\(\renewcommand{\AA}{\text{Å}}\)

# fix wall/lj93 command

Accelerator Variants: *wall/lj93/kk*

# fix wall/lj126 command

# fix wall/lj1043 command

# fix wall/colloid command

# fix wall/harmonic command

# fix wall/lepton command

# fix wall/morse command

# fix wall/table command

## Syntax

```
fix ID group-ID style [tabstyle] [N] face args ... keyword value ...
```

ID, group-ID are documented in fix command

style =

*wall/lj93*or*wall/lj126*or*wall/lj1043*or*wall/colloid*or*wall/harmonic*or*wall/lepton*or*wall/morse*or*wall/table*tabstyle =

*linear*or*spline*= method of table interpolation (only applies to*wall/table*)N = use N values in

*linear*or*spline*interpolation (only applies to*wall/table*)one or more face/arg pairs may be appended

face =

*xlo*or*xhi*or*ylo*or*yhi*or*zlo*or*zhi*

args for styles

*lj93*or*lj126*or*lj1043*or*colloid*or*harmonic*args = coord epsilon sigma cutoff coord = position of wall = EDGE or constant or variable EDGE = current lo or hi edge of simulation box constant = number like 0.0 or -30.0 (distance units) variable = equal-style variable like v_x or v_wiggle epsilon = strength factor for wall-particle interaction (energy or energy/distance^2 units) epsilon can be a variable (see below) sigma = size factor for wall-particle interaction (distance units) sigma can be a variable (see below) cutoff = distance from wall at which wall-particle interactions are cut off (distance units)

args for style

*lepton*args = coord expression cutoff coord = position of wall = EDGE or constant or variable EDGE = current lo or hi edge of simulation box constant = number like 0.0 or -30.0 (distance units) variable = equal-style variable like v_x or v_wiggle expression = Lepton expression for the potential (energy units) cutoff = distance from wall at which wall-particle interactions are cut off (distance units)

args for style

*morse*args = coord D_0 alpha r_0 cutoff coord = position of wall = EDGE or constant or variable EDGE = current lo or hi edge of simulation box constant = number like 0.0 or -30.0 (distance units) variable = equal-style variable like v_x or v_wiggle D_0 = depth of the potential (energy units) D_0 can be a variable (see below) alpha = width factor for wall-particle interaction (1/distance units) alpha can be a variable (see below) r_0 = distance of the potential minimum from the face of region (distance units) r_0 can be a variable (see below) cutoff = distance from wall at which wall-particle interactions are cut off (distance units)

args for style

*table*args = coord filename keyword cutoff coord = position of wall = EDGE or constant or variable EDGE = current lo or hi edge of simulation box constant = number like 0.0 or -30.0 (distance units) variable = equal-style variable like v_x or v_wiggle filename = file containing tabulated energy and force values keyword = section identifier to select a specific table in table file cutoff = distance from wall at which wall-particle interactions are cut off (distance units)

zero or more keyword/value pairs may be appended

keyword =

*units*or*fld*or*pbc**units*value =*lattice*or*box**lattice*= the wall position is defined in lattice units*box*= the wall position is defined in simulation box units*fld*value =*yes*or*no**yes*= invoke the wall constraint to be compatible with implicit FLD*no*= invoke the wall constraint in the normal way*pbc*value =*yes*or*no**yes*= allow periodic boundary in a wall dimension*no*= require non-perioidic boundaries in any wall dimension

## Examples

```
fix wallhi all wall/lj93 xlo -1.0 1.0 1.0 2.5 units box
fix wallhi all wall/lj93 xhi EDGE 1.0 1.0 2.5
fix wallhi all wall/harmonic xhi EDGE 100.0 0.0 4.0 units box
fix wallhi all wall/morse xhi EDGE 1.0 1.0 1.0 2.5 units box
fix wallhi all wall/lj126 v_wiggle 23.2 1.0 1.0 2.5
fix zwalls all wall/colloid zlo 0.0 1.0 1.0 0.858 zhi 40.0 1.0 1.0 0.858
fix xwall mobile wall/table spline 200 EDGE -5.0 walltab.dat HARMONIC 4.0
fix xwalls mobile wall/lepton xlo -5.0 "k*(r-rc)^2;k=100.0" 4.0 xhi 5.0 "k*(r-rc)^2;k=100.0" 4.0
```

## Description

Bound the simulation domain on one or more of its faces with a flat wall that interacts with the atoms in the group by generating a force on the atom in a direction perpendicular to the wall. The energy of wall-particle interactions depends on the style.

For style *wall/lj93*, the energy E is given by the 9-3 Lennard-Jones potential:

For style *wall/lj126*, the energy E is given by the 12-6 Lennard-Jones potential:

For style *wall/lj1043*, the energy E is given by the 10-4-3 Lennard-Jones potential:

For style *wall/colloid*, the energy E is given by an integrated form
of the pair_style colloid potential:

For style *wall/harmonic*, the energy E is given by a repulsive-only harmonic
spring potential:

For style *wall/morse*, the energy E is given by a Morse potential:

Added in version 28Mar2023.

For style *wall/lepton*, the energy E is provided as an Lepton
expression string using “r” as the distance variable. The Lepton
library, that the *wall/lepton*
style interfaces with, evaluates this expression string at run time to
compute the wall-particle energy. It also creates an analytical
representation of the first derivative of this expression with respect
to “r” and then uses that to compute the force between the wall and
atoms in the fix group. The Lepton expression must be either enclosed
in quotes or must not contain any whitespace so that LAMMPS recognizes
it as a single keyword.

Optionally, the expression may use “rc” to refer to the cutoff distance
for the given wall. Further constants in the expression can be defined
in the same string as additional expressions separated by semicolons.
The expression “k*(r-rc)^2;k=100.0” represents a repulsive-only harmonic
spring as in fix *wall/harmonic* with a force constant *K* (same as
\(\epsilon\) above) of 100 energy units. More details on the Lepton
expression strings are given below.

Added in version 28Mar2023.

For style *wall/table*, the energy E and forces are determined from
interpolation tables listed in one or more files as a function of
distance. The interpolation tables are used to evaluate energy and
forces between particles and the wall similar to how analytic formulas
are used for the other wall styles.

The interpolation tables are created as a pre-computation by fitting
cubic splines to the file values and interpolating energy and force
values at each of *N* distances. During a simulation, the tables are
used to interpolate energy and force values as needed for each wall and
particle separated by a distance *R*. The interpolation is done in
one of two styles: *linear* or *spline*.

For the *linear* style, the distance *R* is used to find the 2
surrounding table values from which an energy or force is computed by
linear interpolation.

For the *spline* style, cubic spline coefficients are computed and
stored for each of the *N* values in the table, one set of splines for
energy, another for force. Note that these splines are different than
the ones used to pre-compute the *N* values. Those splines were fit
to the *Nfile* values in the tabulated file, where often *Nfile* <
*N*. The distance *R* is used to find the appropriate set of spline
coefficients which are used to evaluate a cubic polynomial which
computes the energy or force.

For each wall a filename and a keyword must be provided as in the examples above. The filename specifies a file containing tabulated energy and force values. The keyword specifies a section of the file. The format of this file is described below.

In all cases, *r* is the distance from the particle to the wall at
position *coord*, and \(r_c\) is the *cutoff* distance at which the
particle and wall no longer interact. The energy of the wall
potential is shifted so that the wall-particle interaction energy is
0.0 at the cutoff distance.

Up to 6 walls or faces can be specified in a single command: *xlo*,
*xhi*, *ylo*, *yhi*, *zlo*, *zhi*. A *lo* face interacts with
particles near the lower side of the simulation box in that dimension.
A *hi* face interacts with particles near the upper side of the
simulation box in that dimension.

The position of each wall can be specified in one of 3 ways: as the EDGE of the simulation box, as a constant value, or as a variable. If EDGE is used, then the corresponding boundary of the current simulation box is used. If a numeric constant is specified then the wall is placed at that position in the appropriate dimension (x, y, or z). In both the EDGE and constant cases, the wall will never move. If the wall position is a variable, it should be specified as v_name, where name is an equal-style variable name. In this case the variable is evaluated each timestep and the result becomes the current position of the reflecting wall. Equal-style variables can specify formulas with various mathematical functions, and include thermo_style command keywords for the simulation box parameters and timestep and elapsed time. Thus it is easy to specify a time-dependent wall position. See examples below.

For the *wall/lj93* and *wall/lj126* and *wall/lj1043* styles,
\(\epsilon\) and \(\sigma\) are the usual Lennard-Jones
parameters, which determine the strength and size of the particle as it
interacts with the wall. Epsilon has energy units. Note that this
\(\epsilon\) and \(\sigma\) may be different than any
\(\epsilon\) or \(\sigma\) values defined for a pair style that
computes particle-particle interactions.

The *wall/lj93* interaction is derived by integrating over a 3d
half-lattice of Lennard-Jones 12/6 particles. The *wall/lj126*
interaction is effectively a harder, more repulsive wall interaction.
The *wall/lj1043* interaction is yet a different form of wall
interaction, described in Magda et al in (Magda).

For the *wall/colloid* style, *R* is the radius of the colloid particle,
*D* is the distance from the surface of the colloid particle to the wall
(r-R), and \(\sigma\) is the size of a constituent LJ particle
inside the colloid particle and wall. Note that the cutoff distance Rc
in this case is the distance from the colloid particle center to the
wall. The prefactor \(\epsilon\) can be thought of as an effective
Hamaker constant with energy units for the strength of the colloid-wall
interaction. More specifically, the \(\epsilon\) prefactor is
\(4\pi^2 \rho_{wall} \rho_{colloid} \epsilon \sigma^6\), where
\(\epsilon\) and \(\sigma\) are the LJ parameters for the
constituent LJ particles. \(\rho_{wall}\) and \(\rho_{colloid}\)
are the number density of the constituent particles, in the wall and
colloid respectively, in units of 1/volume.

The *wall/colloid* interaction is derived by integrating over
constituent LJ particles of size \(\sigma\) within the colloid
particle and a 3d half-lattice of Lennard-Jones 12/6 particles of size
\(\sigma\) in the wall. As mentioned in the preceding paragraph,
the density of particles in the wall and colloid can be different, as
specified by the \(\epsilon\) prefactor.

For the *wall/harmonic* style, \(\epsilon\) is effectively the spring
constant K, and has units (energy/distance^2). The input parameter
\(\sigma\) is ignored. The minimum energy position of the harmonic
spring is at the *cutoff*. This is a repulsive-only spring since the
interaction is truncated at the *cutoff*

For the *wall/morse* style, the three parameters are in this order:
\(D_0\) the depth of the potential, \(\alpha\) the width
parameter, and \(r_0\) the location of the minimum. \(D_0\) has
energy units, \(\alpha\) inverse distance units, and \(r_0\)
distance units.

For any wall that supports them, the \(\epsilon\) and/or \(\sigma\) and/or \(\alpha\) parameter can be specified as an equal-style variable, in which case it should be specified as v_name, where name is the variable name. As with a variable wall position, the variable is evaluated each timestep and the result becomes the current epsilon or sigma of the wall. Equal-style variables can specify formulas with various mathematical functions, and include thermo_style command keywords for the simulation box parameters and timestep and elapsed time. Thus it is easy to specify a time-dependent wall interaction.

Note

For all of the styles, you must ensure that r is always > 0 for
all particles in the group, or LAMMPS will generate an error. This
means you cannot start your simulation with particles at the wall
position *coord* (r = 0) or with particles on the wrong side of the
wall (r < 0). For the *wall/lj93* and *wall/lj126* styles, the energy
of the wall/particle interaction (and hence the force on the particle)
blows up as r -> 0. The *wall/colloid* style is even more
restrictive, since the energy blows up as D = r-R -> 0. This means
the finite-size particles of radius R must be a distance larger than R
from the wall position *coord*. The *harmonic* style is a softer
potential and does not blow up as r -> 0, but you must use a large
enough \(\epsilon\) that particles always reamin on the correct side of
the wall (r > 0).

The *units* keyword determines the meaning of the distance units used
to define a wall position, but only when a numeric constant or
variable is used. It is not relevant when EDGE is used to specify a
face position. In the variable case, the variable is assumed to
produce a value compatible with the *units* setting you specify.

A *box* value selects standard distance units as defined by the
units command, e.g. Angstroms for units = real or metal.
A *lattice* value means the distance units are in lattice spacings.
The lattice command must have been previously used to
define the lattice spacings.

The *fld* keyword can be used with a *yes* setting to invoke the wall
constraint before pairwise interactions are computed. This allows an
implicit FLD model using pair_style lubricateU
to include the wall force in its calculations. If the setting is *no*,
wall forces are imposed after pairwise interactions, in the usual
manner.

The *pbc* keyword can be used with a *yes* setting to allow walls to be
specified in a periodic dimension. See the boundary
command for options on simulation box boundaries. The default for *pbc*
is *no*, which means the system must be non-periodic when using a wall.
But you may wish to use a periodic box. E.g. to allow some particles to
interact with the wall via the fix group-ID, and others to pass through
it and wrap around a periodic box. In this case you should ensure that
the wall is sufficiently far enough away from the box boundary. If you
do not, then particles may interact with both the wall and with periodic
images on the other side of the box, which is probably not what you
want.

Here are examples of variable definitions that move the wall position in a time-dependent fashion using equal-style variables. The wall interaction parameters (epsilon, sigma) could be varied with additional variable definitions.

```
variable ramp equal ramp(0,10)
fix 1 all wall xlo v_ramp 1.0 1.0 2.5
variable linear equal vdisplace(0,20)
fix 1 all wall xlo v_linear 1.0 1.0 2.5
variable wiggle equal swiggle(0.0,5.0,3.0)
fix 1 all wall xlo v_wiggle 1.0 1.0 2.5
variable wiggle equal cwiggle(0.0,5.0,3.0)
fix 1 all wall xlo v_wiggle 1.0 1.0 2.5
```

The *ramp(lo,hi)* function adjusts the wall position linearly from *lo* to
*hi* over the course of a run. The *vdisplace(c0,velocity)* function does
something similar using the equation *position = c0 + velocity*delta*,
where *delta* is the elapsed time.

The *swiggle(c0,A,period)* function causes the wall position to
oscillate sinusoidally according to this equation, where *omega = 2 PI
/ period*:

position = c0 + A sin(omega*delta)

The *cwiggle(c0,A,period)* function causes the wall position to
oscillate sinusoidally according to this equation, which will have an
initial wall velocity of 0.0, and thus may impose a gentler
perturbation on the particles:

position = c0 + A (1 - cos(omega*delta))

## Lepton expression syntax and features

Lepton supports the following operators in expressions:

+ |
Add |
- |
Subtract |
* |
Multiply |
/ |
Divide |
^ |
Power |

The following mathematical functions are available:

sqrt(x) |
Square root |
exp(x) |
Exponential |

log(x) |
Natural logarithm |
sin(x) |
Sine (angle in radians) |

cos(x) |
Cosine (angle in radians) |
sec(x) |
Secant (angle in radians) |

csc(x) |
Cosecant (angle in radians) |
tan(x) |
Tangent (angle in radians) |

cot(x) |
Cotangent (angle in radians) |
asin(x) |
Inverse sine (in radians) |

acos(x) |
Inverse cosine (in radians) |
atan(x) |
Inverse tangent (in radians) |

sinh(x) |
Hyperbolic sine |
cosh(x) |
Hyperbolic cosine |

tanh(x) |
Hyperbolic tangent |
erf(x) |
Error function |

erfc(x) |
Complementary Error function |
abs(x) |
Absolute value |

min(x,y) |
Minimum of two values |
max(x,y) |
Maximum of two values |

delta(x) |
delta(x) is 1 for x = 0, otherwise 0 |
step(x) |
step(x) is 0 for x < 0, otherwise 1 |

Numbers may be given in either decimal or exponential form. All of the following are valid numbers: 5, -3.1, 1e6, and 3.12e-2.

As an extension to the standard Lepton syntax, it is also possible to use LAMMPS variables in the format “v_name”. Before evaluating the expression, “v_name” will be replaced with the value of the variable “name”. This is compatible with all kinds of scalar variables, but not with vectors, arrays, local, or per-atom variables. If necessary, a custom scalar variable needs to be defined that can access the desired (single) item from a non-scalar variable. As an example, the following lines will instruct LAMMPS to ramp the force constant for a harmonic bond from 100.0 to 200.0 during the next run:

```
variable fconst equal ramp(100.0, 200)
bond_style lepton
bond_coeff 1 1.5 "v_fconst * (r^2)"
```

An expression may be followed by definitions for intermediate values that appear in the expression. A semicolon “;” is used as a delimiter between value definitions. For example, the expression:

```
a^2+a*b+b^2; a=a1+a2; b=b1+b2
```

is exactly equivalent to

```
(a1+a2)^2+(a1+a2)*(b1+b2)+(b1+b2)^2
```

The definition of an intermediate value may itself involve other
intermediate values. Whitespace and quotation characters (’'’ and ‘”’)
are ignored. All uses of a value must appear *before* that value’s
definition. For efficiency reasons, the expression string is parsed,
optimized, and then stored in an internal, pre-parsed representation for
evaluation.

Evaluating a Lepton expression is typically between 2.5 and 5 times slower than the corresponding compiled and optimized C++ code. If additional speed or GPU acceleration (via GPU or KOKKOS) is required, the interaction can be represented as a table. Suitable table files can be created either internally using the pair_write or bond_write command or through the Python scripts in the tools/tabulate folder.

## Table file format

Suitable tables for use with fix *wall/table* can be created by the
Python code in the `tools/tabulate`

folder of the LAMMPS source code
distribution.

The format of a tabulated file is as follows (without the parenthesized comments):

```
# Tabulated wall potential UNITS: real
HARMONIC (keyword is the first text on a line)
N 100 FP 200 200
(blank line)
1 0.04 1568.16 792.00 (index, distance to wall, energy, force)
2 0.08 1536.64 784.00
3 0.12 1505.44 776.00
...
99 3.96 0.16 8.00
100 4.00 0 0
```

A section begins with a non-blank line whose first character is not a
“#”; blank lines or lines starting with “#” can be used as comments
between sections. The first line begins with a keyword which identifies
the section. The line can contain additional text, but the initial text
must match the argument specified in the fix *wall/table* command. The
next line lists (in any order) one or more parameters for the table.
Each parameter is a keyword followed by one or more numeric values.

The parameter “N” is required and its value is the number of table
entries that follow. Note that this may be different than the *N*
specified in the fix *wall/table* command. Let Ntable = *N* in the fix
command, and Nfile = “N” in the tabulated file. What LAMMPS does is a
preliminary interpolation by creating splines using the Nfile tabulated
values as nodal points. It uses these to interpolate as needed to
generate energy and force values at Ntable different points. The
resulting tables of length Ntable are then used as described above, when
computing energy and force for wall-particle interactions. This means that
if you want the interpolation tables of length Ntable to match exactly
what is in the tabulated file (with effectively no preliminary
interpolation), you should set Ntable = Nfile.

## Restart, fix_modify, output, run start/stop, minimize info

No information about this fix is written to binary restart files.

The fix_modify *energy* option is supported by
this fix to add the energy of interaction between atoms and all the
specified walls to the global potential energy of the system as part
of thermodynamic output. The default setting
for this fix is fix_modify energy no.

The fix_modify *virial* option is supported by
this fix to add the contribution due to the interaction between atoms
and all the specified walls to both the global pressure and per-atom
stress of the system via the compute pressure and compute stress/atom commands. The former can be accessed by
thermodynamic output. The default setting for
this fix is fix_modify virial no.

The fix_modify *respa* option is supported by this
fix. This allows to set at which level of the r-RESPA integrator the fix is adding its forces. Default is the
outermost level.

This fix computes a global scalar energy and a global vector of forces,
which can be accessed by various output commands.
Note that the scalar energy is the sum of interactions with all defined
walls. If you want the energy on a per-wall basis, you need to use
multiple fix wall commands. The length of the vector is equal to the
number of walls defined by the fix. Each vector value is the normal
force on a specific wall. Note that an outward force on a wall will be
a negative value for *lo* walls and a positive value for *hi* walls.
The scalar and vector values calculated by this fix are “extensive”.

No parameter of this fix can be used with the *start/stop* keywords of
the run command.

The forces due to this fix are imposed during an energy minimization, invoked by the minimize command.

Note

If you want the atom/wall interaction energy to be included in
the total potential energy of the system (the quantity being
minimized), you MUST enable the fix_modify *energy*
option for this fix.

Styles with a *gpu*, *intel*, *kk*, *omp*, or *opt* suffix are
functionally the same as the corresponding style without the suffix.
They have been optimized to run faster, depending on your available
hardware, as discussed on the Accelerator packages
page. The accelerated styles take the same arguments and should
produce the same results, except for round-off and precision issues.

These accelerated styles are part of the GPU, INTEL, KOKKOS, OPENMP, and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package page for more info.

You can specify the accelerated styles explicitly in your input script by including their suffix, or you can use the -suffix command-line switch when you invoke LAMMPS, or you can use the suffix command in your input script.

See the Accelerator packages page for more instructions on how to use the accelerated styles effectively.

## Restrictions

Fix *wall/lepton* is part of the LEPTON package and only enabled if
LAMMPS was built with this package. See the Build package page for more info.

## Default

The option defaults units = lattice, fld = no, and pbc = no.

**(Magda)** Magda, Tirrell, Davis, J Chem Phys, 83, 1888-1901 (1985);
erratum in JCP 84, 2901 (1986).