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\begin{document}
\makeatletter
\renewcommand{\@oddhead}{Mater.Phys.Mech. {\bf 1} (2000) 00-00\hfil\thepage}
\renewcommand{\@oddfoot}{\copyright 2000 Advanced Study Center Co. Ltd.\hfil}
\makeatother

\vspace*{2cm}
\begin{center}
{\Huge\sf\bf TEM and STM investigations on the disclination nature
of fragment boundary triple junctions}
\end{center}
\vspace{0.3cm}


\centerline{\bf M. Seefeldt}\noindent
\begin{center}
{\small
Freiberg University of Mining and Technology,
Institute of Physical Metallurgy,
Gustav-Zeuner-Stra{\ss}e 5,
D-09596 Freiberg, Germany\\
{\it Present address}:
Catholic University of Leuven,
Department of Metallurgy and Materials
Engineering,
Willem de Croylaan 2,
B-3001 Heverlee, Belgium\\
{\it E-mail}: Marc.Seefeldt@mtm.kuleuven.ac.be}
\end{center}
\begin{quote}
{\small Received October 1999}
\end{quote}
\begin{quote}
{\small{\bf Abstract.}
After a short review of recent TEM measurements to characterize
partial
 disclinations in fragment boundary triple junctions in
cold-rolled copper, STM results
 from the same specimens are presented
which support a disclination interpretation of
 the observed fragment
structure. The role of the fragment boundary mosaic or of
 the
corresponding immobile partial disclination network in work-hardening
is discussed.
 Although the orientational mismatches around the
fragment boundary triple junctions
 are only small, the disclination
contribution dominates work-hardening behaviour at
 large strains.}
\end{quote}


\section{Introduction}

The substructure development of crystalline metals under cold
deformation up to large strains is characterized by the formation
of a fragment structure on the mesoscopic scale [1].
Typically, with increasing strain the mean fragment size decreases
monotonously starting from the micrometer range, whereas the mean
misorientation across the fragment boundaries increases monotonously
reaching several degrees [1-3]%[\cit{Ryb86}-\cit{Liu98}]
.\footnote{Hansen
and coworkers use the term "cell block" instead of fragment.}
In contrast to the cell structure, the fragment structure does not
saturate\footnote{Saturation means in this context that average
cell structure parameters like cell size, cell wall width, or
misorientation do not change any more.} and its formation and
evolution are quite universal phenomena which are observed in fcc,
hdp, and bcc single- as well as polycrystals [1].

While the coexistence of the cell and fragment structures and the
evolutions of their characteristic parameters are well investigated
[1-3], the defect composition of the fragment
boundary mosaic and its influence on work-hardening are not yet
completely understood. Especially, to the authors' knowledge, there
is no work-hardening model up to now which takes the fragment
structure explicitly into account.

Following the Russian school of plasticity, the authors propose the
triple junctions of the fragment boundaries to have a disclination
character [4-6]
 and, thus, to be sources of
long-range stresses hindering dislocation motion and, thereby,
contributing to work-hardening at large strains.

\section{Noncompensated nodes of fragment boundaries}

\makeatletter
\renewcommand{\@evenhead}{\thepage\hfil {\it M.Seefeldt}}
\renewcommand{\@oddhead}{\thepage\hfil {\it Disclinations at triple junctions}}
\renewcommand{\@oddfoot}{}
\makeatother
According to experimental observations, a model describing the fragment
structure development as well as its coupling to the macroscopic
mechanical behaviour requires tools taking into account such collective
behaviour of dislocations that results into the formation, spread-out
and enlargement of misorientations and to the presence of long-range
stresses in the fragment interiors. A model on the substructure
development and work-hardening under cold deformation up to large
strains suggested by the present authors [7,8]
uses disclinations [9,10] as such a tool. Especially,
the formation of partial disclination dipoles (PDD) is modelled
through incidental non-balanced trapping of neighbouring mobile
dislocations of the same Burgers vector into cell walls and by
incidental trapping of mobile dislocations with "matching" Burgers
vector into preexisting fragment boundaries [8]. The first
PDD configuration corresponds to a both-side terminated excess
dislocation wall, the second one to a tilt or twist boundary segment 
with enlarged misorientation. To describe the non-conservative
propagation of the partial disclinations of such dipoles along cell
walls or fragment boundaries, respectively, the capturing mechanism
proposed by Vladimirov and Romanov [10,11] is used.

If a partial disclination propagates like that and approaches a
preexisting fragment boundary, then it will rotate the crystal
regions to the left and to the right of the growing backward
fragment boundary against each other (Fig.~1 b). When it is
finally locked in the preexisting fragment boundary, an
orientational mismatch is left behind (Fig.~1 c). This mismatch
can be accomodated in two different ways: either "plastically" by
generation of an additional dislocation boundary or "elastically"
by rotational distortion. In the latter case, the triple junction
is a non-compensated node containing a line defect of the
disclination type.

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{\small{\it Figure 1:} A propagating partial disclination approaches a
preexisting fragment boundary, is locked in this boundary and
leaves an orientational mismatch behind.}
\vspace*{2ex}
\end{figure}

%\begin{figure}
%\centerline{\epsfig{file=fig1.eps, height=3.5in,width=3.5in}}
%\vspace*{8cm}
%{\small Figure 1. }
%\end{figure}

\section{TEM results}

Since partial disclinations locked in the triple junctions of
the fragment boundaries should -- at least at low homologous
temperatures -- be stable under load relaxation, they are
suitable objects for an experimental confirmation of the
disclination approach used in the above mentioned model.
TEM microdiffraction measurements around fragment boundary
triple junctions in copper single and polycrystals rolled down
to 70\% and 50\% thickness reduction at room temperature,
respectively, showed
[12]
\begin{itemize}
\item that the product of the three misorientation matrices does
      not give the identity and
\item that the local orientation varies continuously when passing 
      with the electron beam by the node.
\end{itemize}
Whereas the first observation proves an orientational mismatch,
the second one points towards its accomodation by elastic distortion
rather than by an additional dislocation boundary. Both items together
indicate that the triple junction includes an additional line defect
which has -- probably beside other contributions [13,14]
-- a disclination nature. The measured orientational mismatches
correspond to Frank vectors of
\boldmath\begin{equation}
\omega\mbox{\unboldmath{$=[-0.0292,\,0.0094,\,0.0052]
\quad\mbox{with}\quad|$}} 
\omega\mbox{\unboldmath{$|=1.78^{\circ},$}}
\end{equation}\unboldmath
for a node in the single crystal specimen and 
\boldmath\begin{equation}
\omega\mbox{\unboldmath{$=[-0.0046,\,-0.0316,\,-0.0004]
\quad\mbox{with}\quad|$}} 
\omega\mbox{\unboldmath{$|=1.83^{\circ},$}}
\end{equation}\unboldmath
for a node in the polycrystal specimen. These PD powers are in good
agreement with the theoretical expectation of
\boldmath$\mbox{\unboldmath{$|$}}\omega
\mbox{\unboldmath{$| \approx 1-3^{\circ}$}}$\unboldmath 
given by Romanov and Vladimirov in [10]. At present, PDD
groups in a copper single crystal rolled down to 60\% thickness
reduction as shown in the TEM micrographs in Figs. 2 and 3 are
analysed.

First results indicate that the PDD groups are not fully compensating
dipole (Fig. 2) and quadrupole (Fig. 3) configurations, respectively.

\section{STM results}

Since it is very difficult to find sufficiently large perfect
crystal volumes to obtain Kikuchi patterns in the surroundings
of the nodes in heavily deformed metals, an additional local
characterization method which does not make use of diffraction
is desirable. Scanning tunnel microscopy (STM) is such a method.
In their pioneering works, Vettegren' et al.
[15-19]
 have studied the profiles
of polished surfaces of loaded copper, gold, and molybdenum specimens
and found clusters of etch pits. The pits have been explained by the
exit of dislocation groups after bursting through barriers on
intersecting glide planes, whereas the clusters have been interpreted
as an early stage of crack formation [20].

Unfortunately, the STM contrast theory is not yet very well
developed. It is known [21] that the strong electric
field between the tip and the surface has an etching effect.
It is generally admitted that this ``field etching" should also
result into preferred evaporation and/or surface diffusion at
high-energy surface regions. From an analogy to Mullins' thermal
etching theory [22], one can conclude that etch grooves
should arise and deepen at the lines where planar defects like
grain boundaries or fragment boundaries reach the surface.
Depth and shape $z(x,t)$ of the grooves scale with the plane
energy $\gamma$ of the planar defect, e.g. for thermal etching
with dominating evaporation and condensation through [22]
\begin{equation}
z(x,t)=-2\tan\beta\sqrt{At}\,\,\mbox{ierfc}\left(\frac{x}{2\sqrt{At}}\right)
\end{equation}
with $\beta$ defined by the mechanical equilibrium between the
planar defect (e.g. grain boundary) tension and the surface tensions,
$2\gamma_{sf}\sin\beta=\gamma_{gb}$, and $A$ given by 
\begin{equation}
A=\frac{p_0\,\gamma_{sf}\,\Omega^2}{(2\pi M)^{1/2}(k_BT)^{3/2}}
\end{equation} with vapour pressure $p_0$ for plain surface,
atomic volume $\Omega$, atomic mass $M$ and temperature $T$.
Fragment boundaries can be approximated as low-angle grain boundaries,
so that their plane energy can be expressed through the misorientation,
e.g. for a tilt boundary [23]
\begin{equation}
\gamma_{fb}=\frac{Gb}{4\pi(1-\nu)}\varphi
\left(\ln\frac{b}{r_0}-\ln\varphi\right)
\end{equation}
with misorientation $\varphi$ and dislocation core radius $r_0$. 
Thus, at least in the thermal etching analogy, the groove depth
and shape are directly related to the fragment boundary misorientation. 
Correspondingly, "field etching" should also result into notching of
etch pits at the points where line defects like dislocations or
disclinations reach the surface. Depth and shape of the pits should
then scale with the line energy, that means with the powers or with
the moduli of the Burgers or Frank vectors, respectively. Although
Romanov has given the elastic fields around PDDs reaching the surface
[24], there is no etch pit theory up to now, not even for the
thermal analogy. Therefore, the following discussion has to remain
a qualitative one: If the triple junctions of fragment boundaries
include additional high-energy line defects, then the corresponding
etch pits should be much deeper than the etch grooves corresponding
to the joining fragment boundaries. 

For the present work, cut surfaces of copper polycrystals rolled at
room temperature have been abraded, polished and treated with the
basic alumina OPS before being investigated with a Nanoscope II STM.
Fig. 4 shows a $5 \times 5 \mu m^2$ surface profile from a copper
polycrystal rolled down to 71\% thickness reduction. The parallelity
of the two grooves on the left (to each other and also to the very
much different structure of a chain of nearly equidistant small pits
marked with the arrows "C"!), the similarity of the defect structure 
with the one observed in TEM (confer e.g. [2,25]),
especially the band width in the micrometer range and the splitting
up of one groove into two neighbouring ones at the point ``A"
(confer splitting up of dense dislocation walls (DDW) into
first generation microbands (MB1), e.g. [2,25])
point to a deformation-induced rather than to a polishing effect.

Fig. 5 gives the scanning profiles for the three lines marked
in Fig. 4, line 1 cutting through the two neighbouring ordinary
grooves, line 2, in addition to this, cutting through the triple
junction marked with "B", and line 3 cutting through the splitting
point marked with "A". Obviously, the pits at the triple junctions
are deeper than the ordinary grooves by about a factor 2 which
might indicate the presence of an additional high-energy line
defect, e.g. a partial disclination.

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%\caption[STM-Tiefenprofil Linie 1]{\label{Profil71-12} Tiefenprofil 
%entlang der Rasterlinie 1 aus Abbildung \ref{STM1}. 400 Rasterpunkte 
%entsprechen $5\,\mu m$.}
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%\caption[STM-Tiefenprofil Linie 2]{\label{Profil72-31} Tiefenprofil
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%entsprechen $5\,\mu m$.}
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\end{picture}
%\caption[STM-Tiefenprofil Linie 3]{\label{Profil73-94} Tiefenprofil
%entlang der Rasterlinie 3 aus Abbildung \ref{STM1}. 400 Rasterpunkte 
%entsprechen $5\,\mu m$.}
{\it Figure 5:} STM surface profiles along the lines 1-3 in
Fig.~4. 400 scanning points correspond to $5 \mu m$. 
\end{center}
\end{figure}


\section{Effect on work-hardening}

In a disclination-based model for the fragment structure development
like the one proposed in [7,8], propagating PDDs are
immobilized as discussed in section 2 and Fig. 1 -- resulting in
a mosaic of fragment boundaries or, correspondingly, in a network of
immobile partial disclinations in the node lines of this mosaic.
The evolutions of the cell wall dislocation density and of the
propagating and immobile disclination densities can be described
with the equations [8]
\begin{eqnarray}
\frac{d\rho_c}{d\epsilon}&=&
I\sqrt{\rho_c}-R\rho_c\label{rhoc}\\
\frac{d\theta_p}{d\epsilon}&=&
\frac{K_c}{N}\sqrt{\rho_c}+\frac{K_f}{N}\sqrt{\theta_i}
-J\sqrt{\theta_i}\theta_p\\
\frac{d\theta_i}{d\epsilon}&=&
J\sqrt{\theta_i}\theta_p.\label{thetai}
\end{eqnarray}
including immobilization ($I$-term) of mobile and dynamic recovery
($R$-term) of mobile with cell wall dislocations, PDD generation
by incidental non-balanced trapping of $N$ neighbouring mobile
dislocations into cell walls ($K_c$-term) and by incidental trapping
of mobile dislocations with the "right" Burgers vector into fragment
boundaries ($K_f$-term), and immobilization of propagating PDDs in
the network of immobile ones ($J$-term). The coefficients can be
estimated from physical models for the elementary processes
[7,8]. Whereas the cell wall dislocation density
saturates according to the Kocks-type equation (\ref{rhoc}), the
immobile disclination density increases monotonously according
to equation (\ref{thetai}) -- corresponding to a monotonously
decreasing mean fragment size.

The cell wall dislocation density is coupled to a flow stress
contribution according to
\begin{equation}
\Delta\sigma_{\rho}=
\frac{\xi \alpha Gb}{m_S}\sqrt{\frac{\rho_r}{\xi}},
\label{flowc}\end{equation}
where $m_S$ is the Schmid factor and $\xi$ denotes the volume
fraction of the cell walls. The effect of the immobile
disclination density on the flow stress is described by
Romanov's and Vladimirov's law [10] \boldmath
\begin{equation}
\mbox{\unboldmath{$\Delta\sigma_{\theta}=\beta G|$}}
\omega\mbox{\unboldmath{$|.$}}
\label{flowf}\end{equation}\unboldmath
The mean disclination power \boldmath
$\mbox{\unboldmath{$|$}}\omega\mbox{\unboldmath{$|$}}\,$
\unboldmath is equivalent to the average
orientational mismatch around the node lines and may
-- following the discussion in section 2 and Fig. 1 --
be approximated by the mean misorientation $\varphi$.
Assuming a Poisson-Voronoi geometry of the fragment
boundary mosaic, the mean misorientation is calculated
by distributing the available excess dislocation density
which develops according to ($E$-term describes capturing
of mobile dislocations by propagating partial disclinations)
\begin{equation}
\frac{d\rho_{exc}}{d\epsilon}=
K_c\sqrt{\rho_c}+K_f\sqrt{\theta_i}+E\theta_p\label{rhoexc}
\end{equation}
on the total fragment boundary plane area, finally giving
[7]
\begin{equation}
\varphi \approx b\rho_{exc}/1.21\sqrt{\theta_i}.
\end{equation}
Integration of the evolution equations (\ref{rhoc}-\ref{thetai})
and the flow stress contributions (\ref{flowc}-\ref{flowf}) shows
that the disclination contribution to flow stress dominates the
dislocation one starting from a von Mises strain of about $0.9$
(parameters for copper at room temperature). Stage III of single
crystal plastic deformation can be attributed to cell structure
or dislocation work-hardening, stage IV to fragment or disclination
work-hardening [7,8].

\section{Conclusions}

TEM microdiffraction measurements indicate the presence of
line defects of disclination nature in the triple junctions
of fragment boundaries. STM shows deep etch pits at the points
where the triple junction lines reach the surface -- pointing 
to high-energy line defects in the triple junctions. Although
the orientational mismatches remain small, the disclination
long-range stress fields do have a significant influence on
work-hardening and even dominate the dislocation contribution
at large strains.
\vspace*{4ex}

{\large\bf References}\\

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\newpage
{\large\bf Figure Captions}\\


Fig. 2. TEM micrograph of a partial disclination
dipole configuration. Copper single crystal rolled down to
60\% thickness reduction at room temperature.
$2.61 \times 1.90 \mu m^2$.\\


Fig. 3. TEM micrograph of a partial disclination
quadrupole configuration. Copper single crystal rolled down to
60\% thickness reduction at room temperature.
$1.59 \times 1.16 \mu m^2$.\\

Fig. 4. STM surface profile with fragment boundary
grooves. Copper polycrystal rolled down to 70\% thickness
reduction at room temperature. Arrow "A" marks a splitting point
of a fragment boundary, arrow "B" a triple junction, arrows "C"
a chain of nearly equidistant shallow etch pits. $5 \times 5 \mu m^2$.
Grey scale from white to black corresponding to $15 nm$ depth.


\end{document}

