# Coexistence of multiple interfacial states at heterogeneous solid/liquid
interface

Jiaojiao Liu  State Key Laboratory of Precision Spectroscopy, School of
Physics and Electronic Science, East China Normal University, Shanghai 200241,
China  Hongtao Liang  State Key Laboratory of Precision Spectroscopy, School
of Physics and Electronic Science, East China Normal University, Shanghai
200241, China  Research and Development Department, Zhangjiang Laboratory,
Shanghai 201204, China  Jinfu Li  School of Materials Science and Engineering,
Shanghai Jiao Tong University, Shanghai 200240, China  Brian B. Laird
Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, United
States  Yang Yang  State Key Laboratory of Precision Spectroscopy, School of
Physics and Electronic Science, East China Normal University, Shanghai 200241,
China

###### Abstract

The growing trend towards engineering interfacial complexion (or phase)
transitions has been seen in the grain boundary and solid surface systems.
Meanwhile, little attention has been paid to the chemically heterogeneous
solid/liquid interfaces. In this work, novel in-plane multi-interfacial states
coexist within the Cu(111)/Pb(l) interface at a temperature just above the Pb
freezing point is uncovered using atomistic simulations. Four monolayer
interfacial states, i.e., two CuPb alloy liquids and two pre-freezing Pb
solids, are observed coexisting within two interfacial layers sandwiched
between the bulk solid Cu and bulk liquid Pb. Through computing the spatial
variations of various properties along the direction normal to the in-plane
solid-liquid boundary lines for both interfacial layers, a rich and varied
picture depicting the inhomogeneity and anisotropy in the mechanical,
thermodynamical, and dynamical properties is presented. The “bulk” values
extracted from the in-plane profiles suggest that each interfacial state
examined has distinct equilibrium values from each other and significantly
deviates from those of the bulk solid and liquid phases, and indicate that the
“complexion (or phase) diagrams” for the Cu(111)/Pb(l) interface bears a
resemblance to that of the eutectic binary alloy systems, instead of the
monotectic phase diagram for the bulk CuPb alloy. The reported data could
support the development of interfacial complexion (or phase) diagrams and
interfacial phase rules and provide a new guide for regulating heterogeneous
nucleation and wetting processes.

###### pacs:

##  I Introduction

Since the term interfacial “complexion” – thermodynamic equilibrium states of
an interface – having been introduced and recognizedTang06a  over the past two
decades, see in Ref.Cantwell14,  and references therein, numerous grain
boundary complexions with different segregation feature and thicknesses in
ceramics and alloys have been unveiled. The manipulation of the interfacial
complexion transition offers potential engineering of various interfacial-
related processes (e.g., sinteringGupta07 , embrittlementLuo11 , and crystal
growthDillon07 ) and interfacial properties (e.g., interfacial mobilityHolm03
, atomic or ionic diffusivitiesDivinski12 ; Bowman17 , electrical or thermal
conductivitiesMarvel18 ; Zong17 .

The majority of recent studies of the multiple interfacial complexion (or
phase) coexistence and transition are made in grain boundary systemsHeckman18
; Brien18 ; Peter18 ; Meiners20 ; Barr21 ; Tuchinda22  and solid
surfacesWynblatt07 ; Pickering18 ; Tang21 ; Wang21 . In contrast, the idea
that the chemically heterogeneous solid/liquid interfaces can also exhibit
multiple interfacial complexion (or phase) coexistence and transitions,
similar to the grain boundary and surface systems, is rare. Such disparity is
because the direct experimental characterization on the interfacial layers for
the solid/liquid interfaces is more challenging than for the free surfaces and
grain boundariesGabrisch01 ; Oh05 ; Moorthy13 ; Schneider17 . With the aid of
advanced computer simulation techniques, several novel interfacial structural
states and interfacial phase transitions are uncovered, such as the lateral
and layer ordering of the interfacial liquidsYang14 ; Kern20 , novel pre-
freezingPalafox16 , roughening and premelting transitionsYang13 . Few existing
studyLiang18  suggests two interfacial states (or phases) can in-plane coexist
within the heterogeneous solid/liquid interface.

Compared with the experimental and computational achievements, the theoretical
frameworks regarding the statistical mechanical nature of the in-plane
interfacial complexion (or phase) transitions remain less developed. For
example, several types of phenomenological models have been proposed to
qualitatively explain the various transitions among interfacial complexions
(or phases) (lattice gas-based modelWynblatt05 ; Luo09 ; Rickman16 , regular
solution-based modelWynblatt08 , diffuse-interface modelTang06b , and force-
balance modelClarke93 ), in addition, a theoretical attempt by Frolov and
Mishin has been proposed to extend thermodynamic formalism of the phase
equilibria and phase rules into semi-2D interfacial systemsFrolov15b .
Meanwhile, efforts have been devoted to developing interfacial complexion (or
phase) diagramsMishin09 ; Luo12 ; Gao19 , see more details in Ref.Cantwell14,
. However, most of these efforts focused on the simplified version of the
interfacial complexion (or phase) diagram, which could not be verified by
experiment yet. Therefore, existing thermodynamical models in predicting
interfacial complexion (or phase) diagrams and transitions have not been
rigorously validated, thus hindering the breakthrough regarding the precise
control of the interfacial structure and various properties affected by the
interfacial complexion (or phase) transitions.

In this study, based on the equilibrium molecular-dynamics (MD) methods and
the embedded atom model, we report an atomistic simulation of a chemically
heterogeneous Cu(111)/Pb(liquid) interface, in which four interfacial liquid
and interfacial solid states, two boundary lines, coexist within two atomic
layers sandwiched between the bulk solid Cu and liquid Pb. Note that the term
“interfacial state” will be used more frequently in describing the current
simulation results, rather than “interfacial complexion (or phase)”, to avoid
potential terminological disagreements. The elegant solid/liquid in-plane
equilibria among these interfacial states and their thermodynamics and
mechanical properties are carefully characterized by computing the in-plane
profiles of the densities, compositions, potential energies, pressure
components, stresses, and diffusivities. In the current heterogeneous
solid/liquid interface system, we confirm that the equilibrium conditions for
the multiple in-plane equilibria among interfacial states can significantly
deviate from the bulk phase diagram, consistent with the fact that GB
complexion diagrams dramatically different from that of the bulk
phasesCantwell14 . We found that the immiscible Cu and Pb (monotectic phase
diagram) can form homogeneous monolayer alloy liquid states, which could
coexist with the interfacial pre-freezing Pb solid and(or) the solid Cu step
islands. The calculated data, especially the local pressure tensor components,
could support the development of interfacial complexion (or phase) diagrams
and offer clues to dig into the fundamental physics of the interfacial
complexion (or phase), thus facilitating the advancement in interface (or the
interfacial complexion) engineering.

##  II Methodology

In this study, the many-body embedded atom method (EAM) potential for CuPb
alloy parameterized by Hoyt et al.Hoyt03b  is employed to describe the
interactions between atoms. It yields a reasonable prediction of the bulk
binary phase diagram (including the melting points of pure Cu (1279K) and Pb
(618K), as well as the mutual immiscibility between two metals in both liquid
and solid phases). This interatomic potential has been used to explore the
wettingWebb03  and spreading dynamics of Pb droplet on Cu substratesHeine05 ,
heterogeneous nucleationsPalafox16  and equilibrium thermodynamic
propertiesPalafox11  across the Cu(solid)/Pb(liquid) interfaces.

The simulations are carried out using the large-scale atomic/molecular
massively parallel simulator (LAMMPS)lammps  at constant $T=620$K, which is 2K
above the freezing(or melting) point of Pb. The time-step and the Nośe-Hoover
thermostat relaxation time are set as 1.0fs and 0.1ps, respectively. The
construction of the equilibrium Cu(111)/Pb(l) interfaces draws on the
techniques described in Refs.Palafox11, ; Yang12, ; Liang18, . The
fcc(111)-oriented Cu crystal sample (84$\times$24$\times$11 unit cells,
133,056 Cu atoms) is constructed with (111) surface parallel to the $x$ and
$y$ axes of the simulation cell using the lattice parameter at 620K
(3.653266Å) determined in a separate $NPT$ simulation. The cross-section
($xy$) dimension of the cell for simulating the crystal sample is
375.84Å$\times 62.00$Å, which is also employed in the separate preparation of
two Pb liquid samples (45,000 Pb atoms in each sample) at the same
temperature. The prepared crystal and two liquid samples are assembled by
applying the periodic boundary conditions (PBC) along $x$ and $y$ to the
initial conglomerate (around 223,000 atoms), and the crystal sample is set in
between the two liquid samples. Along $z$ axis, the normal direction of the
solid/liquid interface, we insert two vapor phases (33Å thickness in $z$ each,
see in Fig.1) next to the two liquid samples, resulting in two liquid-vapor
interfaces to aid in modeling the hydrostatic conditions both in bulk solid Cu
and liquid Pb, in the following equilibration $NVT$ MD simulation. Note that
the thickness of the solid and liquid samples is over 60Å, which is thick
enough to avoid the short-range interaction between solid/liquid interfaces
and liquid-vapor interfaces. The linear momentum for the bulk Cu is fixed as
zero to avoid its Brownian motion.

As based on the phase diagram (computationally verified by us using the semi-
grand canonical Monte-CarloBecker06 ), the equilibrium mole fractions of Cu in
bulk liquid Pb is around $X_{\mathrm{Cu}}^{l}(620$K$)=0.007(1)$, of Pb in bulk
crystal Cu is negligible, or $X_{\mathrm{Cu}}^{s}=1$. So, prior to the
equilibration $NVT$ MD simulation for the initial conglomerate, we randomly
replace 310 Pb atoms in two bulk liquid sample parts into Cu atoms, yielding a
composition value nearly equal to $X_{\mathrm{Cu}}^{l}(620$K$)$. Using this
simple setup, i.e., put liquid Pb (with dilute solute atoms) on a clean Cu
surface, two interfacial hexagonal pre-freezing Pb layers (rotated 6∘ relative
to the underlying Cu(111) lattice) are easily formed within a few hundred
picoseconds during the equilibration $NVT$ MD simulation, as reported in
Refs.Palafox11, ; Palafox16, .

Until this work, the Cu(111)/Pb(l) interface at different temperatures had
only been modeled in a “clean” fashion, i.e., two layers of pure Pb next to
the complete Cu (111) surfaceWebb03 ; Heine05 ; Palafox11 ; Palafox16 .
Enlightened by recent grand-canonically optimized atomistic simulation
methodologies devoted to searching new GB complexion transitions with more
possible sets of local structures, densities, and compositionsCantwell14 ;
Cantwell20 . We examine in this work more possible equilibrium Cu(111)/Pb(l)
interfacial states, which have different local composition values in the
interfacial layers. Since the mass transportation of the solid/liquid
interface is significantly greater than that of GBs, sufficient structural and
compositional equilibration in both the interfacial and bulk regions can be
easily achieved within several tens of nanoseconds via MD simulation, and
there is no need to carry out additional grand-canonical Monte-Carlo
simulation. Based on the pre-freezing Cu(111)/Pb(l) interface obtained as
described in the previous paragraph, in the first pre-freezing Pb layer
adjacent to the Cu surface, we randomly replace a portion of Pb atoms into Cu
atoms, yielding a variety of in-plane Cu composition valves – ranging from 0%
to 60%, separated by a few percents. This operation is followed by an
equilibration $NVT$ MD simulation lasting for 80 ns, in which both the
microstructure and composition at the first two interfacial layers next to the
Cu(111) surface are evolving and reaching new equilibrium interfacial states
beyond the known interfacial pre-freezing phase.

As illustrated in the schematic Fig.1, the final heterogeneous Cu(111)/Pb
solid/liquid interface system that contains the coexistences of multiple
interfacial states are constructed by assembling two pre-equilibrated
conglomerates along $x$ axis and applying PBC in $x$ and $y$ axis of the
combined simulation cell. One (of the two) conglomerates contains two
interfacial pre-freezing Pb layers, and the second equilibrated conglomerate
contains two interfacial alloy liquid layers. In this way, in each
Cu(111)/Pb(l) interface, there are two parallel boundary lines between the
coexisting pre-freezing Pb and interfacial CuPb alloy liquid states in the
first two interfacial layers next to the Cu(111) surface, along $x$ axis. The
dimension of the combined cell is $L_{x}\times L_{y}=751.68$Å$\times 62.00$Å,
containing around 446,000 atoms. The two coexistence boundary lines are
separated by over 370Å in the distance, which is large enough to avoid the in-
plane line-line interaction, which may cause the finite size effect of the
boundary line free energy and their in-plane capillary fluctuationsLiang18 .
The $NVT$ simulation up to 30 ns ensures multiple interfacial states coexist,
reaching structural and compositional equilibrium within the Cu(111)/Pb(l)
interface.

After the combined simulation cell is equilibrated, a following-up 50 ns $NVT$
simulation is carried out to collect data. We divide the MD trajectories of
this 50 ns long run into five sections for block-averaging. The simulation
cell has two independent Cu(111)/Pb(l) interfaces, and each contains two
interfacial layers interested. There are two parallel in-plane interfacial
solid-liquid coexistence boundary lines within each interfacial layer
interested. In total, we have twenty in-plane interfacial solid-liquid
coexistence boundary lines samples (we randomly choose ten of them), and ten
Cu(111)/Pb(l) interface samples for quantitative characterization and
calculation.

The atomistic characterization methodologies developed for the chemically
heterogeneous solid/liquid interfacesPalafox11 ; Yang12 , the interfacial
layer containing the steps at chemically heterogeneous solid/liquid
interfacesLiang18 , and the coexistence of mono-molecular layer confined
crystal and its melt phase with coexistence boundary lineDu18 , are employed
to study various structural and thermodynamic properties for the equilibrium
bulk Cu, bulk Pb, Cu(111)/Pb interface, interfacial CuPb alloy liquids,
interfacial pre-freezing Pb, and the boundary lines between the interfacial
solids and liquids.

The calculations include the fine- and coarse-scale profiles of various
properties and their spatial variations along the $x$ axis for interfacial
states and their in-plane transitional boundary line regions (or along the $z$
axis for bulk phases and their transitional regions). The structural and
thermodynamic properties involved in this study include atomic number
densities ($\rho_{\mathrm{Pb}}$ and $\rho_{\mathrm{Cu}}$), compositions
($X_{\mathrm{Pb}}$ and $X_{\mathrm{Cu}}$), potential energies
($Pe_{\mathrm{Pb}}$ and $Pe_{\mathrm{Cu}}$), pressure components ($p_{xx}$,
$p_{yy}$ and $p_{zz}$), stresses (s), and diffusion coefficient components
($D_{xx}$, $D_{yy}$ and $D_{zz}$).

In order to distinguish the quantities of the interfacial layers from the bulk
phases, we add single-tilde (∼) and double-tilde (≈) over the symbols of
various parameters to denote the first and the second interfacial layers next
to the Cu(111) surface, respectively. We then extract different sets of
fundamental parameters describing different phases and interfacial states from
the uniform regions in the calculated profiles and employ superscript “s” and
“l” to stand for (bulk or interfacial) solids and liquids, respectively.

##  III Results and Discussions

Fig.2 shows the representative snapshots of five equilibrated single
conglomerates containing interfacial layers with different values of Cu
compositions. Fig.2(a1-a3) demonstrate the obtained two stable interfacial
pre-freezing Pb layers with nearly zero in-plane Cu compositions. We verify
that i) the two pre-freezing Pb planes have a $\sim$1.9% compressed lattice
spacing compared to its bulk fcc lattice spacing, and ii) rotate 6∘ relative
to the underlying Cu(111) lattice, identical to those reported by Palafox-
Hernandez et al.Palafox11 . From Fig.2(b1,b2,b3) to (e1,e2,e3), more
equilibrated conglomerates containing interfacial layers with higher Cu
composition values. Although the increasing numbers of Cu atoms from (a1) to
(e1), each equilibrium bulk liquid phase in these cases has a relatively
constant saturation solubility, i.e., $X_{\mathrm{Cu}}^{l}(620K)=0.007(1)$,
suggesting that the varying excess Cu atoms in difference equilibrated
interfacial systems are majorly concentrated within the first two interfacial
layers. Numbers in parenthesis are the 95% confidence level error-bars.

Fig.2(b1-b3) demonstrate snapshots of the equilibrated interface from an
initial setup with the Cu composition of the first Pb layer as 12%. During the
re-establishment of the equilibrium interfacial states, in-plane nucleation of
the pre-freezing Pb island from the disordered binary alloy layer is observed.
The pre-freezing islands grow laterally via a similar fashion of step growth
of faceted silicon crystal from SiAl alloy meltsSaidi17  until the Cu
compositions in the remainder disordered interfacial CuPb mixture reach
saturation values, i.e., $\sim 20\%$ at the first layer adjacent to the
Cu(111) surface and $\sim 1.5\%$ at the second layer. Noticeable boundary
lines between the interfacial pre-freezing Pb and interfacial CuPb alloy are
seen. It is also observed here that the coupled parallel capillary waves of
the neighboring boundary lines, probably bounding with the short-range
disjoining potential among themLiang19 , are beyond the scope of the current
study.

In the following panels of Fig.2, panels (c1,c2,c3) to (e1,e2,e3), as the Cu
composition in the interfacial layer increases, the nucleation and growth of
the pre-freezing Pb island become rare. For example, for the conglomerates in
which Cu composition $\sim$20% in the first layer next to the Cu surface,
Fig.2(c3), the interfacial layers evolve into uniform Pb-rich solutions with
individual Cu atoms and dynamic Cu clusters (mostly less than ten atoms)
suspended within. In Fig.2(d3),(e3), small solid Cu islands (with the size of
several tens atoms) within the first interfacial layer (of the disordered CuPb
alloy) are identified in the conglomerates with interfacial Cu composition
approaching $30\%$ and higher. These localized Cu islands inherit the crystal
lattice of the underlying Cu(111) surface and are relatively immobile compared
with the suspending small Cu clusters not attached to the crystalline
substrate. However, it is noticed that, in all the values of the interfacial
Cu composition investigated in this work, the solid Cu islands are only found
single-layer-wise, unlike the bi-layer pre-freezing Pb. This could be ascribed
to the significant gradient in the Cu composition as the two interfacial
layers are traversed from the crystal Cu side to the bulk liquid Pb side,
i.e., the in-plane Cu composition decreased over one order of magnitude from
the first interfacial layer to the second interfacial layer.

From the above-selected snapshots, one can observe that the interfacial
disordered alloy states coexist well with the interfacial solids (i.e., pre-
freezing Pb or solid Cu island). In order to further confirm the liquid nature
of the disordered interfacial alloy states and study their fluidity as the
function of the Cu composition. We examine the in-plane mean diffusion
coefficients and the in-plane mean bond angle orientational order
parametersDavidchack98  for both interfacial layers, i.e.,
$\langle{\tilde{D}_{xy}}\rangle$, $\langle\tilde{\tilde{D}}_{xy}\rangle$,
$\langle\tilde{q_{6}}\rangle$, and $\langle\tilde{\tilde{q_{6}}}\rangle$,
versus the equilibrium value of the Cu composition
$\langle{\tilde{X}_{\mathrm{Cu}}}\rangle$ of the first interfacial layer
adjacent to the Cu(111) surface. Over the whole composition range
investigated, all diffusion coefficients of the first interfacial layer are
greater than zero, although these interfacial data range in between 1/10 and
1/3 of the magnitude of $D$ for bulk liquid Pb phase at 620K (1.540(2)$\times
10^{-5}$cm2/s), they are at least four orders of magnitude greater than the
diffusion coefficient in the bulk solid phase, indicating the disordered CuPb
alloy states within the first interfacial layers have liquid-like dynamics.
The maximum diffusivity and the minimum in-plane structural ordering of the
first interfacial layer are found around the Cu composition value of 22% –
corresponding to the uniform alloy liquid layer, which is free of pre-freezing
Pb or solid Cu islands. The fluidity decease by either decreasing or
increasing the Cu composition due to the rapidly developing of the pre-
freezing Pb or the solid Cu island, respectively. The second interfacial layer
shows more significant fluidity, which is reasonable as this layer is adjacent
to the bulk Pb liquid.

Next, we present the equilibrium simulation results of the equilibrium
heterogeneous Cu(111)/Pb(l) interface, which contains the coexistence of
interfacial pre-freezing Pb domain and the CuPb alloy liquid domain(maximum
fluidity, ${\tilde{X}_{\mathrm{Cu}}}\sim$22%,
$\tilde{\tilde{X}}_{\mathrm{Cu}}\sim$1.8%,) within the two interfacial layers.
Fig.4(a) demonstrate a full-size sideview snapshot of such equilibrium
Cu(111)/Pb(l) interface from the viewpoint along [10$\bar{1}$]. In contrast to
the rough type liquid Pb-vapor surface subjected to the capillary fluctuation,
the Cu(111)/Pb(l) interface is flat, supporting a faceted nature for this
heterogeneous solid/liquid interface at $T=620$K. Three local interfacial
regions in boxes in Fig.4(a) are zoomed in panels (b1,c1,d1), respectively,
together with the corresponding fine-grained Cu and Pb density profiles along
$z$, $\rho_{\mathrm{Cu}}(z)$ and $\rho_{\mathrm{Pb}}(z)$, in panels
(b2,c2,d2). In (b1) and (b2), the Cu and Pb density profiles are resemble as
previously reported data at $T=625$KPalafox11 . No interfacial alloying or no
overlap of two density profiles at this region of the Cu(111)/Pb(l) interface
is seen.

In the zoomed interfacial region shown in Fig.4(c1) and (c2), the first liquid
layer has more mixing of the Cu and Pb atoms, corresponding to the visible
compositional overlap of the Cu and Pb density peaks at the first interfacial
layer. Note, it has been reported at the Cu(100)/Pb(l) interface at 625K, the
Cu and Pb density peaks also overlap, but in the surface Cu solid
layerPalafox11 , rather than in the first interfacial Pb-rich alloy liquid
layer, as zoomed in the region of current constructed Cu(111)/Pb(l) interface.
In addition, the interface exhibits a less pronounced layer ordering structure
(lower amplitude of the density peaks) in $\rho_{\mathrm{Pb}}(z)$ extending
into the bulk liquid than is seen in panel (b1).

In panels (d1) and (d2) of Fig.4, the zoomed interfacial region contains two
coexisting interfacial states, i.e., the pre-freezing Pb and the CuPb alloy
liquid states. $\rho_{\mathrm{Cu}}(z)$ profile barely changes from (b1) or
(c1) to (d1). Nevertheless, there are minor differences related to composition
or the oscillation amplitudes of the density profiles for $z>0$ due to the
counting in the mixing of the two coexisting interfacial states.

In-plane snapshots of the two interfacial layers extracted from the zoomed
region in Fig.4(d1) are shown in Fig.5(a) and (c), along with the contour
plots of the 2D time-averaged densities, as well as the corresponding 2D
Fourier structure factorsYang12 , see in panel (b) and (d) of Fig.5 and the
insets therein.

With the aid of color coding using the calculated bond angle orientational
order ($q_{6}$) for each atom, the rough-type boundary lines between the in-
plane coexisting pre-freezing Pb and the CuPb alloy liquid in both two
interfacial layers can be tracked. In the vicinity of the equilibrium boundary
lines, the perpetual in-plane growth and melt of the pre-freezing Pb edge via
frequent (ps timescale) attachment/detachment of small Pb clusters are
observed. In principle, the limiting kinetics that governs the line
fluctuations and the line free energies could be further determined by
examining the wave vector dependences of the line fluctuation relaxation times
and the line fluctuation amplitudesLiang18 . These properties will be reported
elsewhere.

The 2D time-averaged density maps of the two interfacial layers demonstrate
that interfacial liquids and solids are well-equilibrated and coexistences of
the multiple interfacial states and the adjacent bulk phases are highly stable
over the entire simulation length. The apparent peak matrix on the left sides
of Fig.5(b) and (d) indicates the bi-layer-wise pre-freezing Pb is a robust
structure as reported by Palafox-Hernandez et al.Palafox11 , even though we
simulate a sample of the Cu(111)/Pb(l) interface with the cross-sectional size
nearly ten times bigger than the work by Palafox-Hernandez et al.

On the right side of Fig.5(b), the first interfacial layer, the modulation of
the underlying solid on the structure of the liquid alloy density field is
hardly observed. However, faint yet visible residual Bragg peaks (identical to
Cu(111) surface lattice) show the weak influence of the underlying crystalline
Cu(111) surface. Correspondingly, such modulation becomes negligible on the
right side of Fig.5(d). No detectable protruding island structures due to the
Cu island steps attached to the underneath Cu(111) surface are seen in the
liquid alloy sides of both interfacial layers. The in-plane transition extents
from pre-freezing Pb to alloy liquids for the two interfacial layers are
similar, approximately over ten Pb atom diameters. The following paragraphs
will show additional atomistic characterization of the thermodynamics
properties for distinguishing differences between two interfacial layers
regarding the in-plane coexistences of interfacial liquid and interfacial
solid.

The profiles of various thermodynamics and dynamics quantities along $x$ axis
for both interfacial layers are shown in Fig.6, panels (a1-g1) show results
for the first interfacial layer next to the Cu(111) surface, panels (a2-g2)
show results for the second interfacial layer. $x=0$ corresponds to the in-
plane Gibbs Dividing Surfaces (GDS)Frolov15b ; Liang18 , where the boundary
line excess number of Pb atoms is zero. $x<0$ for interfacial pre-freezing Pb
solid and $x>0$ for interfacial alloy liquid.

The equilibrium coarse-scaled in-plane density profiles shown in
Fig.6(a1),(a2) count the particle number densities within the volumes for the
two interfacial layers, the thickness of the interfacial layers are defined as
the distances between the adjacent minima of the two fine-grained Pb density
peaks next to the Cu(111) surface, as shown in Fig.4(d2). All the density
values of these two interfacial layers vary between the density values of the
bulk Cu solid and bulk Pb liquid at the same temperature $T=620$K. The
equilibrium values of the “bulk” regions of the interfacial solids and liquids
for the two layers are listed in Tab.1. Interestingly, in the first
interfacial layer, the interfacial alloy liquid has a greater density value
than the coexisting pre-freezing Pb due to the mixing of $\sim$22% copper
atoms, which has a smaller atomistic volume. In the second interfacial layer,
because the Cu composition drops one order of magnitude, pre-freezing Pb in
this layer owns larger density values than that of the dilute CuPb alloy
liquids in the same layer, which has a nearly identical density value of the
bulk Pb liquid. The in-plane decays of the densities for both layers span over
a length scale of roughly 2.6 nm, i.e., the values of 10-90 widthsZhang22
extracted from the density profiles,
${\tilde{\delta}}_{\mathrm{10-90}}=2.7(2)$nm and
${\tilde{\tilde{\delta}}}_{\mathrm{10-90}}=2.4(3)$nm in Tab.1. In contrast to
the 10-90 width ($\delta_{\mathrm{10-90}}=0.8(1)$nm) of the Cu(111)/Pb(l)
interface at this temperature, the relatively broader extents of the
interfacial density decays can be attributed to the noticeable in-plane
capillary waves along the boundary lines.

The equilibrium composition and potential energy proﬁles are shown in
Fig.6(b1,b2) and (c1,c2), the extracted equilibrium solubilities of Cu in
interfacial liquids are 22.0(6)% (first layer) and 1.8(1)% (second layer),
both show significant increases (around 34 times and three times greater)
compared with the tiny solubility value of Cu in bulk Pb liquid phase
(0.64(1)%). Cu atoms are nearly immiscible for both interfacial layers with
pre-freezing Pb, i.e., negligible solid state miscibilities ($\sim$0.2%) for
both layers, suggesting that the interfacial solid (pre-freezing) states are
pure rather than solid-solutions. Despite that the monotectic phase diagram
for bulk CuPb alloy (at $T=620$K) says that pure Cu solid coexists with nearly
pure Pb liquid, it is interesting to note that the interfacial “complexion (or
phase) diagrams” for the Cu(111)/Pb(l) interfacial layers bear a resemblance
to that of the eutectic binary alloy systems, e.g., high liquid state
miscibility and negligible solid state miscibility. The potential energy
profiles (panels (c1) and (c2)) depict that atoms in the first interfacial
layer have shallower potential minima than in the second. Within each layer,
each atom in the solid state has a deeper potential minimum than in the alloy
liquid state. The detailed potential energies of the Cu and Pb atoms belonging
to both interfacial states within the two interfacial layers are listed in
Tab.1.

The pressure components proﬁles in Fig.6(d1-d2) present the in-plane
coexistences of pre-freezing Pb and CuPb alloy liquid within the planar
heterogeneous Cu(111)/Pb(l) interfacial layers, are under non-hydrodynamic
conditions in which the pressure components are highly anisotropic and
spatially non-homogeneous. All proﬁles in (d1) and (d2) show non-zero
pressure, with absolute values on the magnitude of kbar at least three orders
of magnitude greater than the hydrostatic pressure applied in bulk Cu solid or
bulk Pb liquid phases. The $xx$ and $yy$ pressure components for the “bulk”
regimes of the pre-freezing Pb (or CuPb alloy liquid) within both interfacial
layers are equal within statistical uncertainties. Therefore only the averaged
values of these two pressure components $p_{xy}=\frac{1}{2}(p_{xx}+p_{yy})$
are listed in Tab.1. For CuPb alloy liquid sides, positive
${\tilde{p}_{xy}}=1.00(1)$kbar (first layer) and
${\tilde{\tilde{p}}}_{xy}=3.55(2)$kbar (second layer) are greater than those
of their corresponding $p_{zz}$ components. For pre-freezing Pb sides,
significant negative ${\tilde{p}_{xy}}=-6.30(4)$kbar (first layer) and
significant positive ${\tilde{\tilde{p}}}_{xy}=9.42(5)$kbar (second layer) are
found lower and much greater than those of their corresponding $zz$
components, respectively. As one travels across the interfacial coexistence
boundary lines, from the pre-freezing Pb side to the alloy liquid side,
${\tilde{p}_{xy}}(x)$ (or ${\tilde{\tilde{p}}}_{xy}(x)$) show more remarkable
relaxation in comparison with ${\tilde{p}_{zz}}(x)$ (or
${\tilde{\tilde{p}}}_{zz}(x)$), the $xx$ pressure components relax slightly
before the $yy$ over pressure components in both interfacial layers, which
give rise to the finite in-plane line lateral stresses shown in panels (f1)
and (f2).

The panels (e1) and (e2) depict the detailed heterogeneity in the built-up
non-hydrostatic mechanical stresses (defined as the difference between the
longitudinal $zz$ and transverse $xy$ pressure components normal and parallel
to the Cu(111)/Pb(l) interface) within interfacial layers. From the pre-
freezing Pb side to the alloy liquid side, the mechanical stress decays from
3.69kbar to -3.16kbar in the first layer and increases from -10.08kbar to
-2.99kbar in the second layer. The positive sign in the mechanical stress
suggests that the first layer pre-freezing Pb is under lateral tension,
whereas its in-plane coexisting alloy liquid is under lateral compressionLu22
. In the second interfacial layer, the coexisting pre-freezing Pb and the Pb-
rich liquid states are both under lateral compression, although the former
interfacial state withstands the highest magnitude of mechanical anisotropy.
The in-plane lateral stress (defined as the difference between the pressure
components normal $xx$ and parallel $yy$ to the in-plane states coexistence
boundary line) profiles are plotted in panels (f1) and (f2) of Fig.6. A single
positive (or negative) peak in the later stress profile of the first (or the
second) interfacial layer suggests the boundary line in the first (or the
second) interfacial layer is in a state of stretching (or squeezing) along $y$
direction. The two peak structures in the two lateral stress profiles in (f1)
and (f2) have close peak positions and relatively symmetric shapes. Therefore
we deduce a strong short-range structural interaction may be binding the two
boundary lines crossing the two interfacial layers, which deserves further
investigations in the future.

To illustrate the spatial variation in the dynamic properties of the in-plane
coexistence of interfacial solids and liquids, we compute the coarse-grained
diffusion coefficients as the function of distances to the in-plane GDS, see
in Fig.6(g1),(g2). As the boundary lines are traversed from pre-freezing Pb
sides to the alloy liquid sides, the diffusion coefficients start to increase
from small values (0.047(2)$\times 10^{-5}$cm2/s for the first interfacial
layer, 0.064(2)$\times 10^{-5}$cm2/s for the second interfacial layer) until
they converge to their plateau “bulk” values, i.e., 0.509(5)$\times
10^{-5}$cm2/s and 0.947(7)$\times 10^{-5}$cm2/s, respectively, which are
identical to the maximum diffusivity values reported in Fig.3(a). Besides, we
identify that the pre-freezing Pb domains in both layers are undergoing a
random migration along $y$, similarly as a semi-2D “crystal” flake undergoes
Brownian motion in liquid, thus explaining their finite value of the diffusion
coefficients.

Based on the above characterization and computation of the multiple
interfacial state coexistences, we obtain information that offers clues to
further elucidate the relationship between the bulk phase diagram and the
interfacial complexion (or phase) diagram. Being inspired by many previous
works on developing GB diagrams, by overlaying the complexion diagram on the
bulk phase diagram for a better illustration of the stability regions from a
complete perspectiveShi11 ; Cantwell14 . Here, instead of overlaying the CuPb
bulk alloy phase diagram, we try to add one additional axis (i.e., transverse
pressure components parallel to the Cu(111)/Pb(l) interface, $p_{xy}$) into
the parameter space in Fig.7 for comparing the interfacial phase behavior with
the bulk system. The employment of $p_{xy}$ is based on the following
considerations. i) The values of $p_{xy}$ represent a majority of the local
mechanical condition for the coexisting interfacial states, closely relating
to their equations of state, chemical potentials (or the Gibbs free energies).
A better option may be determining the local (including the interactions from
adjacent materials) configurational integration of the partition function,
which is exceedingly difficult. ii) There are experimentalEmuna16 ; Hudon04
and theoreticalMa21  evidence shows substantial pressure-induced variation in
the mutual miscibilities of heterogeneous binary alloy systems. The two
schematic color $X_{\mathrm{Pb}}$–$p_{xy}$ lines for the two interfacial
layers, respectively go through the interfacial “phase” equilibria
compositions and local pressure components, labeled as the filled blue (pre-
freezing Pb) and green (alloy liquid) circles. One could expect a “liquidus”
lines (containing more in-plane coexistence composition-local pressure data)
which in equilibria with pre-freezing Cu (the corresponding “solidus” lines)
at $T$ higher than 620K (note that Palafox-Hernandez and Laird observed that
pre-freezing layer is no longer present at 750KPalafox16 ).

Finally, with the current simulation results, it is interesting to discuss the
validity of the recent thermodynamic formalism of the phase equilibria and
phase rules into interfacial systems by Frolov and MishinFrolov15b . Based on
their interfacial phase rule, the maximum possible number of interface phases
$\nu_{\mathrm{max}}$ can coexist within an interface system composed of
$\varphi=2$ bulk phases equal to $\nu_{\mathrm{max}}=k+3-\varphi$, in which
$k$ is the number of chemical potentials $\mu_{i}$ for $k$ chemical
components. For example, it predicts that an interface of a binary alloy
system can support a maximum of three coexisting interface phases. Frolov and
Mishin pointed out that their analysis was based on simplified isotropic
treatment, which could not immediately apply to solid phases. Besides, the
chemical potential of each chemical component was assumed as constant
throughout the entire system, including bulk and interfacial phases. If we
accept the idea that four interfacial states within the two interfacial layers
coexist in Cu(111)/Pb(l) interface at 620K, as plotted in Fig.7, we will find
the predicted $\nu_{\mathrm{max}}=3$ contradicts the our current result.

However, if one considers the Cu and Pb atoms have distinct values of chemical
potentials different from bulk values within different interfacial states and
interfacial layers (note: this might be reasonable since the local symmetry
broken structure for the interfacial layers should have unique configurational
integrals contributing to their partition functions), for example, four more
values of chemical potentials for interfacial Cu and Pb atoms, one would
obtain a prediction of $\nu_{\mathrm{max}}=7$ maximum interfacial phases
coexist in the Cu(111)/Pb(l) interface at 620K. In this way, the result of
four coexisting interfacial states coexist in Cu(111)/Pb(l) interface in Fig.7
is in agreement with such phase rule prediction. The $\nu_{\mathrm{max}}=7$
prediction also implies more possible in-plane coexistences of interfacial
states in Cu(111)/Pb(l) interface, which has not been substantially explored
yet and deserves a independent future study.

##  IV Summary

Using the MD simulations, we uncover new interfacial states within the
Cu(111)/Pb(l) at 620K. We predict the interfacial alloy liquid states dominate
over the previously reported pre-freezing Pb solids with higher values of
interfacial excess number of Cu atoms, and their potentially in-plane
coexistences with the interfacial solid states (pre-freezing Pb or solid Cu
step). The majority of previous discoveries of the multiple interfacial
complexion (or phase) coexistence and transition are made in GBs and solid
surface systems, rather than the heterogeneous solid/liquid interfacial
systems. We provide here an atomistic simulation demonstration on the in-plane
coexistences of multiple interfacial states within a planar heterogeneous
Cu(111)/Pb(l) interface, including the interfacial pre-freezed metal Pb, the
interfacial binary CuPb alloy liquids, and two boundary lines.
Characterization of the proﬁles of a series of fundamental properties (e.g.,
densities, compositions, potential energies, pressure components, stresses,
and diffusivities) for the in-plane equilibria of the interfacial solid and
liquid states in the first two interfacial layers adjacent to the Cu(111)
surface, are carefully carried out. Overall, the current construction of the
constant temperature equilibrium Cu(111)/Pb(l) interface under ambient
pressure conditions presents rich and in-plane varied mechanical, structural,
and thermodynamical characteristics.

Four sets of the fundamental parameters for the four coexisting states are
extracted from the two sets of the in-plane profiles (Fig.6) of the two
interfacial planes. These parameter sets suggest that each interfacial state
examined has distinct values from each other and significantly deviates from
those of the bulk solid and liquid phases. Before this study, there has been
little discussion of interfacial complexion (or phase) diagrams in the
solid/liquid interface systems. Our findings suggest that the interfacial
“complexion (or phase) diagrams” for the Cu(111)/Pb(l) interface at a
temperature just above the melting point of Pb bears a resemblance to that of
the eutectic type binary alloy systems, in stead of the monotectic phase
diagram for the bulk CuPb alloy. The current simulation results also enable us
to evaluate the validity of the recent formalism of phase rules for the
semi-2D interfacial systems. We propose that a more rigorous interface phase
rule may need the consideration that atoms within different interfacial states
and interfacial layers have distinct values of chemical potentials different
from bulk values.

Preparation and tuning of the multi-interfacial state coexistences and
transformations within the heterogeneous solid/liquid interfaces will
facilitate the effective control of related processes with essential
applications. For example, i) dynamics of wetting and spreading of molten
metal on a heterogeneous solid metal substrateYang17  can be strongly altered
by the occurrence o f the interfacial pre-freezingHeine05  transitions. ii)
The heterogeneous nucleation of Pb crystals on the Cu surface with pre-
freezing Pb is found considerably more favorable than the Cu/Pb solid-liquid
interface exhibits significant surface alloyPalafox16 . Our current simulation
observation indicates a potential new method for quantitatively tuning the
heterogeneous wetting and spreading of Pb droplet on the Cu(111) surface, as
well as the heterogeneous nucleation of Pb crystal on the Cu(111) surface if
one could effectively tune the allocation of the multiple-interfacial state
coexistences and transformations within the Cu(111)/Pb(l).

##  V Acknowledgements

This work is supported by the Chinese National Science Foundation (Grant No.
11874147), the Natural Science Foundation of Shanghai, the Natural Science
Foundation of Chongqing (Grant No. cstc2021jcyj-msxmX1144).

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Figure 1: The schematic diagram for the Cu(s)/Pb(l) interface system built
with equilibrium MD simulations. Two Cu(s)/Pb(l) solid-liquid interfaces and
two Pb(l)-vapor interfaces are parallel to the $xy$ plane. Besides the bulk
solid, liquid, and vapor phases, the coexistences of multiple interfacial
states are constructed by assembling two pre-equilibrated conglomerates half
size of the whole simulation cell. Figure 2: MD snapshots of five pre-
equilibrated Cu(111)/Pb(l) interfaces containing interfacial layers with
different values of Cu compositions at 620K. A portion of the system is
captured from the side view and plotted in the left-side panels. The direction
along $x$, $y$, and $z$ axes are [1$\bar{2}$1], [10$\bar{1}$], and [111],
respectively. Cu atoms are shown as smaller red circular plates in all panels,
while Pb atoms are shown as brown circular plates with larger diameters in the
left side panels. In the $xy$ plan views of the interfacial layers (in the
right-side panels), Pb atoms are color-coded based on the values of
“bond”-angle orientational order parameter $q_{6}$Davidchack98 , blue for the
interfacial solid state with higher values of $q_{6}$, green for the
interfacial liquid state with lower values of $q_{6}$. Figure 3: Interfacial
Cu composition $\langle{\tilde{X}_{\mathrm{Cu}}}\rangle$ dependences of (a)
the in-plane mean diffusion coefficients, $\langle{\tilde{D}_{xy}}\rangle$,
$\langle\tilde{\tilde{D}}_{xy}\rangle$, and (b) the in-plane mean bond angle
orientational order parameters, $\langle\tilde{q_{6}}\rangle$,
$\langle\tilde{\tilde{q_{6}}}\rangle$, for the first two interfacial layers
adjacent to the Cu(111) surfaces at pre-equilibrated Cu(111)/Pb(l) interfaces
at 620K. Figure 4: (a) Full-size sideview snapshot of one equilibrium
heterogeneous Cu(111)/Pb(l) interface at 620K, which contains the coexistence
of interfacial pre-freezing Pb and the CuPb alloy liquids. Three local
interfacial regions in boxes are zoomed in (b1), (c1), and (d1), respectively.
(b2), (c2), and (d2) plot the corresponding fine-grained Cu (red) and Pb
(brown) density proﬁles for the Cu(111)/Pb(l) interface along $z$ axis. $z<0$
for bulk Cu solid phase, $z>0$ for interfacial layers and bulk Pb liquid
phase. Figure 5: (a) and (c) show instantaneous snapshots of the first and the
second Cu(111)/Pb(l) interfacial layers extracted from the zoomed region in
Fig.4(d1), representing in-plane coexistence structure of interfacial solid
states and interfacial liquid states. Cu atoms are represented with red color,
Pb atoms are color-coded based on the values of $q_{6}$ order parameter, blue
for interfacial solid states, green for interfacial liquid states. The
boundary lines are plotted as black solid lines. (b) and (d) show the “long
exposure photography” of the time-averaged 2D density contour maps, the 2D
Fourier structure factors for the corresponding layers are plotted as insets
in the two panels. Figure 6: In-plane coarse-scaled profiles of densities
(a1.a2), compositions (b1,b2), potential energies (c1,c2), pressure components
(d1,d2), interfacial stresses (e1,e2), lateral stresses (f1,f2), diffusion
coefficients (g1,g2), for the two interfacial layers of the Cu(111)/Pb(l)
interface at 620K. $x<0$ for interfacial pre-freezing Pb solids and $x>0$ for
interfacial alloy liquids. Figure 7: Phase diagram of CuPb binary alloy
system. Filled and open circles are determined by a semi-Grand-Canonical MC
method reported by Hoyt et al.Hoyt03b , Filled and open squares are determined
from the equilibrium MD simulations by Laird et al.Palafox16  at $T$=750K, and
by current study at $T$=620K. One additional axis ($p_{xy}$) is added into the
parameter space of the diagram for comparing the interfacial phase behavior
with the bulk phase diagram. The dark and light blue circles represent the
interfacial pre-freezing Pb states in the first and the second interfacial
layers, respectively. The dark and light green circles represent the
interfacial alloy liquid states in the first and the second interfacial
layers, respectively.

Table 1: Summary of the thermodynamic parameters extracted from an equilibrium 620K Cu(111)/Pb(l) interface with in-plane coexistence structure of interfacial solid and interfacial liquid states. ${\rm[Interfacial\ layer\ 1]}\ {\rm Pb}({\rm s}){\rm-alloy}({\rm l})$ | ${\rm[Interfacial\ layer\ 2]}\ {\rm Pb}({\rm s}){\rm-alloy}({\rm l})$ | ${\rm[Bulk\ interface]}\ {\rm Cu}(111){\rm/Pb}({\rm l})$
---|---|---
${\tilde{\rho}}{\hskip 0.80002pt}^{s{\rm Pb}}{\rm(nm^{-3})}$ | 40.90(1) | ${\tilde{\tilde{\rho}}}{\hskip 0.80002pt}^{s{\rm Pb}}{\rm(nm^{-3})}$ | 33.36(2) | $\rho{\hskip 0.80002pt}^{s{\rm Cu}}{\rm(nm^{-3})}$ | 83.063(1)
${\tilde{\rho}}{\hskip 0.80002pt}^{l{\rm alloy}}{\rm(nm^{-3})}$ | 45.03(8) | ${\tilde{\tilde{\rho}}}{\hskip 0.80002pt}^{l{\rm alloy}}{\rm(nm^{-3})}$ | 31.51(5) | $\rho{\hskip 0.80002pt}^{l{\rm Pb}}{\rm(nm^{-3})}$ | 31.406(7)
$\tilde{X}{\hskip 0.80002pt}^{s{\rm Pb}}_{\rm Cu}(\%)$ | 0.2(1) | ${\tilde{\tilde{X}}}{\hskip 0.80002pt}^{s{\rm Pb}}_{\rm Cu}(\%)$ | 0.2(1) | ${X}{\hskip 0.80002pt}^{s{\rm Cu}}_{\rm Cu}(\%)$ | 100
$\tilde{X}{\hskip 0.80002pt}^{l{\rm alloy}}_{\rm Cu}(\%)$ | 22.0(6) | ${\tilde{\tilde{X}}}{\hskip 0.80002pt}^{l{\rm alloy}}_{\rm Cu}(\%)$ | 1.8(1) | ${X}{\hskip 0.80002pt}^{l{\rm Pb}}_{\rm Cu}(\%)$ | 0.64(1)
${\tilde{Pe}}{\hskip 0.80002pt}^{s{\rm Pb}}_{\rm Cu}{\rm(eV)}$ | -3.951(2) | ${\tilde{\tilde{Pe}}}{\hskip 0.80002pt}^{s{\rm Pb}}_{\rm Cu}{\rm(eV)}$ | -4.156(1) | ${Pe}{\hskip 0.80002pt}^{s{\rm Cu}}_{\rm Cu}{\rm(eV)}$ | -3.4547(1)
${\tilde{Pe}}{\hskip 0.80002pt}^{l{\rm alloy}}_{\rm Cu}{\rm(eV)}$ | -3.905(2) | ${\tilde{\tilde{Pe}}}{\hskip 0.80002pt}^{l{\rm alloy}}_{\rm Cu}{\rm(eV)}$ | -4.094(4) | ${Pe}{\hskip 0.80002pt}^{l{\rm Pb}}_{\rm Cu}{\rm(eV)}$ | -4.1033(2)
${\tilde{Pe}}{\hskip 0.80002pt}^{s{\rm Pb}}_{\rm Pb}{\rm(eV)}$ | -1.076(1) | ${\tilde{\tilde{Pe}}}{\hskip 0.80002pt}^{s{\rm Pb}}_{\rm Pb}{\rm(eV)}$ | -1.914(1) | ${Pe}{\hskip 0.80002pt}^{s{\rm Cu}}_{\rm Pb}{\rm(eV)}$ | N/A
${\tilde{Pe}}{\hskip 0.80002pt}^{l{\rm alloy}}_{\rm Pb}{\rm(eV)}$ | -0.933(6) | ${\tilde{\tilde{Pe}}}{\hskip 0.80002pt}^{l{\rm alloy}}_{\rm Pb}{\rm(eV)}$ | -1.788(8) | ${Pe}{\hskip 0.80002pt}^{l{\rm Pb}}_{\rm Pb}{\rm(eV)}$ | -1.8873(3)
${\tilde{p}}{\hskip 0.80002pt}^{s{\rm Pb}}_{xy}{\rm(kbar)}$ | -6.30(4) | ${\tilde{\tilde{p}}}{\hskip 0.80002pt}^{s{\rm Pb}}_{xy}{\rm(kbar)}$ | 9.42(5) | ${p}{\hskip 0.80002pt}^{s{\rm Cu}}_{xy}{\rm(kbar)}$ | 0.002(4)
${\tilde{p}}{\hskip 0.80002pt}^{s{\rm Pb}}_{zz}{\rm(kbar)}$ | -2.61(1) | ${\tilde{\tilde{p}}}{\hskip 0.80002pt}^{s{\rm Pb}}_{zz}{\rm(kbar)}$ | -0.66(1) | ${p}{\hskip 0.80002pt}^{s{\rm Cu}}_{zz}{\rm(kbar)}$ | 0.001(1)
${\tilde{p}}{\hskip 0.80002pt}^{l{\rm alloy}}_{xy}{\rm(kbar)}$ | 1.00(1) | ${\tilde{\tilde{p}}}{\hskip 0.80002pt}^{l{\rm alloy}}_{xy}{\rm(kbar)}$ | 3.55(2) | ${p}{\hskip 0.80002pt}^{l{\rm Pb}}_{xy}{\rm(kbar)}$ | 0.003(6)
${\tilde{p}}{\hskip 0.80002pt}^{l{\rm alloy}}_{zz}{\rm(kbar)}$ | -2.16(2) | ${\tilde{\tilde{p}}}{\hskip 0.80002pt}^{l{\rm alloy}}_{zz}{\rm(kbar)}$ | 0.56(2) | ${p}{\hskip 0.80002pt}^{l{\rm Pb}}_{zz}{\rm(kbar)}$ | -0.004(4)
${\tilde{D}}{\hskip 0.80002pt}^{s{\rm Pb}}_{xy}\cdot 10^{5}(\rm{cm}^{2}\cdot{s}^{-1})$ | 0.047(2) | ${\tilde{\tilde{D}}}{\hskip 0.80002pt}^{s{\rm Pb}}_{xy}\cdot 10^{5}(\rm{cm}^{2}\cdot{s}^{-1})$ | 0.064(2) | ${D}{\hskip 0.80002pt}^{s{\rm Cu}}\cdot 10^{5}(\rm{cm}^{2}\cdot{s}^{-1})$ | 0.002(1)
${\tilde{D}}{\hskip 0.80002pt}^{l{\rm alloy}}_{xy}\cdot 10^{5}(\rm{cm}^{2}\cdot{s}^{-1})$ | 0.509(5) | ${\tilde{\tilde{D}}}{\hskip 0.80002pt}^{l{\rm alloy}}_{xy}\cdot 10^{5}(\rm{cm}^{2}\cdot{s}^{-1})$ | 0.947(7) | ${D}{\hskip 0.80002pt}^{l{\rm Pb}}\cdot 10^{5}(\rm{cm}^{2}\cdot{s}^{-1})$ | 1.540(2)