# A constant of motion for ideal grain growth in three dimensions

E. Eren  Department of Materials Science and Engineering, University of
California, Davis, CA, 95616, USA.  J. K. Mason<EMAIL_ADDRESS>Department of Materials Science and Engineering, University of California,
Davis, CA, 95616, USA.

###### Abstract

Most metallic and ceramic materials are comprised of a space-filling
collection of crystalline grains separated by grain boundaries. While this
grain structure has been studied for more than a century, there few rigorous
results regarding its global properties available in the literature. We
present a new, rigorous result for three-dimensional grain structures that
relates the integral of the Gaussian curvature over the grain boundaries to
the numbers of grains and quadruple junctions. The result is numerically
verified for a grain structure consisting of periodic truncated octahedra.

The grain structure of polycrystalline materials is deceptively simple, and
for that reason has been the subject of intense and ongoing study. For
specificity, consider a model system where the grain boundary energy and
mobility are constants, i.e., do not depend on grain misorientation or the
boundary plane normal. The phenomenological Turnbull equation [1] relates the
normal velocity of a grain boundary in such a system to the driving pressure,
and along with the Young–Laplace equation [2] suggests that the normal
velocity is directly proportional to the mean curvature of the grain boundary.
The migration of individual boundaries induces the evolution of the grain
structure, a process known as grain growth, where the total area of grain
boundaries and the number of grains decrease with time.

There are surprisingly few rigorous results known about grain structures, even
for the two-dimensional version of this system. Energy considerations require
that grain boundaries only meet at triple junctions with internal angles of
$2\pi/3$ [3, 4]. A consequence of this and curvature-driven grain growth is
that a grain’s area changes at a rate that depends only on the number of
bounding vertices [5, 6]. Globally, topological arguments require that the
average number of such bounding vertices be precisely six [7]. There are
natural analogues to several, but not all, of these results in three
dimensions. Grain boundaries only meet at triple junction lines with dihedral
angles of $2\pi/3$, and triple junction lines only meet at quadruple junction
points with angles of $\cos^{-1}(-1/3)$ [3, 4]. The rate of volume change of a
grain depends not only on the total length of the bounding triple lines, but
on a measure of the linear dimension known as the mean width [8, 9]. For both
the two- and three-dimensional systems, the hypothesis that the structure
reaches a statistically self-similar state implies that that the average grain
diameter increases as the square-root of time [10, 11]. This is the effective
extent of current knowledge.

There have been a variety of inexact relationships proposed as well, usually
for grain structures in the conjectured self-similar state [12]. Ones that
relate to the global properties of the three-dimensional system include
proposed distributions for the effective radius of a grain [13, 14, 15] and
the number of faces bounding a grain [15]. Recent advances in several
microscopy techniques promise to make three-dimensional grain structure data
more readily available, possibly allowing such relationships to be further
refined. Three-dimensional electron backscatter diffraction [16, 17]
destructively images the grain structure by a serial sectioning process,
whereas three-dimensional X-ray diffraction microscopy [18, 19] is non-
destructive but generally offers poorer spatial resolution. Given this
situation, additional rigorous results for the global properties of the grain
structure of the model three-dimensional system would be valuable, both to
measure deviations of the experimental systems from the model one, and to
verify the accuracy of grain structures generated by computational means. This
article proves one such result, relating the integral of the Gaussian
curvature over the grain boundaries to the numbers of grains and quadruple
junction points, and thereby to the numbers of grain boundaries and triple
junction lines.

Figure 1: A grain structure in a cubic volume, with several grains removed to
reveal the interior. Color indicates the individual grains, internal curved
surfaces are grain boundaries, internal black lines are triple junction lines,
and four triple junction lines intersect at quadruple junction points.

Let $\Omega$ be a space-filling grain structure composed of grains that meet
in twos on grain boundaries, grain boundaries that meet in threes at triple
junction lines, and triple junction lines that meet in fours at quadruple
junction points, as in Fig. 1. Further suppose that $\Omega$ satisfies
Plateau’s laws (i.e., grain boundaries meet at dihedral angles of $2\pi/3$ and
triple junction lines meet at angles of $\cos^{-1}(-1/3)$), and that $\Omega$
is defined in a three-dimensional region with periodic boundary conditions. If
$G$ is a grain in $\Omega$, then our main result is that the expectation value
of the Gaussian curvature $K$ integrated over the interiors of the grain
boundaries of $G$ and the expectation value of the number of quadruple
junction points $f_{0}(G)$ of $G$ are related by:

$\bigg{\langle}\int_{\partial
G}{K\>\mathrm{d}A}\bigg{\rangle}=4\pi-\alpha\langle f_{0}(G)\rangle.$ (1)

The angle brackets indicate an average performed over all grains in $\Omega$,
$\partial G$ indicates the interiors of the grain boundaries of $G$, and
$\alpha=2\pi-3\cos^{-1}(-1/3)$ is the angular defect at a quadruple junction
point. This result is exact (given a few technical assumptions that are
usually satisfied and are discussed in the supplemental material [20]), and to
our knowledge does not appear in the literature; a related result by Kusner
[21] requires that all the grain boundaries be minimal surfaces, and one by
Glicksman [22] applies only to unconstructable grain structures of average
$n$-polyhedra.

The Gaussian curvature of a surface is defined as the product of the principal
curvatures at any point. The appearance of this quantity in Eq. 1 could be
surprising, since the mean curvature (the sum of the principal curvatures) is
the one that controls the dynamics of the grain boundary network [2, 1]. That
said, the Gaussian curvature is in some ways the more fundamental of the two
quantities, being an intrinsic property of the surface that does not depend on
the way the surface is embedded in Euclidean space. For example, the Gaussian
curvature of a sheet of paper is zero at every point whether the sheet is laid
flat or rolled up, though the same is not true for the mean curvature. This
invariance to the embedding is reflected in the celebrated Gauss–Bonnet
theorem:

$\int_{\partial G}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{2}(G)}\int_{\partial
F_{i}}{\kappa_{g}\>\mathrm{d}s}+\sum_{i=1}^{f_{0}(G)}{\alpha_{i}}=2\pi\chi(\partial
G).$

While this version specifically applies to the surface of a grain, all
versions relate the integrated Gaussian curvature of a surface to its Euler
characteristic $\chi(\partial G)$ (equal to two when the surface can be
smoothly deformed into a sphere without cutting or gluing). The terms on the
left include the integrated Gaussian curvature over the grain boundary
interiors, the sum of the integrated geodesic curvature $\kappa_{g}$ over the
interiors of the bounding triple junction lines $\partial F_{i}$ of all grain
boundaries $F_{i}$, and the sum of the angular defects $\alpha_{i}$ of the
quadruple junction points of $G$.

Figure 2: Grain boundaries $F_{i}$ and $F_{j}$ meet at the triple junction
line in bold, and $\bm{v}$ bisects the dihedral angle between $F_{i}$ and
$F_{j}$. Arrows indicate the tangent direction, and the second Frenet vector
$\bm{e}_{2}$ points along the triple junction line’s normal direction.

If $G$ can be smoothly deformed into a sphere and belongs to a grain structure
$\Omega$ that obeys Plateau’s rules, then this can be simplified to:

$\int_{\partial
G}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{1}(G)}\int_{E_{i}}{\kappa\bm{e}_{2}\cdot\bm{v}\>\mathrm{d}s}+\alpha
f_{0}(G)=4\pi$

where the most significant change is to the middle term on the left; this is
now the sum of the integrated curvature of the triple junction lines of $G$,
weighted by the dot product of the second Frenet vector $\bm{e}_{2}$ of the
curve and a unit vector $\bm{v}$ that bisects the dihedral angle between the
adjoining grain boundaries; see Fig. 2. Summing this equation over all grains
in $\Omega$ results in a remarkable cancellation (previously noted by DeHoff
[23]) where the contribution of the second term on the left vanishes.
Specifically, every triple junction line is integrated over three times, once
for each adjoining grain. $\kappa\bm{e}_{2}$ is an inherent quantity of the
triple junction line that is the same for all three integrals, but the three
$\bm{v}$ are all unit vectors in a plane with mutual angles of $2\pi/3$. That
is, the sum of the three $\bm{v}$ vanishes identically for each triple
junction line, leaving an alternative version of the main result:

$\sum_{i=1}^{f_{2}(\Omega)}\int_{F_{i}}{K\>\mathrm{d}A}=2\pi
f_{3}(\Omega)-2\alpha f_{0}(\Omega)$ (2)

where $F_{i}$ is the $i$th grain boundary of $\Omega$ and $f_{0}(\Omega)$,
$f_{2}(\Omega)$ and $f_{3}(\Omega)$ are the numbers of quadruple junction
points, grain boundaries, and grains of $\Omega$. Dividing through by
$f_{3}(\Omega)$ and multiplying by a constant gives Eq. 1. More detailed
derivations of both Eqs. 1 and 2 are provided in the supplemental material
[20].

Although Eq. 1 appears to be simpler, there are at least two observations that
are more clearly made by means of Eq. 2. The first is that the integral of $K$
over the grain boundaries of $\Omega$ depends only on the numbers of grains
and quadruple junction points of $\Omega$, and not on the geometry of the
grain structure. That is, the left-hand side of Eq. 2 is invariant to any
deformation of $\Omega$ that preserves the numbers of grains and quadruple
junction points. The second is that a sufficiently accurate measurement of the
integral of $K$ over the grain boundaries of $\Omega$ in principle specifies
the numbers of all components of $\Omega$. Observe that since there is no
rational number that relates the coefficients of $f_{3}(\Omega)$ and
$f_{0}(\Omega)$ in Eq. 2, the numbers of grains and quadruple junctions can be
inferred if the left-hand side is known sufficiently accurately. The number of
triple junction lines can then be found from $2f_{1}(\Omega)=4f_{0}(\Omega)$
by a counting argument, and the number of grains from
$0=f_{0}(\Omega)-f_{1}(\Omega)+f_{2}(\Omega)-f_{3}(\Omega)$ which follows from
the domain of $\Omega$ being a three-torus with $\chi(\Omega)=0$. The
necessary modifications to Eqs. 1 and 2 for grain structures in other domains
(e.g., ones with free boundaries) are discussed in the supplemental material
[20].

Figure 3: An infinite periodic grain structure that satisfies Plateau’s laws
can be constructed by repeating a relaxed truncated octahedron (left). This
grain was found by starting with a periodic unit of a grain structure
consisting of unrelaxed truncated octahedra, fixing the location of the
interior quadruple junction points, and minimizing the grain boundary area
(right).

As numerical confirmation of Eq. 1, consider a grain structure consisting of
periodic truncated octahedra, relaxed under the action of surface tension to
satisfy Plateau’s laws; one such grain is shown on the left of Fig. 3. The
shape of this grain was found by starting with a grain structure consisting of
unrelaxed truncated octahedra and identifying the periodic unit shown on the
right of Fig. 3, with a single grain at the center and corners at the centers
of the neighboring grains. The periodic unit was computationally represented
by a volumetric finite element mesh with linear elements, with the average
number of triangles per hexagonal face $n_{t}$ depending on a characteristic
length and the details of the mesh adaptation algorithm. The locations of the
interior quadruple junction points were fixed, and the structure was relaxed
by allowing the vertices on grain surfaces to move according to equations of
motion known to reproduce curvature-driven grain growth [24] until the
magnitude of the vertex forces fell below a threshold. The grain structure did
not reach a steady-state configuration when the locations of the interior
quadruple junction points were not fixed, owing to a known instability of this
grain structure to volumetric perturbations [25]. While quadratic elements
would allow the steady-state geometry to be more accurately represented, a
convergence analysis with an increasing number of linear elements is
sufficient for the present purpose.

The simulations were performed with a modified version of a recently-developed
microstructure evolution code [27] that usually uses SCOREC [28] for mesh
management and maintenance, but the mesh adaptation operations were found to
interfere with the convergence of the grain geometry. Instead, artificial
vertex forces defined by Kuprat [29] were used to maintain the mesh element
quality during the structure relaxation. Since the artificial forces only
acted on vertices on the grain interiors, it is expected that they did not
substantially affect the grain geometry. The boundary conditions were defined
to make the simulation cell behave as a periodic unit in a grain structure
consisting of periodic truncated octahedra. Whereas the grain boundaries on
the simulation cell interior had a constant nonzero energy per unit area, the
external surfaces of the simulation cell were assigned zero energy per unit
area; this is consistent with viewing them as the result of intersecting
grains in the underlying grain structure with the boundary of the periodic
unit. Vertices on the external surfaces were constrained to remain on the
external surfaces during relaxation by projecting away any displacement in the
normal direction, effectively imposing a Neumann boundary condition. The
integrated Gaussian curvature was calculated as the sum of the angular defects
at the vertices on the grain boundary interiors, where the angular defect is
defined as $2\pi$ minus the sum of the interior angles of the grain boundary
triangles meeting at the vertex.

Table 1: The percent grain boundary area change of the relaxed truncated octahedron relative to the unrelaxed one and the integrated Gaussian curvature of the relaxed truncated octahedron as functions of $n_{t}$. $n_{t}$ | 61 | 129 | 477 | 665 | 1885 | 2610 | 3250 | 4314 | 5322 | 6532 | 7649 | 8906
---|---|---|---|---|---|---|---|---|---|---|---|---
$\Delta A(\%)$ | 0.0863 | 0.129 | 0.135 | 0.136 | 0.136 | 0.137 | 0.136 | 0.136 | 0.136 | 0.136 | 0.136 | 0.136
$\int K\>\mathrm{d}A$ | -0.219 | -0.349 | -0.461 | -0.477 | -0.544 | -0.554 | -0.563 | -0.568 | -0.578 | -0.588 | -0.593 | -0.597
Figure 4: The lines defined by Eq. 2 for the data in Table 1 in the feasible
region $3\leq f_{3}(\Omega)\leq 5$ and $20\leq f_{0}(\Omega)\leq 28$. The
lines are colored from red to blue with decreasing error and the black line
passing through $(24,4)$ corresponds to the exact solution. The closest
integer lattice point to the line is $(24,4)$ for $n_{t}\geq 477$.

Table 1 shows the results of this analysis for increasing refinement of the
mesh, i.e., as a function of $n_{t}$. The geometric accuracy of the
representation can be evaluated by means of the percent reduction in grain
boundary area $\Delta A$ of the relaxed truncated octahedron relative to the
unrelaxed one. A detailed analysis [26] suggests a value of $0.159\%$ for the
continuous system; that $\Delta A$ does not converge to this value is likely
due to the irregularity of the mesh. As for the integrated Gaussian curvature,
the average quantities in Eq. 1 are equivalent to those for a single grain by
periodicity. This implies that the integral of the Gaussian curvature over the
interiors of the grain boundaries should be:

$\int_{\partial G}{K\>\mathrm{d}A}=4\pi-24\alpha\approx-0.664484.$

A conjugate gradient minimization algorithm and bootstrapping were used to fit
the model $\int K\mathrm{d}A=a+bn_{t}^{c}$ to the data in Table 1, giving
$a=-0.661\pm 0.022$, $b=2.08\pm 0.51$, and $c=-0.378\pm 0.060$ (reported as
the medians and half the interquartile range). This implies that the
integrated Gaussian curvature would be $-0.661\pm 0.022$ in the
$n_{t}\rightarrow\infty$ limit, and is interpreted as numerically confirming
Eq. 1 given the degree of approximation of the grain geometry. That the
integrated Gaussian curvature converges to the expected value even though the
percent area reduction does not confirms the assertion that Eqs. 1 and 2 are
invariant to geometric perturbations of the structure, provided the numbers of
grains and quadruple junction points remain the same and Plateau’s laws are
satisfied.

Alternatively, one could consider the feasibility of inferring $f_{3}(\Omega)$
and $f_{0}(\Omega)$ by means of a sufficiently accurate measurement of the
integral of $K$ over the grain boundaries of $\Omega$ in Eq. 2. This can be
done by a graphical construction in the plane with $f_{3}(\Omega)$ and
$f_{0}(\Omega)$ on the vertical and horizontal axes. Given the integral of the
Gaussian curvature over the grain boundaries of $\Omega$, Eq. 2 defines a line
in this plane with an irrational slope. Since the actual values of
$f_{3}(\Omega)$ and $f_{0}(\Omega)$ are necessarily positive integers, this
line passes through exactly one point on the integer lattice in the positive
quadrant. In practice, any error in the measurement of the integrated Gaussian
curvature would change the intercept with the vertical axis and shift the line
off of the lattice point; whether this is an issue or not depends on the
magnitude of the error and any a priori bounds that can be placed on
$f_{3}(\Omega)$ and $f_{0}(\Omega)$. For example, Fig. 4 shows this
construction for the data in Table 1 with the constraints $3\leq
f_{3}(\Omega)\leq 5$ and $20\leq f_{0}(\Omega)\leq 28$. Since the magnitude of
the error is assumed to be unknown, it is reasonable to suppose that the
correct values of $f_{3}(\Omega)$ and $f_{0}(\Omega)$ correspond to the
integer lattice point closest to the line within the feasible region. This
procedure correctly identifies the relevant integer lattice point as $(24,4)$
for $n_{t}\geq 477$; in general, the effect of integrated Gaussian curvature
error is reduced as the area of the feasible region decreases.

Apart from advancing our fundamental understanding of grain structures, there
remains the question of the practical utility of Eqs. 1 and 2 (and the
analogues in the supplemental material [20]). This question is made more
pressing by there being few experimental systems that actually evolve by the
relevant ideal grain growth process. The authors propose two possible
applications based on the differences of the left and right sides of Eqs. 1
and 2:

$\displaystyle e_{1}=\bigg{\langle}\int_{\partial
G}{K\>\mathrm{d}A}\bigg{\rangle}-4\pi+\alpha\langle f_{0}(G)\rangle$
$\displaystyle
e_{2}=\sum_{i=1}^{f_{2}(\Omega)}\int_{F_{i}}{K\>\mathrm{d}A}-2\pi
f_{3}(\Omega)+2\alpha f_{0}(\Omega)$

First, $e_{1}$ and $e_{2}$ could be used as rough measures of the deviation of
a physical system from an ideal one, along with other quantities like the
grain growth exponent and the grain size distribution. Second, $e_{1}$ and
$e_{2}$ could be used to evaluate the accuracy of the geometric representation
of a grain structure in a microstructure evolution code; the derivation above
suggests that these quantities should be particularly sensitive to the
geometry around triple junction lines and quadruple junction points. Since the
angle conditions around triple junction lines are directly implicated in the
rates of area and volume change of two-dimensional [5, 6] and three-
dimensional [8, 9] grains, any deviations from Eqs. 1 and 2 could function as
bounds on the maximum achievable accuracy of simulations of mean-curvature
driven grain growth.

## Acknowledgments

###### Acknowledgements.

The authors are grateful to R.D. MacPherson for enlightening discussions about
grain structure geometry. E.E. and J.K.M. were supported by the National
Science Foundation under Grant No. 1839370.

## Author contributions

J.K.M. conceived the study, developed the mathematics, and analyzed the
numerical results. E.E. designed and implemented the simulation model and
performed the numerical experiments. J.K.M. and E.E. jointly prepared the
manuscript.

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Supplementary information:
A constant of motion for ideal grain growth in three dimensions

##  I Derivation of Eqs. 1 and 2

Suppose that $\Omega$ is a grain structure in a domain with periodic boundary
conditions, that $\Omega$ satisfies Plateau’s laws (i.e., grain boundaries
meet at dihedral angles of $2\pi/3$ and triple junction lines meet at angles
of $\cos^{-1}(-1/3)$), and that in $\Omega$ every triple junction line is
bounded by two distinct quadruple junction points, every grain boundary is
homeomorphic to a disk and bounded by two or more distinct triple junction
lines, and every grain is homeomorphic to a ball and bounded by three or more
distinct grain boundaries.

The initial objective is to establish the following version of the
Gauss–Bonnet theorem for a grain $G$ belonging to $\Omega$:

$\int_{\partial G}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{2}(G)}\int_{\partial
F_{i}}{\kappa_{g}\>\mathrm{d}s}+\sum_{i=1}^{f_{0}(G)}{\alpha_{i}}=2\pi\chi(\partial
G)$ (1)

where $K$ is the Gaussian curvature of a grain boundary, $\partial G$
indicates the grain boundary interiors, $\kappa_{g}$ is the geodesic curvature
of a triple junction line, $\partial F_{i}$ indicates the interiors of the
triple junction lines bounding the $i$th grain boundary, $f_{2}(G)$ is the
number of grain boundaries of $G$, $\alpha_{i}$ is the angular defect at the
$i$th quadruple junction point, $f_{0}(G)$ is the number of quadruple junction
points of $G$, and $\chi(\partial G)$ is the Euler characteristic of the
boundary of $G$. The standard version of the Gauss–Bonnet theorem applies to
an oriented surface $F_{i}$ [1]:

$\int_{F_{i}}{K\>\mathrm{d}A}+\int_{\partial
F_{i}}{\kappa_{g}\>\mathrm{d}s}+\sum_{j=1}^{f_{0}(F_{i})}{(\pi-\gamma_{j})}=2\pi\chi(F_{i})$
(2)

where $\gamma_{j}$ is the interior angle at the $j$th corner along $\partial
F_{i}$. As suggested by the notation, let $F_{i}$ be the $i$th grain boundary
of $G$. Since every grain boundary is homeomorphic to a disk, $\chi(F_{i})=1$
and summing Eq. 2 over the grain boundaries of $G$ gives:

$\int_{\partial G}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{2}(G)}\int_{\partial
F_{i}}{\kappa_{g}\>\mathrm{d}s}-\sum_{i=1}^{f_{2}(G)}\sum_{j=1}^{f_{0}(F_{i})}\gamma_{j}=2\pi
f_{2}(G)-\pi f_{0,2}(G)$

where $f_{0,2}(G)$ is the number of distinct pairs of adjacent quadruple
junction points and grain boundaries. Adding $2\pi f_{0}(G)$ to each side of
the above equation, using the identity $f_{0,2}(G)=f_{1,2}(G)=2f_{1}(G)$, and
applying the definition $\chi(\partial G)=f_{0}(G)-f_{1}(G)+f_{2}(G)$ for the
Euler characteristic of a surface gives

$\int_{\partial G}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{2}(G)}\int_{\partial
F_{i}}{\kappa_{g}\>\mathrm{d}s}+\bigg{(}2\pi
f_{0}(G)-\sum_{i=1}^{f_{2}(G)}\sum_{j=1}^{f_{0}(F_{i})}\gamma_{j}\bigg{)}=2\pi\chi(\partial
G).$

Since $2\pi$ minus the internal angles around the $i$th quadruple junction
point of $G$ is the angular defect $\alpha_{i}$ at that point, this reduces to
Eq. 1 above.

Figure S1: The respective arrangement of the curve $\bm{c}_{k}(s)$, the unit
vectors $\bm{u}_{i}$, $\bm{u}_{j}$, and $\bm{v}$, and the angle $\beta$ at a
point along $E_{k}$.

The next objective is to simplify Eq. 1 by means of the initial assumptions.
Initially observe that $\chi(\partial G)=2$ since $G$ is homeomorphic to a
ball, and that the angular defect at every quadruple junction point is
$\alpha=2\pi-3\cos^{-1}(-1/3)$; this allows Eq. 1 to be reduced to:

$\int_{\partial
G}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{2}(G)}\sum_{j=1}^{f_{1}(F_{i})}\int_{E_{j}}{\kappa_{g}\>\mathrm{d}s}+\alpha
f_{0}(G)=4\pi$

where $E_{j}$ is the $j$th triple junction line and $f_{1}(F_{i})$ is the
number of triple junction lines of $F_{i}$. Now consider the summation of the
integrals of the geodesic curvature over the triple junction lines of $G$.
Every triple junction line is integrated over twice, once for each of the
adjoining grain boundaries. Suppose that all of the grain boundaries have
outward-pointing unit normal vector fields, that the grain boundaries $F_{i}$
and $F_{j}$ intersect on the $k$th triple junction line $E_{k}$ of $G$, and
that $\bm{c}_{k}(s)$ is the curve parameterized by arc length that travels
along $E_{k}$ in the positive orientation on $F_{i}$ and the negative
orientation on $F_{j}$. Let $\bm{\nu}_{i}(s)$ and $\bm{\nu}_{j}(s)$ be the
restrictions of the unit normal vector fields of $F_{i}$ and $F_{j}$ to points
on $\bm{c}_{k}(s)$, and
$\bm{u}_{i}(s)=\bm{\nu}_{i}(s)\times\bm{c}^{\prime}_{k}(s)$ and
$\bm{u}_{j}(s)=\bm{\nu}_{j}(s)\times\bm{c}^{\prime}_{k}(s)$ be the tangent
normals of $\bm{c}_{k}(s)$ on $F_{i}$ and $F_{j}$ as indicated in Fig. S1,
where the prime indicates differentiation with respect to arclength. Finally,
write the geodesic curvatures $\kappa_{g}$ of $\bm{c}_{k}(s)$ on $F_{i}$ and
$F_{j}$ as $\bm{c}^{\prime\prime}_{k}(s)\cdot\bm{u}_{i}(s)$ and
$\bm{c}^{\prime\prime}_{k}(s)\cdot\bm{u}_{j}(s)$. This allows the middle term
on the left of the above equation to be written as:

$\displaystyle\sum_{i=1}^{f_{2}(G)}\sum_{j=1}^{f_{1}(F_{i})}\int_{E_{j}}{\kappa_{g}\>\mathrm{d}s}$
$\displaystyle=\sum_{k=1}^{f_{1}(G)}\bigg{[}\int_{0}^{s_{k}}{\bm{c}^{\prime\prime}_{k}(s)\cdot\bm{u}_{i}(s)\>\mathrm{d}s}-\int_{0}^{s_{k}}{\bm{c}^{\prime\prime}_{k}(s)\cdot\bm{u}_{j}(s)\>\mathrm{d}s}\bigg{]}$
$\displaystyle=\sum_{k=1}^{f_{1}(G)}\int_{0}^{s_{k}}{\kappa\bm{e}_{2}\cdot(\bm{u}_{i}(s)-\bm{u}_{j}(s))\>\mathrm{d}s}$

where $s\in[0,s_{k}]$ and $\bm{c}^{\prime\prime}_{k}(s)=\kappa\bm{e}_{2}$, or
the product of the curvature $\kappa$ and the second Frenet vector
$\bm{e}_{2}$ of $\bm{c}_{k}(s)$. Since $\bm{u}_{i}(s)$ and $\bm{u}_{j}(s)$ are
unit vectors, $\bm{u}_{i}(s)-\bm{u}_{j}(s)$ can be written as
$2\sin(\beta/2)\bm{v}(s)$ where $\bm{u}_{i}(s)\cdot\bm{u}_{j}(s)=\cos\beta$
and $\bm{v}(s)$ is the unit vector pointing along
$\bm{u}_{i}(s)-\bm{u}_{j}(s)$ as in Fig. S1. Since $\beta=\pi/3$, this
simplifies further to $\bm{u}_{i}(s)-\bm{u}_{j}(s)=\bm{v}(s)$ and Eq. 1
reduces to:

$\int_{\partial
G}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{1}(G)}\int_{E_{i}}{\kappa\bm{e}_{2}\cdot\bm{v}\>\mathrm{d}s}+\alpha
f_{0}(G)=4\pi.$ (3)

This is perhaps the extent of simplification that is possible for a single
grain.

Equations 1 and 2 in the main text are found by summing Eq. 3 over all the
grains in $\Omega$. As described in the main text, the contribution of the
second term on the left vanishes since every triple junction line is
integrated over three times, once for each adjoining grain. $\kappa\bm{e}_{2}$
is an inherent quantity of the triple junction line that is the same for all
three integrals, but the three $\bm{v}$ are all unit vectors in a plane with
mutual angles of $2\pi/3$. This causes the sum of the three $\bm{v}$ to vanish
identically for each triple junction line, leaving:

$\sum_{i=1}^{f_{3}(\Omega)}\int_{\partial
G_{i}}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{3}(\Omega)}\alpha f_{0}(G_{i})=4\pi
f_{3}(\Omega)$

where $f_{3}(\Omega)$ is the number of grains of $\Omega$. Dividing through by
$f_{3}(\Omega)$ gives Eq. 1 of the main text. Alternatively, observing that
every grain boundary is included two times in the first sum, that every
quadruple junction point is included four times in the second sum, and
dividing through by two gives Eq. 2 of the main text.

##  II Other boundary conditions

If the grain structure $\Omega$ exists in a domain without periodic boundary
conditions, then the elements of $\Omega$ intersecting the external surfaces
need to be handled differently. The derivation of the counterparts of Eqs. 1
and 2 of the main text proceeds as above, starting with Eq. 1 for a grain $G$:

$\displaystyle\sum_{i=1}^{\mkern 3.0mu\overline{\mkern-3.0muf\mkern
0.0mu}\mkern 0.0mu_{2}(G)}\int_{\mkern 3.0mu\overline{\mkern-3.0muF\mkern
0.0mu}\mkern 0.0mu_{i}}{K\>\mathrm{d}A}+\sum_{i=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern
0.0mu_{2}(G)}\sum_{j=1}^{\mkern 3.0mu\overline{\mkern-3.0muf\mkern
0.0mu}\mkern 0.0mu_{1}(\mkern 3.0mu\overline{\mkern-3.0muF\mkern 0.0mu}\mkern
0.0mu_{i})}\int_{\mkern 3.0mu\overline{\mkern-3.0muE\mkern 0.0mu}\mkern
0.0mu_{j}}{\kappa_{g}\>\mathrm{d}s}+\sum_{i=1}^{f_{2}(G)}\sum_{j=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{1}(F_{i})}\int_{\mkern
3.0mu\overline{\mkern-3.0muE\mkern 0.0mu}\mkern
0.0mu_{j}}{\kappa_{g}\>\mathrm{d}s}+\sum_{i=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{0}(G)}{\alpha_{i}}$
$\displaystyle+\sum_{i=1}^{f_{2}(G)}\int_{F_{i}}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{2}(G)}\sum_{j=1}^{f_{1}(F_{i})}\int_{E_{j}}{\kappa_{g}\>\mathrm{d}s}+\sum_{i=1}^{f_{0}(G)}{\alpha_{i}}$
$\displaystyle=2\pi\chi(\partial G)$

where quantities relating to grain structure elements on external surfaces are
indicated by overlines (e.g., the second term on the left concerns triple
junction lines on the external surface that bound grain boundaries on the
external surface, whereas the third term on the left concerns triple junction
lines on the external surface that bound internal grain boundaries). Using
that $\chi(\partial G)=2$ and that the angular defect at every internal
quadruple junction point is $\alpha$, repeating the steps involving integrals
over triple junction lines, and summing over all the grains of $\Omega$ gives:

$\displaystyle\sum_{i=1}^{f_{3}(\Omega)}\sum_{j=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{2}(G_{i})}\int_{\mkern
3.0mu\overline{\mkern-3.0muF\mkern 0.0mu}\mkern
0.0mu_{j}}{K\>\mathrm{d}A}+\sum_{i=1}^{f_{3}(\Omega)}\sum_{j=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{1}(G_{i})}\int_{\mkern
3.0mu\overline{\mkern-3.0muE\mkern 0.0mu}\mkern
0.0mu_{j}}{\kappa\bm{e}_{2}\cdot[2\cos(\lambda/2)\bm{v}]\>\mathrm{d}s}+\sum_{i=1}^{f_{3}(\Omega)}\sum_{j=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{0}(G_{i})}{\alpha_{j}}$
$\displaystyle+\sum_{i=1}^{f_{3}(\Omega)}\sum_{j=1}^{f_{2}(G_{i})}\int_{F_{j}}{K\>\mathrm{d}A}+\alpha\sum_{i=1}^{f_{3}(\Omega)}f_{0}(G_{i})$
$\displaystyle=4\pi f_{3}(\Omega)$ (4)

where $\cos(\lambda/2)=\sin(\beta/2)$ and $\lambda$ is the interior angle
along the triple junction line.

Dividing through by $f_{3}(\Omega)$ gives:

$\displaystyle\bigg{\langle}\sum_{i=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{2}(G)}\int_{\mkern
3.0mu\overline{\mkern-3.0muF\mkern 0.0mu}\mkern
0.0mu_{i}}{K\>\mathrm{d}A}+\sum_{i=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{1}(G)}\int_{\mkern
3.0mu\overline{\mkern-3.0muE\mkern 0.0mu}\mkern
0.0mu_{i}}{\kappa\bm{e}_{2}\cdot[2\cos(\lambda/2)\bm{v}]\>\mathrm{d}s}+\sum_{i=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern
0.0mu_{0}(G)}{\alpha_{i}}\bigg{\rangle}$
$\displaystyle+\bigg{\langle}\int_{\partial
G}{K\>\mathrm{d}A}\bigg{\rangle}+\alpha\langle f_{0}(G)\rangle$
$\displaystyle=4\pi$

which is the counterpart to Eq. 1 of the main text and emphasizes the
properties of individual grains. This version groups the contributions of
grain structure elements on external surfaces in the first average over all
grains of $\Omega$. Since $f_{3}(\Omega)$ should increase as the volume
whereas the number of terms in the average should increase as the surface
area, the influence of the first average should decrease with decreasing
surface area to volume ratio of $\Omega$. Alternatively, starting with Eq. 4
and observing that every internal grain boundary is included two times in the
fourth term, that every quadruple junction point is included four times in the
fifth term, and dividing through by two gives:

$\displaystyle\frac{1}{2}\bigg{\\{}\sum_{i=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{2}(\Omega)}\int_{\mkern
3.0mu\overline{\mkern-3.0muF\mkern 0.0mu}\mkern
0.0mu_{i}}{K\>\mathrm{d}A}+\sum_{i=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{1}(\Omega)}\int_{\mkern
3.0mu\overline{\mkern-3.0muE\mkern 0.0mu}\mkern
0.0mu_{i}}{\kappa\bm{e}_{2}\cdot\bigg{[}2\sum_{j=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern 0.0mu_{1,3}(\mkern
3.0mu\overline{\mkern-3.0muE\mkern 0.0mu}\mkern
0.0mu_{i})}\cos(\lambda_{j}/2)\bm{v}_{j}\bigg{]}\>\mathrm{d}s}+\sum_{i=1}^{\mkern
3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern
0.0mu_{0}(\Omega)}\sum_{j=1}^{\mkern 3.0mu\overline{\mkern-3.0muf\mkern
0.0mu}\mkern 0.0mu_{0,3}(\mkern 3.0mu\overline{\mkern-3.0muV\mkern
0.0mu}\mkern 0.0mu_{i})}{\alpha_{j}}\bigg{\\}}$
$\displaystyle+\sum_{i=1}^{f_{2}(\Omega)}\int_{F_{i}}{K\>\mathrm{d}A}+2\alpha
f_{0}(\Omega)$ $\displaystyle=2\pi f_{3}(\Omega)$

where $\mkern 3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern
0.0mu_{1,3}(\mkern 3.0mu\overline{\mkern-3.0muE\mkern 0.0mu}\mkern 0.0mu_{i})$
is the number of grains adjacent to the $i$th triple junction line $\mkern
3.0mu\overline{\mkern-3.0muE\mkern 0.0mu}\mkern 0.0mu_{i}$ on the external
surface and $\mkern 3.0mu\overline{\mkern-3.0muf\mkern 0.0mu}\mkern
0.0mu_{0,3}(\mkern 3.0mu\overline{\mkern-3.0muV\mkern 0.0mu}\mkern 0.0mu_{i})$
is the number of grains adjacent to the $i$th quadruple junction point $\mkern
3.0mu\overline{\mkern-3.0muV\mkern 0.0mu}\mkern 0.0mu_{i}$ on the external
surface. This is the counterpart to Eq. 2 of the main text. This is as far as
the equations can be developed without specifying the shape of the domain of
$\Omega$ and the nature of the boundary conditions.

## References

  * Gray  _et al._ [2017] A. Gray, E. Abbena, and S. Salamon,  _Modern differential geometry of curves and surfaces with Mathematica®_ (Chapman and Hall/CRC, 2017).