text:
\section{Introduction}

In metallic alloys obtained by solidification processing, dendritic microstructures are common \cite{flemings}.
From a fundamental standpoint, dendritic growth theory and modeling stands as a challenge to combine phenomena across a wide range of scales, from microscopic capillarity at dendritic tips to macroscopic transport of heat and solute in the melt \cite{Langer1980,Trivedi1994}.
Due to this multiscale aspect, numerous approaches have emerged over the years that aim at bridging length scales in solidification modeling \cite{boettinger2000solidification,Asta2009a,karma2016atomistic}.
Each of these approaches operates within a different range of scales, which makes them valuable tools from a technological innovation perspective, since a key hurdle on the way to effective ICME (Integrated Computational Materials Engineering) implementations relies upon our ability to couple models at different length scales \cite{allison2011integrated, panchal2013key}.
Yet, most modeling approaches to dendritic growth have thus far been developed separately, with little effort to compare them and to discuss their capabilities and limitations on a quantitative basis.

In this article, we focus on models of solidification that operate from the microscopic to an intermediate, or {\it mesoscopic}, scale between that of the {\it microscopic} solid-liquid interface pattern and that of the {\it macroscopic} solidification process.
At this intermediate scale, interactions between grains occur that determine microstructural features such as grain size, shape, morphology, and crystal texture.
Mesoscopic models are more recent than their microscopic and macroscopic counterparts, such that a rigorous quantitative assessment of their relative advantages and limitations remains to be performed.
We also focus on moderate solute supersaturation, $\Omega\leq 0.25$, which is a common regime for most processes that do not involve rapid solidification.

Here, we focus on three prominent models for dendritic crystal growth --- namely phase-field, dendritic needle network, and mesoscopic grain envelope models.
Using a set of benchmark simulations for isothermal equiaxed growth in a binary alloy at given undercoolings (i.e. at different solute supersaturations), we compare the predicted growth velocities of the dendrite tips in both steady and transient growth regimes.
The objective of this comparison is to highlight the advantages and limitations of the different methods, and to suggest how to select model parameters in order to achieve quantitative predictions while retaining profitable computational efficiency.
While this study is still ongoing, the preliminary results presented here already provide useful insight into the capabilities of the models and the choice of parameters.

\section{Models}

\begin{figure*}[b]
\centering
 \includegraphics[width=.78\textwidth]{Fig_Schematics.png}
\caption{
Schematic illustration of the phase field (a), dendritic needle network (b), and envelope (c) models, with characteristic model parameters, namely the PF diffuse interface width $W$, the DNN integration radius $r_{i}$ and bounding radius $r_{max}$, and the stagnant-film thickness $\delta$.}
\label{fig:schematics}
\end{figure*}

\subsection{Phase-field}
\label{sec:phasefield}

The phase-field (PF) method is a powerful tool for the resolution of free-boundary problems.
The method can handle moving interfaces of arbitrary morphological complexity.
Interfaces and boundaries are not tracked explicitly, but instead different phases or grains are represented by a ``phase field'' that varies continuously through an interface of finite width $W$ (Figure~\ref{fig:schematics}a).

The free energy of the system is formulated, typically using a Ginzburg-Landau functional~\cite{ginzbuglandau,halperin1974renormalization}, which describes the energetic contributions of interfaces, transformations, and local state (e.g. temperature, concentration, stresses, etc.)~\cite{chen2002phase,boettinger2002phase,Moelans2008,steinbach2009phase,Plapp2016,tonks2019phase}.
The time evolution of the fields derives directly from the functional derivatives of the constructed free energy functional, using the so-called Cahn-Hilliard equation \cite{cahn1958free} for a conserved field, such as the solute concentration, or a time-dependent Ginzburg-Landau (also referred to as Allen-Cahn \cite{allen1979microscopic}) equation for non-conserved fields.
A number of PF models have been proposed, including some that consider a grand potential functional in place of a free energy description \cite{Plapp2011,choudhury2012grand}, but the basic recipe typically follows these same common traits for every phase-field model.
Hence, what makes the distinction between different PF models is essentially the choice of the number of phase fields (or order parameters) necessary to describe the system and the transformation(s), the mathematical/physical formulation of the functional, the interpolation functions used for different properties between different phases, and the matching of the model parameters to physical parameters relevant to the phenomena one aims to describe.

In the context of solidification, the PF method has been particularly successful, in part due to the presence of well-defined sharp interface problems to be matched by the PF formulations~\cite{boettinger2002phase,steinbach2009phase}.
In particular, ``thin-interface'' asymptotic analyses have permitted using a diffuse interface width $W$ much higher than the actual width of the solid-liquid interface \cite{karma1998quantitative,Echebarria2004}.
The introduction of a so-called ``anti-trapping current'' has allowed quantitative simulations of alloys by eliminating spurious numerical solute trapping for wide interfaces \cite{Karma2001a,Echebarria2004}.
In the model considered here, the temperature is assumed homogeneous and constant in time, solid-state diffusion and kinetic undercooling are neglected, a binary alloy is considered within its dilute limit where its phase diagram can be linearized with a constant interface solute partition coefficient $k$, and solute transport in the liquid is limited to diffusion with a constant diffusivity $D$.
These assumptions result in the following evolution equations \cite{karma1998quantitative,Karma2001a,Echebarria2004,Tourret2015}
\begin{align}
\label{eq:psi}
a_s({\mathbf n})^2 \frac{\partial\psi}{\partial t} =
\nabla \cdot \left[a_s({\mathbf n})^2 \nabla\psi \right]
 + \varphi \sqrt 2
 - \lambda \sqrt 2 (1-\varphi^2) U
\nonumber\\
 + \sum_{m=x,y,z}\left[ \partial_m\left( |\nabla\psi|^2 a_s({\mathbf n}) \frac{\partial a_s({\mathbf n})}{\partial(\partial_m\psi)} \right) \right] \\
\label{eq:u}
\Big( 1+k-(1-k)\varphi \Big) \frac{\partial U}{\partial t} =
\widetilde{D} \; \vec\nabla\cdot\left[ (1-\varphi)\vec\nabla U\right]
\nonumber\\
 +\Big[ 1+(1-k)U \Big] \frac{(1-\varphi^2)}{\sqrt{2}} \frac{\partial\psi}{\partial t} \nonumber\\
 +\vec\nabla\cdot\left[ \Big(1+(1-k)U\Big) \frac{(1-\varphi^2)}{2} \frac{\partial\psi}{\partial t} \frac{\vec\nabla\psi}{|\vec\nabla\psi|} \right] ,
\end{align}
where the dimensionless solute field is
\begin{equation}
U = \frac{1}{1-k}\left[ \frac{c/c_l^0}{(1-\varphi)/2 + k(1+\varphi)/2 }-1\right]
\end{equation}
with $c$ the solute concentration field, $c_l^0$ the equilibrium solute concentration of a flat interface at the considered temperature $T_0$.
In these equations, space is scaled with the diffuse interface width $W$ and time with the relaxation time $\tau_0$ at $T_0$. Non-dimensional values of the liquid diffusion coefficient $\widetilde{D}=D\tau_0/W^2=a_1a_2W/d_0$ and the coupling factor $\lambda=a_1W/d_0$, with $a_1=5\sqrt2/8$, and $a_2=47/75$ from thin-interface asymptotic analysis~\cite{karma1998quantitative}.
We use the standard cubic symmetry of the surface tension with $a_s(\mathbf{n}) = (1-3\epsilon_4) + 4\epsilon_4(n_x^4+n_y^4+n_z^4)$, where $n_x$, $n_y$, and $n_z$ are the components of the unit vector $\mathbf{n}$ normal to the interface, and $\epsilon_4$ is the strength of the surface tension anisotropy.
The classical phase field $\phi$, with $\phi=+1$ in the solid and $\phi=-1$ in the liquid phase, was replaced by a preconditioned phase field $\psi$, defined by $\phi=\tanh(\psi/\sqrt{2})$.
This change of variable stabilizes the numerical resolution of the equations, thus allowing the use of a coarser grid spacing with negligible loss in accuracy~\cite{glasner2001nonlinear}.
The model is solved using finite differences on a homogeneous grid of cubic elements, and an explicit Euler time stepping scheme.
The code is written in cuda (Compute Unified Device Architecture) programming language, which allows its acceleration through multi-threading on Nvidia graphics processing units (GPUs).

\subsection{Dendritic Needle Network}
\label{sec:dnn}

The DNN approach aims at bridging the scale of the dendrite tip radius $\rho$ and the larger scale of transport phenomena, e.g. the solutal diffusion length $l_D=D/V$, with $V$ the crystal growth velocity.
The model, rigorously derived in the limit of small P\'eclet numbers, $P=\rho/(2l_D)\ll 1$, represents a dendritic crystal as a network of thin needles that approximate the primary stems and higher order side-branches of the grain.
In earlier descriptions of the model, in two dimensions, the needle-like branches were treated as infinitely sharp \cite{Tourret2013}.
In later versions of the model, branches were given a parabolic shape, which lead to the same governing equations, while also allowing a three-dimensional extension of the model \cite{Tourret2016a}.

The DNN model was shown to quantitatively reproduce primary dendritic spacings measured in directional solidification experiments in different Al-based alloys \cite{tourret2015threea,tourret2015threeb,Tourret2016a}. Implementations have been developed that include fluid flow in the liquid \cite{tourret2019multiscale,isensee2020} and the interaction between nucleation and growth~\cite{Sturz2016,geslin2016numerical}.

The evolution of the network during solidification is described by the growth dynamics of each needle tip and the generation of new sidebranches.
For a binary alloy, typical assumptions of the model include a linear phase diagram with an interface partition coefficient $k$, negligible fluid flow in the liquid phase (at least in the vicinity of the dendrite tips~\cite{tourret2019multiscale}), no solute transport in the solid phase, and negligible release of latent heat during solidification.
Under these conditions, for an isothermal domain at a given solute supersaturation $\Omega$, in three dimensions, the needle tip radius, $\rho$, and its velocity, $V$, are given by combining the following set of equations \cite{Tourret2016a}: a solvability condition at the needle tip scale $\approx\rho$
\begin{align}
\label{eq:r2v}
\rho^2V=2Dd_0/\sigma ,
\end{align}
a solute conservation at intermediate scale much larger than $\rho$ but much smaller than $l_D$
\begin{equation}
\label{eq:rv}
\rho V =D \cal{F} ,
\end{equation}
and solute diffusion in the bulk liquid at a scale $\approx l_D$
\begin{equation}
\partial_t u =D \nabla^2 u ,
\end{equation}
with
\begin{equation}
u=\frac{c_l^0-c}{(1-k)c_l^0}
\end{equation}
the normalized solute concentration, $c_l^0$ the liquid equilibrium concentration and $d_0$ the chemical capillary length, both at the fixed temperature $T_0$, and $\sigma$ the tip selection parameter.
The flux intensity factor $\cal F$ is a measure of the solute flux towards the needle in the vicinity of its tip up to a distance $a$ behind the needle tip (Figure~\ref{fig:schematics}b), and captures intra- and intergranular solutal interaction.
It is defined as
\begin{align}
\label{eq:fif}
{\cal F} =~&
\frac{1}{2\pi a} \; \iint_{\Gamma'}\frac{\partial u}{\partial n}{\rm d}\Gamma'
\nonumber\\
\approx~& \frac{1}{2\pi a} \; \left( \iint_{\Gamma}\frac{\partial u}{\partial n}{\rm d}\Gamma' + \frac{V}{D} \iiint_{\Sigma}\frac{\partial u}{\partial x'}{\rm d}\Sigma
 \right),
\end{align}
where $a$ is a chosen integration distance behind the tip, typically a few tip radii, and $\Gamma'$ is the resulting surface of the paraboloid up to a distance $a$ from the tip (Figure~1b). In Eq.~\eqref{eq:fif}, the last equality assumes that the field $u$ is Laplacian in a moving frame of velocity $V$, i.e. $D\nabla^2 u = -V \partial u/\partial x'$, at the scale of the integration domain, with $x'$ the principal growth direction of the branch.
This assumption allows calculating the normal solute flux along the surface $\Gamma'$ using the normal solute flux through any surface $\Gamma$ that intersects the parabolic tip at a length $a$ behind the tip, with an enclosed volume $\Sigma$ (Figure~\ref{fig:schematics}b)~\cite{Tourret2016a}.
In earlier model implementations \cite{Tourret2013,Tourret2016a,tourret2015threea,tourret2015threeb}, the outer surface $\Gamma$ was set as a square or a cuboid for their ease of implementation onto a square numerical grid.
In later works, a circular or spherical shape of the integration domain is preferred \cite{tourret2019multiscale,isensee2020}, which facilitates the simulation of grains of arbitrary orientations regardless of the numerical grid.

With $c_\infty$ the alloy nominal solute concentration, the supersaturation $\Omega=(c_l^0-c_\infty)/[(1-k)c_l^0]$ acts as boundary condition far from the interface, where $u\to\Omega$, while thermodynamic equilibrium neglecting capillary correction is applied along the dendritic branches, with $u=0$.

We solve the equations numerically on a 3D finite difference mesh with a homogeneous spacing $\Delta x$, and an explicit time stepping scheme with a constant time step $\Delta t$.
At each time step, we integrate the value of ${\cal F}(t)$ using Eq.~\eqref{eq:fif} with a spherical integration domain of radius $r_{i}$ centered on the tip of the needle~\cite{tourret2019multiscale}, which provides the value of the tip velocity $V(t)$ and radius $\rho(t)$ from Eqs~\eqref{eq:r2v} and \eqref{eq:rv}.

Far behind its tip, we bound the maximum thickness of a branch to a cylinder of radius $r_{max}$ (Figure~\ref{fig:schematics}b).
While it does not affect the paraboloid shape in the tip vicinity, this allows a tip to slow down and stop with $V\to0$ without numerical instabilities that would otherwise be induced by the constancy of $\rho^2V$.
Sidebranches normal to the growth direction (here assuming $\langle100\rangle$ growth directions) are generated at positions $\delta=(N_{br}+\delta_{br})\times\rho_s$ behind the tip in all possible directions, when the tip has advanced by a distance $N_{br}\times\rho_s$ since the last branching event.
A random fluctuation $\delta_{br}$ is generated for every branching event within a range of amplitude $\Delta_{br}$ .
The average distance between sidebranches $N_{br}$ and its fluctuation $\Delta_{br}$, both expressed in units of the theoretical steady-state tip radius $\rho_s$, are user input parameters.
Both truncating and branching methods are similar to those presented and discussed in previous works~\cite{Tourret2013,Tourret2016a,tourret2015threea,tourret2015threeb,tourret2019multiscale,isensee2020,Sturz2016}.

\subsection{Mesoscopic Grain Envelope Method}
\label{sec:envelope}

The Mesoscopic Grain Envelope Model (GEM)~\cite{Steinbach1999,Steinbach2005} bridges the scales from the dendrite tip, over the internal grain structure, to the mesoscopic scale of an ensemble of grains that interact by diffusion and convection of solute. The core idea of the GEM is that solutal interactions at the mesoscopic scale can be accurately simulated without a detailed representation of the ramified structure of the solid-liquid interface. Instead, the description of a dendritic grain can be simplified and represented by its envelope, an artificial smooth surface that links the tips of the actively growing dendrite branches (Figure~\ref{fig:schematics}c). Individual branches are not represented by the model, although, by definition, the growth of the envelope depends on the growth speed of the branch tips. The growth of the tips is controlled by the solute flux that they reject into the liquid and is therefore determined by the local solute concentration of the surrounding liquid. In the envelope model a constitutive model of tip kinetics is used that gives the tip growth speed as a function of the solute concentration field adjacent to the envelope.

The model was shown to be capable of quantitatively predicting grain envelope shapes, growth velocities, and internal solid fraction for a single grain growing into an infinite melt~\cite{Steinbach1999,Souhar_CMS_2016,Viardin_AM_2017} and for transient interactions among several grains~\cite{Steinbach2005, Olmedilla_AM_2019}.
The envelope model has been successfully used for simulations of {\it in situ} synchrotron X-ray imaging experiments~\cite{Delaleau2010, Olmedilla2019a, Olmedilla_AM_2019},
for modeling spacing adjustments and growth competition in columnar growth~\cite{Viardin_AM_2017,Li_ICASP_2019},
and for characterizing grain shapes and growth kinetics under the influence of solutal interactions used in upscaling to macroscopic models~\cite{TorabiRad_Mtl_2019}. The model was also extended to include fluid flow and solute advection~\cite{Delaleau2011, Olmedilla2019a}.

Two key assumptions are made: (i) that the phenomena controlling the growth of a dendrite tip are universal and are therefore valid for tips of any order (primary, secondary, tertiary,\ldots) and (ii) that the characteristic time of the interactions at the small (tip) scale is much smaller than that of the interactions at the large (grain) scale. In this work we further assume an isothermal domain at a given solute supersaturation, $\Omega$, in three dimensions, and we formulate the model as follows. The normalized concentration at the tip interface is that of thermodynamic equilibirium, neglecting capillary corrections: $u=0$. A stagnant-film formulation of the Ivantsov solution is written that relates the normalized concentration difference, $ u_\delta$, between the liquid at the tip and that at a finite distance, $\delta$, from the tip~\cite{Cantor1977} to the growth P\'eclet number of the tip, $P=\rho V/(2D)$, as
\begin{equation}
\label{eq:cantorvogel}
u_\delta = P \exp(P) \left\{ {\rm E}_1(P) - {\rm E}_1\left( P \left[ 1+\frac{2\delta}{\rho} \right]\right) \right\} .
\end{equation}
Eq.~\eqref{eq:cantorvogel} is used to solve for $P$. The concentration $u_\delta$ is obtained from the concentration field in the liquid around the envelope, which is resolved numerically as explained below.
The tip selection criterion $\rho^2V=2Dd_0/\sigma$, is a supplementary relation that gives the tip radius, $\rho = d_0/(\sigma P)$, needed in Eq.~\eqref{eq:cantorvogel}, and the tip speed, $V = 2\sigma P^2 D/d_0$.
Dendrite tips are assumed to grow in predefined growth directions.
In the present study, the cubic crystal dendrites are given six possible growth directions perpendicular to one another.
The normal envelope growth velocity, $V_{n}$, is calculated from the local tip speed, $V$, by the relation $\vec V_{n}=V\vec n\cos(\theta)$, where $\theta$ is the angle between the outward normal to the envelope, $\vec n$, and the preferential growth direction that forms the smallest angle with $\vec n$ (Figure~\ref{fig:schematics}c). To track the envelope on a numerical grid we use a phase-field-like interface tracking method~\cite{Sun2007}, combined with a surface reconstruction method for better accuracy~\cite{Souhar_CMS_2016}.
Combining these methods, one can avoid explicit tracking of the envelope, which is defined throughout the whole domain with a level set (iso-surface) of a continuous indicator field. Details are given in references \cite{Souhar_CMS_2016,Sun2007}.

Solute transport at the mesoscopic scale, here by diffusion, is described by volume-averaged equations that are valid in the whole domain, both inside and outside the envelopes.
Solidification inside the envelope uses the Gulliver-Scheil assumptions of thermodynamic equilibrium at the solid-liquid interface, negligible diffusion in the solid, and instantaneous diffusion in the liquid \cite{Scheil_ZM_1942}.
The resulting liquid fraction field, $g_l$, in the interior of the envelopes, was shown to realistically represent the distribution of solid and liquid within an actual dendritic grain envelope~\cite{Steinbach1999,Souhar_CMS_2016,Viardin_AM_2017}.
The liquid inside the envelope is at the equilibrium concentration, $u_l=0$. These assumptions lead to the conservation equation for the solute in the liquid phase
\begin{equation}
\label{eq:solute}
g_l \partial_t u_l = D \nabla\cdot\left( g_l\nabla u_l\right) + \partial_t g_l .
\end{equation}
The solution of Eq.~\eqref{eq:solute} gives the liquid concentration $u_l$ outside the envelope and $g_l$ inside the envelope. Outside the envelope, $g_l=1$ and Eq.~\eqref{eq:solute} reduces to a single phase diffusion equation.
Inside the envelope, $u_l=0$ is the known equilibrium concentration and Eq.~\eqref{eq:solute} gives the evolution of the liquid fraction inside the envelope. The concentration of the solid growing inside the envelope at constant $\Omega$ is $u_s=1$.

The supersaturation across the stagnant film, $u_\delta$, needed to calculate the envelope growth velocity (Eq.~\eqref{eq:cantorvogel} and additional relations), is obtained from the concentration field in the liquid around the envelope, which is fully resolved numerically (Eq.~\eqref{eq:solute}). This means that the locally valid Ivantsov solution for diffusion around a dendrite tip is matched to the solution of the mesoscopically valid solute transport equation at a distance $\delta$ from the envelope (Figure~\ref{fig:schematics}c).

The solute conservation equation~\eqref{eq:solute} and the equation of the evolution of the indicator field used for envelope tracking are solved by a finite volume solver with the OpenFOAM\textsuperscript{\textregistered} toolbox. Second and higher order discretization schemes are used for the spatial operators. Implicit Euler time-stepping is used for the solute conservation equations and explicit time-stepping for the envelope indicator field equation. Adaptive time-stepping is employed and is useful during fast transients, such as the initial transient after nucleation. During other growth stages the time step is essentially limited by the characteristic diffusion time, $\Delta x^2/D$, at the scale of the mesh cell, $\Delta x$. In the present work uniform meshes with cubic cells are used. The mesh size needs to be adapted for an accurate description of the diffusion layer adjacent to the envelope, i.e. $\Delta x \lesssim 0.2 D/V$, where $V$ is a typical primary tip speed. A special search and interpolation algorithm is designed to provide the concentration at the distance $\delta$ from the envelope, needed for the calculation of the envelope velocity~\cite{Souhar_CMS_2016}.

\section{Benchmarks}
\label{sec:benchmark}

Here, we compare the predictions of the three methods described above with a similar benchmark test case for undercooled isothermal equiaxed growth of a binary alloy.
The phase field simulation uses well-converged numerical parameters, i.e. small enough grid spacing and diffuse interface width $W$, such that the PF results will be used as a reference for the discussion of the effect and choice of numerical parameters in the other two methods.

We set a solid seed (about as small as the method allows) in the corner of a cubic domain initialized at a given solute supersaturation $\Omega$.
Thus, we simulate the growth of 1/8 of an equiaxed grain, using no-flux (mirror symmetry) Neumann boundary conditions on every face of the cube.
We choose a value of the domain size $L$ (edge length of the cube) that is large enough to reach (or closely approach) a steady growth velocity, but also small enough to make simulations computationally achievable within a few days (less than two weeks for the longest) using quantitative phase-field using the current single-GPU-parallelized implementation.
Hence, we selected $L/l_D\approx2$, 4, 6, 8, and 10 for $\Omega=0.05$, 0.10, 0.15, 0.20, and 0.25, respectively.

In the PF simulations, we use a small interface energy anisotropy $\epsilon_4=0.007$ and a solute partition coefficient $k=0.1$.
In the DNN and GEM simulation we consider a tip selection constant $\sigma\approx0.0264$.
This value was estimated from preliminary PF simulations of isothermal growth for $0.05\leq\Omega\leq0.25$ in a domain with $512^3$ grid points with tip velocities and radii measured as soon as the diffusion field gets affected at the side of the domain opposite to the dendrite. This value agrees with results from joint directional solidification simulation with similar $\epsilon_4$ and $k$. We noted a typical uncertainty of the order of 10 to 15\% depending upon the procedure chosen to extract the dendrite tip radius \cite{karma2000three}. Here, we follow the method described in reference \cite{clarke2017microstructure} (figure 5 therein), i.e. by 2D cross-section least-square fitting to a second order parabola over a fitting range that spans a distance of one tip radius behind the tip. A deeper analysis in terms of selection parameter is underway, such that the value used here only provides a reasonable estimation of $\sigma$ to use consistently in DNN and GEM simulations.

This value was estimated from independent directional solidification PF simulations with similar $\epsilon_4$ and $k$ parameters.
The selection parameter $\sigma$ estimated from the present isothermal PF simulations falls within 10\% of this value. However, directional solidification simulations reach a sustained steady-state even at lower undercooling, unlike the current simulations (see Figure~\ref{fig:comparison} discussed in section~\ref{sec:transient}).

The objective is to compare predicted growth velocities, not only in the steady-state, which should closely match the Ivantsov solution, but also and most importantly the initial transient, as the velocity tends towards the free dendrite steady-state, and the final transient, as the velocity declines to zero when the dendrite approaches the no-flux boundary wall.

For a given solute supersaturation $\Omega$, the theoretical P\'eclet number $P_s=\rho_s V_s/(2D)$ at steady-state (hence marked with a subscript ``$s$'')
is given by the Ivantsov solution~\cite{Ivantsov1947a}.
It directly gives the ratio between the steady-state diffusion length $D/V_s$ and dendrite tip radius $\rho_s$.
In this paper, most results are presented scaled with respect to the steady-state tip radius $\rho_s$ and velocity $V_s$.
For each value of $\Omega$ considered here, Table~\ref{tab:steady} summarizes the correspondence between these theoretical steady-state values and material parameters in physical units, such as the liquid solute diffusivity $D$ and the interface capillarity length $d_0$.
Within the selected supersaturation range, there is an order of magnitude of variation in the theoretical steady-state values of the P\'eclet and tip radius, and two orders of magnitude of variation of the tip velocity.

In the phase-field simulations, the finite difference grid element size is set equal to the diffuse interface width with $\Delta x=W\approx\rho_s/10$, which was found to yield well-resolved converged results.
The initial spherical seed has a radius $r_0=2.5\Delta x$, with a preconditioned phase field $\psi=r-r_0$ with $r=\sqrt{x^2+y^2+z^2}$ and a solute field $U=-\Omega$, which is equivalent to setting $c=c_\infty$ (i.e. $u=\Omega$) in the liquid and $c=kc_\infty$ in the solid.

In the DNN simulations, we initialize a solid seed with $u=0$ in the corner of the domain and setting $u=\Omega$ in the liquid.
The seed consists of three quarters of paraboloids along the three normal grid directions with initial tip velocity $V=0$, radius $\rho=\rho_s$, and length $L_0$ larger than $r_{i}$ and $r_{max}$.
In particular, we study the effect of mesh spacing, $\Delta x$, and truncation radius, $r_{max}$.
During the final transient, the integration sphere may be truncated if it exceeds the limits of the domain.

In the GEM simulations, we initialize a spherical envelope seed with $u=0$ and $g_l=1-\Omega$ in the corner of the domain and set $u=\Omega$ in the liquid. The size of the envelope seed is given by the smallest radius of curvature that can be accurately represented by the envelope tracking method on a mesh with spacing $\Delta x$~\cite{Sun2007,Souhar_CMS_2016}: $r_0=4.46 \Delta x$. The mesh spacing and time step needed for well-resolved converged results were determined by systematic convergence studies. Remaining mesh dependence, due to the initial seed size, which can be reduced on finer grids, is perceptible only during the initial growth transient and is discussed in Section~\ref{sec:resu:env}.

\begin{center}
\begin{table}[t]
\caption{
Theoretical steady-state P\'eclet number $P_s$, tip radius $\rho_s$, and velocity $V_s$, for different $\Omega$.
\label{tab:steady}
}
\centering
\begin{tabular}{c c c c}
\hline
\vspace{2pt}
$\Omega$ & $P_s=\frac{\rho_sV_s}{2D}$ & $\frac{\rho_s}{d_0} = \frac{1}{\sigma P_S}$ & $\frac{V_sd_0}{D} = 2\sigma P_s^2$\\
\hline
0.05 & 0.0131 & {2895} & {$9.03\cdot 10^{-6}$} \\
0.10 & 0.0341 & {1111} & {$6.14\cdot 10^{-5}$} \\
0.15 & 0.0624   & {607} & {$2.06\cdot 10^{-4}$} \\
0.20 & 0.0988   & {383} & {$5.15\cdot 10^{-4}$} \\
0.25 & 0.1446   & {261} & {$1.10\cdot 10^{-3}$} \\
\hline
\end{tabular}
\end{table}
\end{center}

\section{Results and Discussions}

\begin{center}
\begin{table}[!b]
\caption{Model parameters used with the three methods.
}
\label{tab:model_params}
\centering
\begin{tabular}{c c c c c c c}
\hline
  & PF & \multicolumn{2}{c}{DNN} & GEM  \\
$\Omega$ & $W/d_0$ & $r_i/\rho_s$ & $r_{max}/\rho_s$ &  $\delta/(D/V_s)$\\
\hline
0.05 & 289 & 2.6 & 3,\,10 & {0.06} \\
0.10 & 110 & 2.4 & 3 & {0.20} \\
0.15 & 60 & 1.6 & 3 & {0.40} \\
0.20 & 38 & 1.9 & 3 & {0.60} \\
0.25 & 26 & 0.6 & 2 & {0.80} \\
\hline
\end{tabular}
\end{table}
\end{center}

Model parameters used for these simulations are summarized in Table~\ref{tab:model_params}.
In order to provide a consistent illustration of the computational cost for all three methods, we also summarize the numerical parameters, namely the grid element size and time step for each $\Omega$ in Table~\ref{tab:num_params}.

The results presented here were obtained using model parameters that gave a good fit to the time evolution of the growth velocities obtained from PF simulations, while conserving a substantially lower computational cost.
Since we used a wide range of computers (e.g. CPU/GPU) and methods (e.g. explicit/implicit time stepping), we did not perform a systematic performance analysis and focused on the potential of each method to yield quantitative predictions. Yet, in the following subsection, we provide values used for grid spacing and time steps, which provide a consistent picture of the relative computational workload to each method

\subsection{Steady-state growth}

First, we compare steady-state growth predictions by all models for the range of solute supersaturation $0.05\leq\Omega\leq0.25$.
Steady-state growth P\'eclet numbers for $\Omega=0.05$ to 0.25, predicted by PF, DNN, and GEM models, are compared to the theoretical Ivantsov solution in Figure~\ref{fig:ivantsov}.
Values reported in figure 2 are averaged in time during steady-state growth, in order to integrate numerical oscillations. Such oscillation may occur as a dendrite tip crosses successive grid points and are due to the coarse grids used in DNN and GEM.
The fluctuations in tip velocity and radius (see e.g. Fig.~\ref{fig:comparison} discussed later) lead to a negligible fluctuation in P\'eclet number (within the symbol size used in Fig.~\ref{fig:ivantsov}).
In summary, all  models are capable of solving numerically the Ivantsov problem, and they thus appropriately predict the steady growth state.

\begin{figure}[!b]
\centering
\includegraphics[width=.775\columnwidth]{Fig_Ivantsov.pdf}
\caption{
Predicted steady-state P\'eclet number for different imposed supersaturation $\Omega$ compared to the analytical Ivantsov solution~\cite{Ivantsov1947a}.
}
\label{fig:ivantsov}
\end{figure}

\begin{center}
\begin{table*}[!t]
\caption{
Normalized mesh size and time step used with the three methods.
}
\label{tab:num_params}
\centering
\begin{tabular}{c c r r r c r r r}
\hline
$\Omega$	& & \multicolumn{3}{c}{$\Delta x/(D/V_s)$} &  &  \multicolumn{3}{c}{$\Delta t/(D/V_s^2)$} \\
			& & PF & DNN & GEM & & PF & DNN & GEM\\
\hline
0.05 & & {0.0026} & 0.0065 & {0.02} 	& &	{$10^{-6}$}	& {$7\cdot10^{-6}$}	&	{$10^{-4}$}\\
0.10 & & {0.0068} & 0.017 & {0.04} 	& &	{$7\cdot10^{-6}$}	& {$4\cdot10^{-5}$}	&	{$4\cdot10^{-4}$}\\
0.15 & & {0.0125} & 0.031 & {0.06} 	& &	{$2\cdot10^{-5}$}	& {$10^{-4}$}	&	{$9\cdot10^{-4}$}\\
0.20 & & {0.0198} & 0.049 & {0.08} 	& &	{$6\cdot10^{-5}$}	& {$4\cdot10^{-4}$}	&	{$10^{-3}$}\\
0.25 & & {0.0289} & 0.072 & {0.10} 	& &	{$10^{-4}$}	& {$8\cdot10^{-4}$}	&	{$2\cdot 10^{-3}$}\\
\hline
\end{tabular}
\end{table*}
\end{center}

The shape of the predicted grain when the length of the primary trunk reaches one half of the domain size is shown in Figure~\ref{fig:shapes} for all three models for $\Omega=0.05$ (top) and $\Omega=0.25$ (bottom).
For the sake of illustration, the DNN microstructure is shown for a simulation performed without sidebranches for $\Omega=0.05$ and no thickness bounding (i.e. with $r_{max}=\infty$), and with sidebranches and $r_{max}=2\rho_s$ for $\Omega=0.25$.
In the latter case, the effect of a finite $r_{max}$ becomes apparent from the cylindrical shape of the branches.
While at this stage the velocity has already started decreasing due to the effect of boundaries, the shape of the microstructures is barely distinguishable from that at steady-state, for all three models.

\begin{figure*}[!t]
\centering
 \includegraphics[width=.8\textwidth]{Fig_Grains.png}
\caption{
Grain shape (gray) and iso-concentration lines,
 from $u=0$ (red) to $u=0.9\Omega$ (blue) with steps of $0.1\Omega$,
 along the ($x=0$), ($y=0$), and ($z=0$) planes when the length of the primary dendritic trunk reaches one half of the simulation domain for $\Omega=0.05$ (a) and 0.25 (b) simulated using phase-field (left), DNN (middle), and Envelope (right) methods.
The total size of the domain, illustrated by the visible bottom left corner of each simulation, is $2D/V_s\approx 221\,400\,d_0$ for $\Omega=0.05$ and $10D/V_s\approx9\,060\,d_0$ for $\Omega=0.25$.
}
\label{fig:shapes}

\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=.775\textwidth]{Fig_Comparison.pdf}
\caption{
Evolution of the primary dendrite tip velocity from the initial to the final transient state (left) and primary trunk length versus time (right) for $\Omega=0.05$ (a) and $\Omega=0.25$ (b).
In order to allow comparing models all together at different supersaturations, time and space are scaled with respect to the theoretical steady-state velocity $V_s$ and tip radius $\rho_s$.
The velocity evolution is plot as a function of the tip location normalized by the domain length.
Note that left-hand-side velocity plots have been smoothed to reduce numerical oscillations and improve readability (right-hand-side length plots are raw).
}
\label{fig:comparison}
\end{figure*}

\subsection{Transient growth}
\label{sec:transient}

Next, we compare the predicted transient evolution of grain growth velocities. We focus on $\Omega=0.05$ and $\Omega=0.25$, which are illustrative of results obtained at intermediate $0.05<\Omega<0.25$.
The evolution of the velocity from initial to final transient appears in Figure~\ref{fig:comparison} for $\Omega=0.05$ (a) and $
\Omega=0.25$ (b).
The velocity is shown as a function of the tip location normalized by the domain length.
This way, we filter out the possible different onset times for the final decrease of the velocity --- such differences arise when the steady-state is maintained at a different velocity.
The right-hand-side panels, showing the evolution of the primary trunk length versus time, illustrate that the oscillations appearing in the left-hand-side velocity plot have a negligible effect on the length of the dendrite, once integrated over time.
These results show that, for appropriately chosen sets of parameters, both DNN and GEM models can reproduce PF results with reasonable accuracy.

DNN simulations were performed with $\Delta x=0.25\rho_s$, $r_i=2.6\rho_s$, $r_{max}=3\rho_s$ for $\Omega=0.05$, and with $\Delta x=0.25\rho_s$, $r_i=0.6\rho_s$, $r_{max}=2\rho_s$ for $\Omega=0.25$ (see Table~\ref{tab:model_params}).
We chose a resolution providing a compromise between accuracy and simulation time.
The chosen grid size is smaller than the typical value used in DNN simulations, i.e. $\Delta x \approx \rho_s$ or higher.
Other parameters are close to typical parameters used in the DNN model, i.e. fulfilling $r_i\ll l_D$ and $r_{max}$ chosen as to not bound the needle thickness within the flux integration domain.
Interestingly, we notice very little influence of sidebranching on the velocity of the primary tip, both in steady and transient states.
It is also worth noting that the integration radius $r_i$ can be taken lower than the tip radius, as long as the integration domain contains a sufficient number of integration points.

GEM simulations were performed with a stagnant film thickness of $\delta=0.06 D/V_s$ and $0.80 D/V_s$, and with a mesh spacing of $\Delta x=0.02 D/V_s$ and $0.10 D/V_s$, for $\Omega=0.05$, and 0.25, respectively. The $\delta$ and the mesh used at high $\Omega$ are close to those already recommended by Souhar et al.~\cite{Souhar_CMS_2016} for steady-state equiaxed growth. At low $\Omega$, significantly smaller $\delta$ were needed to accurately match both the initial and the final transient of the tip evolution. The same conclusion was recently made by Olmedilla et al.~\cite{Olmedilla_AM_2019} when comparing the GEM to experiments on transient equiaxed growth at comparably low undercoolings. The mesh spacing is dictated by the diffusion length at high $\Omega$ ($\Delta x\approx0.1 D/V_s$ here), however at low $\Omega$ the small stagnant-film thickness, $\delta$, becomes the limiting factor. For an accurate calculation and interpolation of $u_\delta$ the mesh spacing should be $\Delta x<0.5 \delta$.

\subsection{Effect of DNN parameters}

Figure~\ref{fig:dnn} shows the tip velocity evolution predicted by DNN simulations for different combinations of grid spacing $\Delta x$ and needle truncation radius $r_{max}$, for a given flux integration radius $r_i=2.6\sim2.7\rho_s$ and $\Omega=0.05$.

Decreasing the grid spacing $\Delta x$ for identical values of $r_{max}$ and $r_i$ results mainly in (i) less fluctuation of the velocity and (ii) higher velocities, in better agreement with PF results.
On the other hand, the effect of increasing the radius of the stem truncation $r_{max}$ for identical values of $\Delta x$ and $r_i$ is to decrease the steady-state tip velocity.

The effect of the initial length of the seed branches needs to be discussed. The initial length $L_0$ of the branches differ among the simulations because the simulations are initialized with paraboloids of length $L_0=r_{max}+\Delta x$.
The difference in initial length most notably affects the growth kinetics during the initial and final transient stages.
All four simulations nonetheless stabilize close to the theoretical steady-state velocity given by the Ivantsov solution.
This plateau region, however, lasts for a shorter time for the simulations with larger $r_{max}$ and hence longer $L_0$.
At this low $\Omega$, in order for PF simulations to be achievable within a few days, we limited the domain size to 2$l_D$.
For this reason, the amount of time spent in steady-state is limited, and for high $L_0$ values, the final transient sets in before steady-state is fully maintained.
In the last stage of the final transient, a velocity plateau may be observed (not represented here but visible by interrupted plots) due to a numerical restriction set to the tip velocity when the distance to the wall becomes lower than $r_i$.
This limitation might be accommodated by either (i) truncating the integration domain, (ii) integrating boundary conditions within the integration of $\cal F$, or (iii) using an adaptive $r_i$ that decreases when approaching a boundary.

\begin{figure}[t]
\centering
\includegraphics[width=.81\columnwidth]{Fig_DNN.pdf}
\caption{
\label{label}
Evolution of the primary dendrite tip velocity for $\Omega=0.05$ from DNN simulations, showing the effect of grid spacing $\Delta x$ and stem truncation radius $r_{max}$.
In all runs, the radius of the integration sphere $r_i/\rho_s$ was kept to 2.6 or 2.7.
For readability, symbols show raw data whereas lines show smoothed data (filtering numerical oscillations as the tip progresses between successive grid points).
}
\label{fig:dnn}
\end{figure}

\subsection{Effect of Envelope parameters}
\label{sec:resu:env}

The main model parameter of the GEM is the stagnant film thickness, $\delta$. Table~\ref{tab:model_params} summarizes the $\delta$ that gave the best match to the tip evolution predicted by phase field. The best $\delta$ clearly increases with $\Omega$. The dependence of the transient growth velocity on $\delta$ at $\Omega=0.05$ is shown in {Figure~\ref{fig:env}a} for a fixed stagnant film thickness $\delta/(D/V_s)=0.08$. The most striking observation is the effect on the steady-state velocity (already pointed out by Souhar et al.~\cite{Souhar_CMS_2016}), which affects the whole evolution. The de- and accelerations during the initial and final transient do not significantly depend on $\delta$.

\begin{figure*}[b]
\centering
\includegraphics[width=.81\textwidth]{Fig_Envelope.pdf}
\caption{
Evolution of the primary dendrite tip velocity for $\Omega=0.05$ from GEM simulations, showing the effect of the model and numerical parameters. (a) The effect of the stagnant film thickness, $\delta$; all simulations with a mesh $\Delta x/(D/V_s)=0.02$. (b) The effect of the mesh spacing $\Delta x$, and of the size of the initial nucleus, $r_0$; all simulations with $\delta/(D/V_s)=0.08$.
}
\label{fig:env}
\end{figure*}

{Figure~\ref{fig:env}b} shows the influence of the mesh spacing on the transient tip growth at $\Omega=0.05$. The influence of the mesh spacing is twofold. In addition to its direct impact on the solution of the PDEs, $\Delta x$ also limits the smallest radius of the initial nucleus, $r_0$, and thus the initial conditions (see Section~\ref{sec:benchmark}). Typically, $r_0 \geq 4.46 \Delta x$. The direct impact of $\Delta x$, \emph{excluding} that of $r_0$, is shown by four simulations in which $r_0$ was kept fixed and only the mesh spacing was varied from $0.013 D/V_s$ to $0.04 D/V_s$. The initial radius was $r_0 = 0.18 D/V_s$ which is the smallest $r_0$ that can be resolved on the coarsest of these four grids. The differences between the four solutions are strictly limited to the initial transient, and perfectly overlap at larger arm lengths. The reason for this is that the thin diffusion layer around the grain during the initial stages of growth requires a fine mesh resolution to be accurately resolved. Once the tip decelerates and the diffusion layer grows wider, coarse meshes are sufficient.

However, the most important influence related to the mesh resolution is actually that of the initial nucleus size. This is shown through a comparison of the evolution of tip velocity for $r_0=0.18 D/V_s$ and $0.09 D/V_s$ with the same $\Delta x=0.02D/V_s$ (open versus full square symbols in Figure~\ref{fig:env}b). We note a much slower deceleration during the initial transient at smaller $r_0$. Furthermore, the steady growth stage is replaced by a smooth transition to the final transient, which is not significantly affected.

In summary, accurate simulation of transient equiaxed growth requires a stagnant film thickness that roughly scales as $\delta \sim \Omega D/V_s$. This does not contradict, but goes beyond prior results~\cite{Souhar_CMS_2016} that established a recommendation for \emph{steady-state} growth: $\delta \sim D/V_s$. If compared, the new recommendation is slightly less accurate in steady-state (with a difference of $<10\%$) but is more accurate during transients.

The mesh spacing and the initial nucleus radius have a significant influence on the results only during the initial growth transient, but not during steady-state and the final transient. Note that because of the relatively small domain size ($L=2 D/V_s$) in the case presented here ($\Omega=0.05$), the impact of the initial transient on the whole evolution is prominent. This is not the case at higher $\Omega$, where larger domains are used and the initial transient is a smaller portion of the growth. Generally, $\Delta x \sim 0.1 D/V_s$ is sufficiently fine for steady-state and for the final growth transient, but should be refined if good accuracy during initial transients is required. Furthermore, $\Delta x < 0.5 \delta$ is an additional constraint that is important at low $\Omega$, where smaller $\delta/(D/V_s)$ must be used.

\section{Summary and Perspectives}

We have compared the results of phase-field, dendritic needle network, and grain envelope models for a simulation of the equiaxed growth of an isothermally undercooled binary alloy.
Using well-converged PF results as the benchmark reference, we discussed the effect and choice of model and numerical parameters in the DNN and GEM methods upon their accurate prediction of steady and transient dendrite growth velocities.

As expected, all models can appropriately predict steady-state growth conditions prescribed by microscopic solvability and Ivantsov solutions.
Moreover, in the transient regimes, both GEM and DNN can accurately describe growth velocities, if appropriate model and numerical parameters are selected.
The final transients (solutal interactions between grains in practical applications) are typically accurate with standard parameters.
The early-stage growth transient, on the other hand, requires a fine tuning of model parameters in order to achieve quantitative predictions, such that the initial seed size and mesh spacing need to be finer than typical values recommended in previous works, in both DNN and GEM approaches.

While the present study provides already some new insight and guidance for the use of these relatively new modeling techniques, these results remain preliminary, and the study is ongoing.
Work in progress on the current benchmarks include:
(i) the extraction of scaling laws for initial and final transient growth stages,
(ii) further systematic exploration of model parameters (e.g. $r_i$ for DNN),
(iii) comparison of envelope shapes in equiaxed growth, which is important in the scope of grain interactions,
and
(iv) exploring the advantages and limitations of models at higher supersaturations.
Perspectives for future benchmarks include:
(i) benchmarks for directional solidification, in particular in the scope of spacing selection,
and
(ii) configurations with convection in the liquid phase.

From the results of these studies, we expect that new numerical techniques will improve the predictive capabilities of the mesoscopic models.
Such techniques could include, for instance, the introduction of adaptive parameters to ensure better treatment of transient regimes, e.g. using adaptive $r_i$ in DNN or $\delta$ in GEM that could depend upon the growth acceleration/deceleration.
We also expect that the present work, and those to follow, will provide useful guidance to potential users and developers of these promising tools for bridging length and time scales in the modeling of dendritic solidification.

\section*{Acknowledgements}
D.T. acknowledges support from the European Union's Horizon 2020 research and innovation programme under a Marie Sk\l odowska-Curie Individual Fellowship (Grant Agreement 842795).
L.S. and A.V. acknowledge support by the German Ministry of Economy via the Deutsche Zentrum f\"ur Luft- und Raumfahrt, DLR (Contract 50WM1743).
MZ acknowledges support by the French State through the program ``Investment in the future'' operated by the National Research Agency (ANR) and referenced by ANR-11 LABX-0008-01 (LabEx DAMAS).

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


Of all the components of a wood staircase, the stair treads and handrail take the most abuse. Hands can carry sticky, dirty substances that degrade the finish, but even clean hands running over a rail will eventually wear it down. The key to refinishing your handrail successfully is to properly prepare the surface for a new finish, then follow up by applying a wear-resistant clear coat with a quality applicator.
Sticky fingers can take their toll on a wood staircase handrail.
Dampen a sponge with premixed all-purpose cleaner. Wipe the rail with the sponge.
Rinse the sponge with plain water and wipe the rail again. Dry the surface with a rag or let it dry naturally.
Apply painter's tape around the newel post and baluster connections and any other surface that contacts the handrail. If you cannot curve the tape to fit around a shapely rail, tape the area well and trim the excess with a utility knife to conform to the curves.
Lay newspaper or a drop cloth on the stair treads to guard against drips.
Put on a dust mask to cover your mouth and nose.
Attach a sheet of 100-grit sandpaper to a handheld sander. Sand the rail following the grain of the wood until the finish wears away, revealing the bare wood.
Work the sandpaper into any fine curves manually. Sand the entire surface again using 220-grit sandpaper to smooth away any roughness.
Sand the rail by hand with a sheet of 320-grit sandpaper until it develops a smooth sheen.
Wipe down the handrail with a tack cloth to completely remove any residual sanding dust.
Check the tape and replace any areas where sanding lifted or moved it.
Open the can of polyurethane varnish and stir it gently with a paint paddle.
Dip a high quality synthetic filament brush into the polyurethane. Brush it onto the rail in long, slow strokes to prevent bubbles.
Allow it to dry two hours, or as directed on the product's label.
Sand out any fine bubbles or visible brush marks using the 320-grit sandpaper. Apply a second coat of polyurethane varnish, as before, and allow it to dry thoroughly.
Peel the painter's tape off the newel posts and balusters. Remove the drop cloths.
If your handrail is fastened to a wall, tape newspaper to the wall behind the rail and tape off the bracket hardware before applying the varnish.
If your sanding produces wood that is unevenly colored, apply a coat of gel stain and allow it to dry before applying the polyurethane varnish.


To create confident learners who use IT safely.
Click here to read guidance on the effect of the internet on pupil's well-being.
Online Safety Guidance for Parents and Carers.
A resource from Hwb on how to use the most popular games safely.
A new resource is available to help raise awareness of the issues surrounding making friends online.
Click here to read more about issues surrounding screen time.


It is grudge match day as my pool team, Leroy's team 12 plays againt the evil Leroy's team for our only scheduled match against each other this season. A lot is at stake, not only because our teams are right next to each other in the league standings but also bragging rights and my own personal vendetta being that the Evil Leroy's team stole two players who had originally committed to my team before the season started. This is a day long awaited.
I like that vest you have on in the photo. I need one of those.
I'll have Bently send you one!


The garden is managed to encourage wildlife with the help of the Staffordshire Wildlife Trust and the Churnet Valley Project. The bee hotel now provides a refuge for several species of bee and other beneficial insects.
The long grass with a mown path in the shape of the Staffordshire Knot is a mini-hay meadow, created by strewing hay from a species-rich hay meadow reserve.
This summer we were thrilled to find a Southern hawker dragonfly zooming around the garden.
The squirrels have of course made the garden and trees beyond their home.


You are visiting the placeholder page for Aaron Lewis. This page is here because someone used our placeholder utility to look for Aaron Lewis. We created this page automatically in hopes Aaron Lewis would find it. If you are not Aaron Lewis, but are an alumni of Fort Meade High School, register on this site for free now.


There is plenty of research showing the link between acidic pH and cancer. Cancer thrives in an acidic environment and cannot survive in an alkaline environment. Cancer cells make your body even more acidic as they produce lactic acid. So if you have cancer, your pH levels are low and your body is too acidic.
Taking action to make your body more alkaline is vital in the battle against cancer.
The majority of food and drinks we consume are acidic, such as meat, grains and sugar, with colas and other soft drinks being highly acidic. So unless you have been eating a very healthy diet, full of fresh fruit and vegetables, chances are good your body is too acidic, providing the perfect environment for cancer to grow.
Actually, too much acidity is an underlying factor in many degenerative diseases — diabetes, arthritis, fibromyalgia and more. A basic maxim of natural physicians is: Balance the biological terrain. Do this first, then everything can come back to normal.
Taking medicines or supplements for cancer while your pH is highly acidic is a bit like washing dishes in a sink of dirty water, even when you put in plenty of soap, you can't get the dishes clean.
According to Keiichi Morishita in his book, Hidden Truth of Cancer, when your blood starts to become acidic, your body deposits acidic substances (usually toxins) into cells to allow the blood to remain slightly alkaline. This causes your cells to become more acidic and toxic, which results in a decrease of their oxygen levels, and harms their DNA and respiratory enzymes.
Over time, he theorizes, these cells increase in acidity and some die. These dead cells themselves turn into acids. However, some of these acidified cells may adapt in that environment. In other words, instead of dying – as normal cells do in an acid environment – some cells survive by becoming abnormal cells.
These abnormal cells are called malignant cells. Malignant cells do not correspond with brain function nor with our own DNA memory code. Therefore, malignant cells grow indefinitely and without order. This is cancer.
As you can see, he is describing, from a different point of view, the link between pH and cancer, a process by which low oxygen levels turn some cells cancerous. Alkaline water (including the water in cells) holds a lot of oxygen. Acidic water holds very little oxygen. So the more acidic your cells are, the less oxygenated they will be. To make matters worse, the fermentation process cancer cells use to produce energy creates lactic acid, further increasing acidity and reducing oxygen levels.
There is a long history of reversing cancer simply by alkalinizing the body. It is one of the basic strategies in the battle against cancer and for improving your health in general.
Virtually everyone with cancer has low pH levels. Low pH causes your body to store more toxins in cells, and reduces oxygen levels, both of which are fundamental to the development of cancer. When cancer cells grow, they produce even more acid, making it very difficult to raise pH levels.
When you take supplementation to increase pH levels, your cells notice the difference and start to dump some of the toxins they are holding back into the bloodstream now that they have a chance to get rid of them. Because the cells are releasing these stored toxins, your pH levels drop again. This detoxification process can take months to get through.
In addition, because cancer cells pump out lactic acid as a byproduct of their energy production, they create even more acidity. It can take a long time to normalize pH because of these factors, but, as you do, you can make your body, in a sense, "cancer proof".
Now that you know about the link between pH and cancer, you can do something about it. There are many sites and books or ebooks promoting various ways to beat cancer by alkalizing the body with their particular product suggestions. Coral calcium and alkaline water machines are a few examples. Also a combination of oxygen catalyst drops and a high quality green drink.
Over the years, as some of the more important cancer fighting suggestions like Zeolite Enhanced with DHQ worked well to increase pH levels, our primary suggestion was to drink 3 teaspoons of baking soda in water every day, an inexpensive alternative. But since pH is so fundamental to fighting cancer, we did some energetic testing to determine which products/methods were best for fighting cancer. The results were pretty interesting.
Alkaline drops, put in your daily liquids, tested at a low 35 — not very good. Getting a machine that makes alkaline water and drinking the water tested at 135. A machine that vortex's water, oxygenates it and adds minerals to increase pH is a bit better at 195. The 3 teaspoons of baking soda in water every day tested at 142. Taking Coral Calcium was better, at 235 for fighting cancer. A high quality green drink marketed for alkalizing the body came in at 214. For adjusting pH only, Zeolite Enhanced with DHQ is 235 when used in the quantities suggested for cancer.
OCMP is a proprietary formula of minerals with some other nutrients, that is effective at alkalizing cells and is a very strong cancer killer. Better still, it works to correct the main underlying cause of cancer, the buildup of toxins in cells which prevents nutrients like oxygen from entering cells. OCMP works by adjusting the ionic charge gradients inside and outside the cell wall, allowing toxins to be better eliminated, and for nutrients to be better absorbed. This makes it one of the best preventive/maintenance products to use. It's healing power for fighting cancer is 1280 when using 1 to 3 bottles per month.
The next best pH adjuster we have found is an herbal formula originally designed to reduce inflammation and support the heart and lungs. But when the developer of this product checked to see how well it adjusted pH levels, he discovered it did a remarkably good job of increasing pH levels in the body. So good, in fact, he renamed the supplement pH Balancer 8.0.
Energetic testing validated his conclusions. pH Balancer 8.0 comes in at 499 when taken in the suggested quantities. It has the ability to potentize – making other products more effective. Most particularly, OxyDHQ and Zeolite Enhanced with DHQ. When using these three supplements in the appropriate quantities for the amount of cancer you have, energetic testing puts their healing power at 3340 for fighting cancer.
BLA (an acronym for Block Lactic Acid effluxion), is an elixir supplying a set of instructions to the body in the form of vibrational frequencies. BLA is designed to stop cancer cells from pumping the lactic acid they produce (as part of the fermentation process) into the body.
What users are finding is that it takes four days for BLA to stop the lactic acid pumps in cancer cells. Lactic acid is constantly being produced by cancer cells as they primarily ferment sugars to make their energy.
Once BLA stops the lactic acid pumps, that lactic acid can no longer be moved out of the cancer cells. Once this happens, the lactic acid starts to build up in the cancer cell.
Cancer cells produce a lot of lactic acid as it is a byproduct of the sugar fermentation they primarily use to make energy. As this lactic acid produced by the cancer cell accumulates, thanks to BLA, it makes the cancer cells too acidic. Over time, the lactic acid builds up to such a level that the cancer cells becomes too acidic and they die a natural death.
This process is the opposite of trying to kill cancer cells by increasing pH levels.
It is much more effective because it uses what the cancer cells produce, lactic acid, to kill the cells, as opposed to trying to make cancer cell pH more alkaline, with alkaline minerals, alkaline water and so on.
And cancer cells produce a lot of lactic acid, as glucose fermentation is their primary source of energy.
Many times, if a significant amount of cancer is in the liver, lungs or stomach, the cancer may produce so much lactic acid in a relatively confined space that the body produces fluid to dilute the acidic buildup.
The liver also plays a role in the development of this fluid buildup in the abdominal area. When the liver is not doing its job of eliminating toxins, they can build up to such an extent that fluid is produced to dilute the toxic overload — which is also acidic.
BLA is fundamental to dealing with toxic fluid overload, even when the liver is not involved. Use six bottles per month. Fulvitea and Immune Force are also important to use when the liver is involved in causing ascites.
The process of alkalizing cancer cells is slower and less effective than making them too acidic, because cancer cells naturally neutralize pH buildup with their lactic acid production. All cells, including cancer cells, die if they get too alkaline or too acidic.
So to explain again, cancer cells produce energy almost entirely by fermentation, breaking down sugars, as opposed to aerobically producing energy with oxygen. This fermentation process produces lactic acid as a byproduct. When you stop the elimination of lactic acid with BLA, lactic acid builds up in the cancer cell. At some point the lactic acid builds up to an amount that the cancer cells become so acidic they can no longer function, and die a natural death.
The buildup of lactic acid works amazingly well for causing cancer cells to die a natural death — fast.
One researcher, Dr. Rosenberg, MD, a leading cancer doctor, using a different process to block the lactic acid pumps in cancer cells, fed a patient a huge amount of sugar by giving him glucose IVs — once the lactic acid pumps were blocked.
Feeding cancer cells sugar caused them to produce much more lactic acid then they normally would. In fact, the more sugar he gave his patient, the faster the cancer cells died because the lactic acid built up faster in the cells.
He brought the tumor markers in one advanced cancer patient down to 0 in 30 days. A second patient's markers dropped 40% in 2 weeks. Truly amazing results.
In use since early 2011, BLA is proving to be one of the top cancer fighting supplements we have researched. Especially when used with PrugX, a sister elixir that causes a build up of hydrogen peroxide in cancer cells. They work synergistically together, creating a toxic soup of lactic acid and hydrogen peroxide in cancer cells.
You can read more about PrugX and the new PrugX Enhancer in the Antioxidants and Cancer section of this site.
The hydrogen peroxide buildup causes free radical damage to cancer cells. PrugX was developed to tell the body to stop the release of hydrogen peroxide by cancer cells after researchers discovered that cancer cells were continually producing hydrogen peroxide and releasing it to damage nearby healthy cells.
When using BLA you can more rapidly increase the build up of lactic acid in the cancer cells by doing what Dr. Rosenberg did, increase consumption of sugars so that the cancer cells eat more and thus make more lactic acid. Any sugar will work, but as refined sugars tend to wipe out the immune system response, natural sugars are better. Early on the recommendation with BLA was to do just this.
However, with the addition of PrugX, PrugX Enhancer, BLA Enhancer and Glutam, additional sugar is not needed or even wanted.
The combination causes cancer cells to die quickly without consuming extra sugar. And feeding cancer cells sugar will increase their energy. This combination is now focused on shutting down the energy production of cancer cells so that they are starving for energy along with suffering an overload of lactic acid and hydrogen peroxide.
Even from the first, without PrugX, BLA was producing impressive results.
Dean Edmonds reported that his PSA had been rising every month, from 940.6 in February of 2011 to 3778.7 on May 11, 2011. This was when he stated using BLA, a couple of other supplements, and honey. On June 7, 2011, less than a month later, his PSA test came in at 77.
"You will know that I started taking [BLA, Custom Elixir pH and Zeolite Enhanced with DHQ] some 7 weeks ago. My local doctor is fully aware of what I am doing and is being very supportive. I had my blood tested at that time ( 7 weeks ago). At that time all my liver functions were normal apart from the tumor marker which showed a level of over 11.000 (eleven thousand) My blood was re-checked last week and yesterday my doctor advised me of the results. The tumor marker had reduced dramatically to 1400 ( fourteen hundred ) back down to the level when the cancer was diagnosed some 12 months ago….
"You should be aware that I have undertaken three separate chemo treatments already none of which showed any improvement in the tumor marker, as I said above it actually increased!
If you are not using BLA, it is best to avoid consuming carbohydrates like sugar that feed the cancer. But when you are using BLA, once the lactic acid pumps have been shut down, the additional lactic acid that is produced by the cancer cells when you feed them sugar stays in the cancer cells. The faster you increase levels of lactic acid, by feeding them sugars, the faster the cancer cells die a natural death.
BLA raises overall body pH levels too.
When the cancer cells have stopped pumping out the lactic acid they produce, your pH levels naturally will start to increase. One Naturopath tested BLA on 10 of her patients with advanced cancer. At the end of two months every one had increased their overall pH levels significantly, most by at least .75, some by a full point some by half a point.
In addition, the CA markers all went down, most by around 20%, one was 10% and another by 25%. All these are significant scores, and they were caused by BLA alone.
This sister elixir to BLA reduces the ability of cancer cells to neutralize the buildup of lactic acid in them. The major minerals, calcium, magnesium, sodium and potassium would do just this if taken in high quantities. (The amount in ESME is a problem.) While cancer cells contain low amounts of calcium, magnesium and potassium, they contain a great deal of sodium along with the extra water that sodium attracts.
BLA Enhancer is also a frequency enhanced elixir. Among the instructions it carries into the body are instructions to reduce as much as possible the amount of minerals entering cancer cells. This keeps to a minimum the absorption of any acid neutralizing minerals.
Additional instructions tell the cancer cells to eliminate sodium and the extra water they contain. Potassium too, as sodium and potassium work together and we don't want extra potassium being brought into cancer cells to replace the sodium being eliminated. The decrease in sodium and water means that less lactic acid will be neutralized — and that it will be more concentrated as there will be less water to neutralize it. So they will die faster. Significantly faster.
L-Glutamine is the most abundant amino acid in the body. Your body needs it for ensuring nitrogen balance, maintaining gut integrity, and keeping your immune system strong.
Unfortunately glutamine also is avidly used by cancer cells in many ways. It is metabolized for fuel. In fact, one reason cancer can grow when you are on a no sugar, low carb diet is because it will burn glutamine in place of glucose sugar to make the energy it needs.
Glutamine also supplies nitrogen to cancer cells so that they can function correctly, and in general, promotes the growth of cancer cells. Even more importantly, it unfortunately helps cancer cells thwart efforts to destroy them.
There are three primary actions using glutamine that protect cancer cells and enable them to grow and thrive.
First, glutamine stimulates production of glutathione in cancer cells. Glutathione is a major antioxidant enzyme. You want to have glutathione, lots of it, in healthy cells. However glutathione protects cancer cells from free radical damage. Many types of chemotherapy and the elixir PrugX, kill cancer cells by causing free radical damage the DNA in cancer cells.
When cells have plentiful glutamine, they can make enough glutathione to prevent destruction.
Secondly, when cancer cells have excess glutamine, they are able to exchange it for essential amino acids needed to prevent cell death when cells are badly damaged and poorly functioning. This process of destroying damaged cells is called autophagy, and is an important component of natural cancer cell death. Glutamine, excess glutamine at any rate, prevent this.
Third and most important, researchers have discovered that glutamine buffers or neutralizes lactic acid buildup in cancer cells. As we have talked about in this article, cancer cells produce most of their energy through the metabolism of glucose sugars (and glutamine too). This produces excess lactic acid that would lower pH in cancer cells to such a low level that they would die.
Cancer cells deal with this in two primary ways. They pump out of the cell as much lactic acid as they can. And they metabolize glutamine. When glutamine is metabolized, ammonia is produced. Researchers have discovered that ammonia will neutralize acid buildup.
Cancer cells grown in a dish that had no glutamine as a food source could not survive low pH levels but would grow rapidly in high pH environments.
Cancer cells that had glutamine as a food source were found to consume more glutamine as pH was lowered — and not die at the low pH levels.
When the glutaminase enzyme used to metabolize glutamine was inhibited, cancer cell survival dropped when pH was lowered — causing the cancer cells to become more acidic.
What this means is that the cancer cells in your body depend on glutamine to survive — especially when BLA is used.
They can pump out lactic acid to remove some of it, (until BLA stops that process). Cancer cells metabolize glutamine to make some of their energy, and the byproduct of this energy production is ammonia which will neutralize the lactic acid.
This mechanism explains why BLA hasn't caused all cancer cells to die quickly. They die, but not as quickly as they would because it takes a longer period of time to build up the lactic acid levels when glutamine is available to metabolize.
The cancer cells metabolize more glutamine to make energy which produces more ammonia which neutralizes the lactic acid buildup caused by BLA. Or at least some of the build up — slowing down the process of cancer cell death by acidosis.
It doesn't do any good to stop consumption of foods that supply glutamine. Researchers note that cancer cells will always get enough from the glutamine that is made by the body. So stopping the consumption of glutamine will only harm our healthy cells. For example, your body will not be able to make as many immune cells to fight cancer.
Stopping glutamine consumption would also create a serious negative nitrogen balance in the body. Nitrogen depletion is tied to Cachexia or muscle wasting that is the principle cause in about a third of all deaths from cancer.
So what can be done to deal with this issue?
Glutam is the solution as it carries instructions to the body to stop the use of glutamine in cancer cells.
It is a frequency enhanced elixir that, like BLA, PrugX and other frequency enhanced elixirs, carries a set of instructions — in the form of vibrational frequencies of the subtle energies of those instructions — into the body. Its instructions turn off the metabolism of glutamine by cancer cells only. They stop production or uptake of the glutaminase enzymes needed to metabolize glutamine.
Glutam also tells the body to reduce the amount of glutamine that is absorbed by cancer cells.
As a consequence of using Glutam, cancer cells will have less glutamine in them, and less of that glutamine will be metabolized. This results in an inability to neutralize the lactic acid buildup caused by BLA and BLA Enhancer. This increased lactic acid buildup will cause faster cancer cell death. This is a natural death that does not initiate an inflammatory response by the immune system.
Glutam will also help PrugX and PrugX Enhancer work more effectively. PrugX causes a buildup of hydrogen peroxide in cancer cells. This hydrogen peroxide causes free radical damage to cancer cells. Glutam, by reducing the amount of glutamine in cancer cells, will reduce the ability of cancer cells to produce glutathione. Glutathione would neutralize the free radicals in hydrogen peroxide.
Between Glutam and BLA Enhancer, Glutam is the more effective supplement. While using both with BLA is excellent, many other elixirs are more important to add on than the BLA Enhancer, starting with Glutam and PrugX.
OCMP, Baking Soda, alkaline water, MSM, and other mineral alkalizing supplements like coral calcium or cesium, interfere with the action of BLA. These work to alkalize cancer cells. This would, to some extent, neutralize the acid buildup in the cancer cells that BLA causes, and thus would protect the cancer cells from some of the acidosis BLA causes.
Fortunately this next supplement is a unique highly alkaline OH water concentrate that actually helps BLA to work better.


It's Summer time, the weather is fine, so get down to Swingin Safari too enjoy the line up at this months XMas Love Respect Music Charity Event! This installment sees us open the doors at midday for all the neighborhood to get down and have some Xmas drinks, food and laughs with friends and family with Dj's and musicians playing through out the afternoon from the balcony overseeing the car park and outdoor area.
This event is a fundraiser for the Non For Profit Organisation "The World As I AM" a facilitator of volunteer programs abroad offering assistance to needy communities in the areas of community sustainability, environmental conservation and cultural exchange.
Sellouts (Gold Coast), Phonic Funk, Fuzion, Lucas D, Freddy Fly Fingaz, Soxy, Black Katz, Lincoln & Plush MC, Chef, Eddy Daly, James Canning, Andrea Tenille, DPN, Jade, Darren Somerville, Malcolm Wood, Ben Cooper, Dan Mada, Stevie Keyse , Daniel Carter and Bigbad.
100% of all profits from of the event will go to "The World As I AM" to aid their development in further community programs abroad.
We couldn't have done this event without your support!


Click the button below to add the Latin X-Press Volume 2 to your wish list.
Latin X-Press Volume 2 on DJ Wholesale Club, Inc.
Cover is NM Some storage wear on the cover and back but not enough to drop the grade. Corner might be water damaged if not creased from storage.


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He has the patience of a saint – and then some. Former French minister of Foreign Affairs Michel Barnier has the unenviable job of explaining to the British government that Brexit means Brexit – to paraphrase Prime Minister Theresa May who first introduced the combative slogan and has lived to regret it; the words keep bouncing back.
In July 2016, Mr Barnier was appointed chief negotiator by the European Commission to help steer the United Kingdom towards the exit whilst safeguarding the interests of the remaining 27 EU member states. To this end, Mr Barnier was given a clear but limited set of instructions by the European Council which includes all heads of state and government leaders of the union. He was also charged with maintaining the full transparency of the negotiating process and decline any and all offers to seal backroom deals.
Whilst the commission may perhaps have displayed a slightly greater sensitivity to the feelings of the British – whose love of all things French is particularly understated – Michel Barnier proved, in the end, an excellent choice. The Frenchman has almost singlehandedly redefined the meaning of stoic. Unperturbed by the firestorm kicked up by the tabloids, Mr Barnier refuses to lose his cool, limiting his periodic interventions to curt, yet elegantly worded, dismissals of the latest scheme produced by his counterparts as they seek to fit their square peg into a round hole.
Mr Barnier is not one to bend the rules, engage in flights of fancy, or deviate from the script he was handed. He also remains perfectly calm and carries on – patiently explaining the workings of the European Union to the uninitiated and delineating the playing field set by his mandate and within which the disentanglement of the United Kingdom from the European Union must take shape.
Sometimes, though, keen observers think they may detect a hint of irritation in Mr Barnier's sporadic utterings such as when he had to explain, yet again, that the militarily useful features of the European navigation system Galileo – a constellation of thirty satellites – is the preserve of member states only as per the agreement that underpins the €10bn undertaking. Ironically, the clause excluding non-members states from accessing the encrypted bits of Galileo – which will feature and estimated position error of less than one centimetre – was, at the time, included at the express insistence of the British.
Mr Barnier would like nothing more than for the United Kingdom to understand that leaving the European Union implies severing all ties to the bloc, including unfettered access to its specialist agencies, markets, and regulatory entities. So far, the Frenchman has discharged his task with considerable aplomb welcoming and appreciating the steps taken by Prime Minister May to infuse the talks with a greater sense of realism.
Born into a Gaullist family and growing up in the Alpine Isère Department next to Grenoble, Michel Barnier graduated from the École Supérieure de Commerce du Paris, one of France's Grandes Écoles, which only admits the exceptionally bright. Touted as the world's oldest business school, Mr Barnier, however, choose politics as his trade. He was elected to parliament, age 27, and went on to rise through the ranks. He helped organise the 1992 Winter Olympics in Albertville and, the following year, joined the cabinet of Prime Minister Edouard Balladur before receiving an appointment as Secretary of State for European Affairs.
In Europe, Mr Barnier served as commissioner for Regional Policy between 1999 and 2004, returning to the French cabinet as minister of Foreign Affairs, before again taking up a seat on the European Commission as commissioner for Internal Market and Services (2010-2014). The experience gained in this position now serves Michel Barnier well as he tries to remind his interlocutors of the need to reserve the benefits EU's single market to member states and associated members who abide by the union's four freedom – the free movement, of capital, services, goods, and labour.
At times, it seems that Mr Barnier keeps to the maxim first pronounced by the 26th US president Theodore Roosevelt who inhabited the White House between 1901 and 1909: Speak softly and carry a big stick. Mr Barnier knows, more than most, that he need never raise his voice, express frustration, or become irritated for he represents the world's largest bloc of democratic nations who have – to the surprise of friend and foe – kept a united front despite their differences in the face of the United Kingdom's likewise democratic decision to exit the European Union. Mr Barnier's job is to make this happen.


text:
'Tapping Indian market will create jobs for Australia'
Melbourne, Oct 19: As Australia and India embark on an ambitious plan to double two-way trade to $40 billion by 2015, a business magnate here has said that tapping the Indian market can create jobs
PTI Updated on: October 19, 2012 14:50 IST
tapping indian market will create jobs for australia
Melbourne, Oct 19: As Australia and India embark on an ambitious plan to double two-way trade to $40 billion by 2015, a business magnate here has said that tapping the Indian market can create jobs and prosperity for Australia.
Lindsay Fox, who runs billion dollar trucking and logistic firm and led the Australian trade delegation during Prime Minister Julia Gillard visit to India this week, said the country with a booming economy and population of 1.2 billion had a great potential to create jobs and prosperity for Australia.
"We're in the right spot at the right time and we've all got to be smart enough to capitalise on it," Herald Sun quoted him as saying.
"That's a hell of a lot of tomatoes, bananas, cows, sheep or anything that we can produce in Australia," he said.
Fox further said that there will be 100 million new people every year with money ready to buy things — five times Australia's population.
It has been noted that in the past Australia did not give top priority to India. Apart from that there were only handful of Australian business leaders doing businesses in India.
ANZ head Mike Smith, who was also part of the business delegation, said he believed that there was already a "can do" spirit of goodwill between the two sides.
Julia Gillard, in her 3—day official visit to the subcontinent, announced enhancing economic relations with India. She said the two countries have set a goal of doubling bilateral trade to USD 40 billion by 2015.
"We have already set a goal of USD 40 billion trade in 2015 - in other words, our trade could double by 2015," she said, adding that currently the trade between the two nations was growing at 13 per cent a year.
"In recent years, our economic relationship has grown stronger. Two—way trade has grown at 13 per cent per annum in the past five years. Indian investment approvals in Australia grew last year alone to over 11 billion dollars.
"This is a one hundred—fold increase from 10 years ago," Gillard said. The Australian Prime Minister is expected to unveil a White Paper on Australia in the Asian Century, a blueprint for the nation's strategic and economic engagement with the region.
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