0
7” dS,,.i,V,,,,,(r,,iL,,i; 0
problem
is equivalent
to
taking all possible dislocation of the dislocation system can be
r,,,iv; i,,,,ir),
(7)
QUALITATIVE
THEORY
OF SOLID-LIQUID
225
TRANSITION
potential between dislocations’) of Burgers V,,,# is the interaction and rO,i = r(S,,i), L,,i = dr,,i/dS,,i. Under the vector b” and b”‘, respectively condition that the following potential inversion can be solved with respect to where
V’. &R”V,+(r,
where
i; r”, S”)V,$mt(r”, i”; r’, i’) = &&‘(r
d6R” = d3r”d3r”, it can be shown
exp{-
- !$g, where
that the following
VILT} = N-’ n / S[@,] exp{-i Cr
the normalization
3
I d6R 1 d6R’@,(r, factor
N is defined
Eqs. (9) and (10) are functional integrals. problem can now be written in the form
r’)6’(i
-
identity
7’ dS,@,(r,,i, 0 L)Vib(r,
- i’),
(8)
holds:
i0.i)
$; r’i’)@Jr’,
if)],
(9)
by
The grand
partition
function
of the
z=1+$
x c Cr,u’
x~dS,idSuidS,r
d6R’oC(r,
i)V$(r,
i; r’, +‘)@,,(r’, i’)
@r’+allpossibleirred.graphsj-11. c(
(11)
where the sums extend over all possible Burgers vectors but excluding the inverse Burgers vectors. The oriented lines drawn in eq. (11) are the propagators of the theory
A. HO12
226
;, = G(r,
;G
S,,; r’, 0; @,,,)
r..s
=
1 d[r(S)Iexp( r’..S=O
-jdSl~i~~ixh”.(rxi)+i~~,(r.i)_i./~Ti}, 0
In eq. (12), f(, is the free energy Burgers
vector
(12)
per link of the core
of the dislocation
with
b”, i.e.. (13)
f(, = y,, - kT In z,,.
where y(, and z,, are defined as in eq. (2). The “kinetic” term in eq. (I?) enters over the Gaussian approximation to path integrals and mJ2 = 3/2 (see refs. 19 and 20). The K-integration in eq. (11) enters over the integral representation of eq. (1) where the right-hand side is set equal to zero for the sake of simplicity. The factors gcrBvappearing in eq. (I 1) are related to branching processes of dislocations.
If the internal
energy
of a node
is t,+
then
%JBY= exp(-•,,,/kT).
(14)
The first loop integral in eq. (I I) carries a factor l/S,, in the integrand to guarantee correct counting of graphs and E = ( (1). Let us point out that in eq. (I 1) only the constraint eq. (1) has been taken into account and besides AN = 0 has been assumed for the sake of simplicity. Constraints entering over torsicn and curvature have not been taken into account either. In order to work with eq. (1 I), eq. (8) should be solved. One observes first that V&r, r; i’, f’) is bilinear in i- and L’. For an isotropic material one can deduce from the expressions given in ref. 5 V,,,+(r,,, i,,; r,,,, r,,,) = K[-(
1 + v)(b” . fi”)(b”
+ Cc,,,(W,d)(R” where
u and /J are Lame
constants,
. a<‘)
. f2”)]i,,f,,,/lr,,
- r,,J.
(15)
K = ~/47~( 1 - U) and
c,,r,(n,,,) = b” * b”’ + (b” . n,,,,‘)(b”’ . n<,<,.), nWd = (r, - r,,,)/ir,, - rrr,/; Because integrals
the “kinetic” in the interval
rr-+3jr(rr),
k = 0 ;.
term in eq. (12) favors r-values in the functional (0, 1) it is sensible in eq. (IS) to make the substitution (16)
where j,(x) is the spherical Besselfunction of order 1. (Notice that 3j,(x < I) = x and that the firs&zero of jr(x) appears for x > rr.) Only in that case eq. (8) is sensible and can be solved. Confining oneself to an iterative solution, i.e., inverting the diagonal terms only one obtains
QUALITATIVE
THEORY OF SOLID-LIQUID
TRANSITION
d3qV;;,(q; r; rJ) eiq.(r-r’),
V,$(r, i; r’, i’) = &
227
(17)
I
where
v;$(q; i,
i’) = $[V;‘(q@j(b”
- d2)(b”’ - 0’) + V;‘(qW
{(2b”‘- (b” - q?')((l V;‘(q’?=
c4T;3K
/
- i2’)1W~Mr+), (174
- v)b”‘- (b” * q’?‘)),
(2bu2 - (b” * q”>‘>,
(17c)
where q” = q/q. The general inversion formula corresponding by means of the ansatz V$(q, r, +‘) = (2/3T)q’j,(+) + T;;3;){2bA
(17b)
to eq. (17a) can be obtained
[ hfliZr’(qO)L?j * b” -(b*
- qq(b”
. 49);;.],
(18)
which represents an orthogonal decomposition of V;$. Here the second term takes care of the right-hand side of eq. (8) and the matrix function Z(q? is obtained via an inhomogeneous system of equations easily obtained from eq. (8) Zr”‘(q? = 9,(,‘~3~)beV~‘(qqb~V6,l(qq x (-(1 + r)b:b;‘V::;(q’))+
I)$‘-‘,
(18a)
where V&(q@)= 2(b” - b@)- (ba . q@)(b’ . qy and I is the unit matrix. From the second term of eq. (18) it can be seen that a linearly independent set of Burgers vectors has to be used in order that Vi: is well behaved. If linearly dependent sets of Burgers vectors are used eq. (8) is not anymore sensible. An extension of the theory to cover this case which certainly gets important at high temperatures is easily accomplished but will not be presented here. For the functional potential Oo(r, i) in eqs. (9) to (12) the following ansatz can be made: @,(r, i’) = CL(r)
- i,
(19)
where d&(r) is an arbitrary vector function. Then the integrals appearing in eqs. (10) and (11) can be evaluated by means of the substitution given by eq. (16) which is necessary in order that these integrals are defined. Let us point out that so far we have taken into account only long range direction dependent interactions. If short range direction independent interactions are also
22x
A. HO12
included
in V,,, then it can be shown
@
short
range
that eq. (19) is substituted
by
* i.
interaction
(19’)
are a consequence
of nonlinear
volume
expan-
sion effects related to the core of dislocations. They are discussed in the non linear elasticity theory of dislocations’) and contribute to the discontinuous volume
expansion
during
the melting
the asymmetry of the atomistic themselves for the discontinuous
transition.
They
interaction forces melting transition
are a consequence
of
but are not responsible by as explained in section 1.
at the end of Section 1 they will not be taken into However, if a melting theory of ionic crystals is
Due to the reasons given account in the following.
formulated then due to the alzpearance of charged dislocations long range Coulomb forces have to be considered and then x(,(r) must be included. The propagators defined by eq. (12) can be represented in the form G(r, S,,; r’; 4,) = where the Schrodinger
ff” =
F
$k(r.
&)q!!;(r’,
&(f
A(r) = n,(r)
H(r)
The
7 - f&,(r) -i
K(b” X
=
to the
r))‘.
of a spinless
(21) charged
particle
moving
in the
+ !j rc(b” X r)
corresponds rot n,(r)
constraint
therefore
(20)
+ f,,/kTSL
set {I&} represents an orthonormal set of eigensolutions like eigenvalue problem with Hamiltonoperator
This is the Hamiltonoperator vector potential
to which
&) expt-(EL
on
a constant
a magnetic
(22)
field
+ Kb”. the
(22’)
conservation
magnetic
of
matter
field in the direction
given
by
eq.
(1) imposes
of the Burgers
vector
b”
and being of variable strength K. In addition one has a space dependent magnetic field which originates from the direction dependent interaction between dislocations. Having n linearly independent Burgers vectors one gets n propagators of the type given by eq. (20), which are coupled among each other through the strength K of the constant magnetic field. Further coupling between them is provided by the V&term in eq. (11). It should be pointed out that eq. (11) is not in the usual form to be treated by the methods of field theory. In that case it should be brought into Siegert’sz4) representation. It is, however, in a form which is useful for the following purposes because all diagrams appearing in a perturbative treatment of the partition function can be identified easily with the physical processes they correspond to.
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
229
Finally it should be pointed out that eq. (11) does not include phonon excitation. As far as the harmonic phonon excitations are concerned this is no problem because the free energy of those is just an additive term in the total free energy of the problem. Anharmonic phonon effects are more severe and lead to T-dependent renormalization of all coupling constants appearing in eq. (11) and in addition to many body interactions which have to be added to V,,,. Such effects have been studied in a similar two-dimensional problemz5). 4. Single loop approximation The evaluation of eq. (11) will be confined to a single loop approximation, i.e., an ideal gas of noninteracting loops. This is not necessarily a good approximation but it is sufficient for the present purposes. Four cases listed below in increasing order of difficulty will be considered. a) The free loop. b) The “magnetically” confined loop. c) The conservative loop. d) The magnetically embottled loop. In the present approximation it is sufficient to calculate the single loop integral in eq. (11) using the eigenfunction spectrum of eq. (21). a) The free loop In this case the eigenfunction spectrum of eq. (21) is continuous and the eigenfunctions can be normalized in a big sphere of radius R. One obtains easily for r+O G(r, S; 0; 0) = y,f 7T
dk k2jo(kr)jo(0) e-v’kT+k2’2m)s,
(23)
0
where jo(x) is a spherical Besselfunction of zeroth order, and 0(x) is the Heaviside function. Further integration occuring in eq. (11) is done using the identity r
I $!
m
emeSg(E) =
J5
e-(‘+x)‘Og(E)
(24)
0
‘0
and letting at the end of the calculation lo+ 0 in such a manner that only the finite part of the integral survives. Then E in the exponential of eq. (24) can be neglected and one obtains for the single loop partition function in the large loop approximation m
=dS
@O, 0) = 1 s
10
G(0, S; 00) = $1
277
dk k* o f/kT + x + k2/2m e-xb’
dx 1 0
(25a)
A.HOLZ
230 Or
(25b)
valid
for f 2 0. The
elementary
lattice
extra
integration
cell. This procedure
in eq. takes
(25b)
is performed
over
care of the probabilistic
one
property
of the path integral where a “closed” loop has to be interpreted as one coming back into the same cell where its initial point is located. In eq. (25a) this fact is taken care of by means of a cut-off in the eigenfunction lattice units are taken as 1. The quantity G-(0,0) increases function of decreasing f/kT and diverges for f + 0. Using
spectrum, monotonically for f + 0
f/kT = IT - T,J a representative heat Cr easily Cr=
where as a
(26)
quantity to describe the divergence of G(O, 0) is the specific obtained over the free energy F = -kT log Z
l/IT - T-,1”?.
(27)
b) The magneticully confined loop In this case only the constant magnetic term described by the vectorpotential eq. (22) is taken into account. For that case the propagator given by eq. (12) has been first calculated by Feynman (see Feynman and Hibb?). However, in order to make the subsequent integrations needed here tractable it is more convenient to start from eq. (20) and use a cut off in momentum space as done in eq. (25a). One obtains
G(O, Kb)
=
!$ j-dx i
(28)
0 where L represents the linear dimension of the normalization eigenfunctions of H are unbounded in b-direction). Furthermore and n and n, ran over all positive
integers
where
the prime
volume (the k; = (2~r/L)n
on the summation
sign confines them to the momentum cut off k, = 2~. The further integrations are straightforward but lead to awkward expressions. For consistency the final K-integrdoII is also confined by the momentum cut off via IK,,,,bI = kf. In case that eq. (25b) is used this is not necessary. The fact that we still talk about a single loop approximation is due to the cumulant expansion used in evaluating the exponential in eq. (11). The divergence when the system approaches T, is best represented by the specific heat singularity corresponding to eq. (27), i.e., c,,
=
1T - T,/“’
(29)
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
231
c) The conservative loop Although the magnetically confined loop does satisfy conservation of matter globally as required by eq. (1) it does not do that locally. This as a matter of fact is also not necessary but if one wants to distinguish between cases where transport of matter is easy or not then also the case where matter is conserved locally is of interest. The motions of the loop are then confined by its possible glide planes which can be reached by cross slipz2). Because the two-dimensional surface on which the loop is located after purely conservative glide motion has no intrinsic curvature it can be laid out on a plane surface as shown in fig. 3. The Greensfunction of the closed loop can then be calculated by first evaluating it in the (x, y)-plane between two walls being a distance 1 apart. In the second step the straight segment 1 is allowed to display its conformations in the (y, z)-plane. This implies crumbling the straightened out surface, or say leaf, back into its physical shape where selfintersections are allowed. Allowing all values of I between 0 and infinity gives all possible conservative loop conformations. The quantity corresponding to G(O,O) eq. (25a) can now be represented in the form G,(O,0) = j-dl 0
z: j- y 42
j- dSz 1’ dS, SI
I
m c
X
dx Gz((x, 1- E, Sz; E, O)Gz(x, 1 - E, S,; E, O), J OY
(30)
where Gz is the two-dimensional Greensfunction corresponding to eq. (12), i.e., fz = y - kT In ~2. Here t2 is the number of possibilities that exist at each step for the loop generating random walk process, i.e., z2 = 3 and m2 appearing in eq. (12) instead of m, is for the two-dimensional problem given by
Fig. 3. Defolded glide plane of a dislocation loop with Burgersvector b. Hatched areas mark inpenetrable walls for reference particle described by Schradinger like boundary value problem. I represents maximal linear extension perpendicular to b of the set of loops covered by this drawing. x,y-coordinates are in plane and z-coordinate points out of plane.
A. HOLZ
232
rnz = 1 (see refs. It cannot the
19 and 20). The quantity
be set to zero in the present
Hamiltonian
ditions, mations
of the
i.e., they
present
E in eq. (30) is of the order scheme
problem
vanish
for y = 0 and to be used in eq. (20) are
are
because subject
y = 1. The
of unity.
the eigenfunctions to the boundary
eigenfunctions
and
sin k,y,
to consum-
(31a)
c -$ j- dk, j- dk,,
(31b)
respectively, where L % 1 is a normalization length for the eigenfunctions. The subsequent calculations are straightforward but lengthy. The essential point of the result is that the specific heat singularity is the same as in case a) (32)
(‘CL= I/IT - T;(“‘, but T:/ T, = Evaluating from
In z In 2~ + (l/S)ln’
eq. (33) for z = 5 and z2 = 3 one obtains
eq. (33) that due to the reduced
loop the problem,
(33)
z2’
number
of freedom
of the
system is stabilized. Observe that for the purely two-dimensional i.e., without cross slip a specific heat singularity c = l/IT ~ T,,,:I is
obtained where T&T,,, = In 513. d) The magnetically embottled loop So far the interaction between loop parts represented f&,(r) in eq. (21) has not been taken into account. function
Tf/T,,, = 1.29. It follows
of degrees
which
appears
in the functional
integration,
by the vector Because n,,(r) the eigenfunctions
potential is some to eq.
(21) are not easily determined. One may. however, choose such functions C&,(r) for which the integrand of the functional integral eq. (1 I) displays a saddle point. Such methods are described in refs. 19 and 20. Because the term in eq. (1 I) involving the inverted interaction V,,,:, is multiplied by kT one expects that such vector potentials which maximise the loop integral may give the main contribution at low temperatures. What is required in that case is a magnetic field which confines the charged particle described by the Hamiltonian eq. (21) to the origin or say a magnetic bottle. From this hypothesis the title of this paragraph derives. The detailed calculations for the many magnetic bottle fields which have been done by the author will be published elsewhere*‘). The main result obtained so far is negative. Setting K = 0, i.e., ignoring conservation of matter no saddle point is found except for the origin
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
233
C&(r) = 0. This result is preliminary because so far only real and purely imaginary vector potentials have been studied and no complex vectorpotentials. If the present result turns out to be true then the small and large N limits studied in section 2 and explained by fig. 2 are not seperated by an activation barrier, but the dash pointed free energy curve applies. This then would also provide a simple explanation why in the planar rotator model a continuous phase transition is observed. An interesting question is if a saddle point takes off the origin once the homogeneous magnetic field representing the mass constraint is turned on, because once two saddle points are present the possibility of a discontinuous phase transition arises. Unfortunately so far no results with respect to that problem have been reached.
5. Discussion
and conclusion
We have studied in this paper various mechanisms defined over dislocation loop conformations which we hope will give some insight into the problem of the nucleation mechanism of the solid liquid transition. In section 2 we have first considered the problem of screening of long range interactions. At first sight it appears obvious that a small loop in order to become a large loop or say an infinitely extended loop has to pass over a free energy barrier. This would allow to explain the discontinuity of the transition as observed empirically. Unfortunately vortex loop systems interacting also via a long range Biot-Savart law as in the three-dimensional planar rotator model do exhibit a continuous transition. This puts the argument into considerable doubt and in fact there exist already on an elementary level reasons that a free energy barrier does not have to exist necessarily. One reason is that the small loop may grow over dipolar configurations of a tree like structure into a large loop. We have studied this problem in the appendix over an integral equation for the number of tree like structures. Because the restriction to tree like structures implies a monotoneous increase of trees with increasing size the possibility to see the development of an activation barrier is not inherent in this approach. The results obtained suggest to a certain degree that the cardinality of dipolar loop conformations approaches steadily the cardinality of the stochastically conformed loops if the constrained degrees of freedom are successively liberated. Because no mechanism is built into this model which allows a change-over from dipolar conformed loops to stochastically conformed loops a more powerful approach to this problem is needed. In section 4 we have studied this problem again under case d, using a functional integral formulation of the problem. The preliminary result obtained is that there exists no activation barrier (see fig. 2).
234
A. HOLZ
The answer
to the problem
of the discontinuity of freedom
which
are operative
at least three additional instance
in the
stated
law of melting
servation
of matter.
Second contrast
the dislocation to the current
rotator
may be simply
that the universality
of the additional
in solids but not in vortex
properties
planar
above
is a consequence of solids not found
model.
This implies
First
transport
that loop coformations
loops in solids are characterized strengths characterizing vortex
systems.
in vortex
degrees There
systems
of matter must
satisfy
are
as for
and
con-
eq. (I).
by a Burgers vector in systems. This allows a
three-dimensional manifold of defect states in solids in contrast to the one-dimensional manifold of defect states described by vortex systems. The third property is that the motion of dislocations changes the metrical properties of solids over torsion and curvature. The consequences
of the first distinguishing
property
stated
above
has been
studied in section 4 under cases b. and c. Case b. showed that taking matter conservation on the single loop level into account disregarding long range interaction weakens the singularity connected with infinite loop extension considerably. As can be shown and as one expects it does not imply a discontinuous transition. Because transport of matter in solids is a thermally activated process and because conservative and nonconservative processes do not have to take place with the same speed it is reasonable to study also the conformations of a loop where already locally matter is conserved, i.e., which is formed by a pure glide process. This has been done in section 4 listed under case c. Clearly such a constraint stabilizes the loop against expanding, and it could be shown that the transition temperature is increased in that case. The essential point of this example is demonstrated by studying the following fictitious process. Consider a loop formed by conservative and non conservative elementary random walk steps each showing y< and qnc possibilities of orientation respectively formations 2 of a loop of N steps
at each step. The total is then given by
number
of con-
(33) Because each step contributing to eq. (33) takes place over an activation barrier it is reasonable to attach the probabilities pc and pnc to the occurrence of conservative and nonconservative steps per time unit respectively, where pc + pnc = 1. Instead of eq. (33) one obtains now
z+
Z’ = (pcq, + pncqnc)N.
In the authors thermodynamic ses contribute
(33’)
opinion eq. (33’) should be prefered to eq. (33) in calculating properties in a system where strongly rate dependent procesto the thermal formation of structures like dislocation loops
QUALITATIVE
THEORY
OF SOLID-LIQUID
TRANSITION
235
requiring extensive matter transport. This is in fact obvious because if the partition function is calculated only with sets of final states like loop conformations then all processes the system has to activate in order to get from one state to another are disregarded. Although on a very large time scale the number of conformations of the loop of N steps is given by eq. (33), in a smaller time interval At the number of realizations counted starting from some arbitrary conformation is given by eq. (33’). This phenomena is also known from the theory of quenched disorder or the theory of glasses where the free energy has to be calculated for one possible sector of the phase space of the system. Applying this idea to the nucleation process of melting we may assume that at some favorable place at the surface or in the bulk an energy fluctuation produces a small loop over a shear process. Because things happen on a small time scale the arguments given above imply that only such loop conformations are reached where the connecting processes to the initial state are the quickest. Hence many possible loop conformations reached only over complicated intermediate processes (e.g. trapped in configurations) will not be activated. In particular we think that the diffusive motion of matter in the solid state occurs on time scales too large to guarantee the occurrence of all possible loop conformations with equal probability per time unit. However, once the loop has reached by chance a critical size the stiffness may be reduced in the area affected by the loop’s polarizability that much that diffusive motion of matter may assume a collective character. Then suddenly many more degrees of freedom are in action per time unit and the statistical description of matters must shift over from eq. (33’) to eq. (33). The example developed under case c, in section 4 demonstrates then easily that such a phenomena may be connected with a discontinuous transition. We would like to emphasize that the present discussion does not rely on any specific properties of the initial state on top of which the nucleus forms but is of purely generic character. Eventually we propose that melting is discontinuous because the activation of conservative and nonconservative processes is such interdependent that none of them can activate its degrees of freedom without the other having them activated as well. Because the polarization potential of a loop depends on the loop having a critical size where it is able to screen long range interactions, and the collective diffusive motion requires just that polarization to get into motion, whereas the polarizability of the loop sets only in once the collective diffusive motion releases trapped in configurations, the strongly coupled qualities of both processes are at once obvious. It follows also that the transition must be discontinuous. That melting usually starts at free surfaces can also easily be expiained within that scheme. Polarization and transport properties are facilitated by the surface
A.HOLZ
236
due to mirror-forces, facilitating conditions requires
additional
Because
volume
consequence
free space and heat transport, whereas in the bulk such do not exist and besides the usual volume expansion
work to be dissipated expansion
of the increased
easy formation
of vacancies
once a nucleus
in the liquid density at free
state
under
of vacancies surfaces
is formed
in the bulk.
ordinary
conditions
is a
as stated
in section
I the
may be considered
as the main
driving force for the nucleation of the liquid phase at such surfaces. Finally we like to mention that the branching processes contributing partition function given by eq. (11) have not been studied in this Because these will lead to a interaction are where the third a simple model
to the paper.
branching processes are entropy generating in all orders they destabilization of the ground state for T > 0 if the elastic not properly taken into account. A free propagator model term of eq. (12) is neglected will therefore make no sense and taking such processes into account is not easily conceived.
Acknowledgements The author likes initiated melting, amorphous state melting in polymer
Appendix
to thank Professor H. Gleiter for discussions on surface Prof. H. Kronmtiller for stimulating discussions on the and Prof. J. Naghizadeh for interesting discussions on systems.
A
The counting problem for dipolur configurations In order
to count
the tree like structures
of a loop having
n links as the one
e.g. shown in fig. 1 we have to add up all possible .trees with the origin fixed and total trunk and branch length N = n/2. To generate the tree we use the building blocks formally depicted in fig. 4a. A simple unit of the tree requires two links of the original loop, a branching element of order (Y3 2 requires 2(a! + 1) links of the original loop. The total number of tree configurations W(N) obeys an integral equation which is graphically depicted in fig. 4b where for the sake of simplicity we have confined ourselves to branching elements of order three. Assume that the simple unit of the tree has q. possibilities and a branching element of order (x, qz” possibilities of orientation respectively. If rno is the number of simple units along the trunk of the trees in fig. 4b and mk the number of branched off trees of total branch length
QUALITATIVE
THEORY
-
Tr
’
TRANSITION
237
&l
+--,---I
c--_(
-
OF SOLID-LlQUID
’
+-$++
a=2
’
a=3
Fig. 4. a) Elementary units used in constructing dipolar structures as depicted in fig. 1. From left to right are drawn a simple tree unit, and branching units of order 2 and 3 respectively. Full drawn lines refer to original oriented links of loop where segments are indicated by vertical dashes. Dashed lines mark tree skeleton. b) Graphical representation of eq. (A. I). The hedgehogs attached to the root or trunk of the graph on the left- and right-hand side of the equation respectively refer to the complete set of possible trees with N and k links respectively.
k 2 1 then fig. 4b is translated
into the equation
(A.11
where each term corresponds to one graph of fig. 4b. The prime on the summation sign implies that such sets {mk}N are used which satisfy N=mo+g
mk(k+3). k=l
(A.21
Next we define the generating function
G(Z) =
N$o ZNWW
C-4.3)
A. H0I.Z
238
where
2
M =
mr, k=O
Zk = &,oq,Z + (I - &.“)W(k)qZZk+3
(A.4)
and Z is a complex number. Eq. (A.3) has been derived using the multinomial theorem and analytical continuation. Writing eq. (A.3) in the form G(z) = (I - [qoz + q;z’(i: observing
h-0
W(k)zk
- W(O)!)i
W(0) = 0 and the definition
G(Z) = (1 - qoz - (q?z)‘G(z)) This equation
can be solved
‘,
of G(z) one obtains (A.5)
‘.
for G(z)
yielding
(A.61
. Because this is an analytical one obtains using the Cauchy W(N)
where
=
the contour
oneself
determined
’ - (1 -
Because
is finite
at the origin
(G(0) = I)
is around
(A.7) the origin
avoiding
the square
for N + I using the saddle
that the leading
contribution
point
comes
root branch method.
from
cuts of
One easily
the saddle
point
by 4(qzz)’
i
which
&f d$y,
G(z). Eq. (A.7) is best solved convinces
function theorem
there
“?
qozY~ = l/N.
N % 1.
is no need to get more than an estimate
just for two reasonable W(N)
= (2qo)““;
W(N)
= ($4”7qo)n”;
numbers
we will give the results
of q:.
qz = q,,/4”‘,
(A.9)
q? = qo.
(A.10)
Here we have used N = n/2 and n is the number of original ing that qo= 5 one notices that the tree like structures cardinality of the stochastic loop structures because in that front of qo in eqs. (A.9) and (A.10) should be of the order however, be pointed out that we have not taken all degrees
links. Rememberdo not have the case the factor in of qo. It should, of freedom of the
QUALITATIVE
THEORY OF SOLID-LIQUID
TRANSITION
239
dipolar structure into account. The two foldedness of the trees allows to a certain degree rotational degrees of freedom to be activated. In addition to each tree characterized by the dashed trunk as in fig. 1 many more dipolar structures (enveloping oriented line in fig. 1) correspond than just the one drawn in fig. 1. These dipolar structures are characterized by crossing points of the oriented lines at the trees branching points in a planar representation. These effects will increase qo_
References
2) 3) 4) 5) 6) 7) 8) 9) 10) ll) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27)
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