# A thermo-hygro computational model to determine the factors dictating cold
joint formation in 3D printed concrete

Michal Hlobil  Institute for Building Materials, ETH Zurich, Switzerland  Luca
Michel  Institute for Building Materials, ETH Zurich, Switzerland  Mohit
Pundir  Institute for Building Materials, ETH Zurich, Switzerland  David S.
Kammer  Institute for Building Materials, ETH Zurich, Switzerland

## Abstract

Cold joints in extruded concrete structures form once the exposed surface of a
deposited filament dries prematurely and gets sequentially covered by a layer
of fresh concrete. This creates a material heterogeneity which lowers the
structural durability and shortens the designed service life. Many factors
concurrently affect cold joint formation, yet a suitable tool for their
categorization is missing. Here, we present a computational model that
simulates the drying kinetics at the exposed structural surface, accounting
for cement hydration and the resulting microstructural development. The model
provides a time estimate for cold joint formation as a result. It allows us to
assess the drying severity for a given structure’s geometry, its interaction
with the environment, and ambient conditions. We evaluate the assessed factors
and provide generalized recommendations for cold joint mitigation.

## Keywords

3D printed concrete; Additive manufacturing; Cold joint; Hydration modeling;
Multiphysics modeling

##  1 Introduction

The adoption of extrusion-based additive manufacturing opened new
possibilities for producing concrete structures. This technology draws from
the synergy between computer-aided design, robotics, and material usage
optimization [1, 2, 3]. Utilizing additive manufacturing, structural elements
are produced by successively stacking thin layers, or filaments, of rapidly
reacting concrete, which are extruded from a robotically-controlled printing
head [4], see Figure 1(a). This technique facilitates the production of shape-
optimized elements with complex structural geometries [5, 6, 7], avoiding
traditional formwork. However, successfully implementing this technology into
common practice faces several challenges, both on the structural and material
design fronts [8, 9].

Figure 1: Schematic illustration of the modeling approach. (a) Sketch of a
wall-like extruded concrete structure consisting of sequentially stacked
filaments. (b) A planar 2D slice from the structure comprising a materially
homogeneous cross-section. A finite element mesh is created over the entire
domain to simulate transient and coupled heat and humidity transport, leading
to a prediction of surface drying rate. The model accounts for i) fresh
concrete hardening, accompanied by internal heat generation, which leads to a
temperature increase within the concrete structure, and ii) material
diffusivity, capturing the state of the pore network formed within the
material.

The material prerequisites for printable concrete call for designing an
initially fluid and pumpable mixture that rapidly hardens upon deposition,
developing both stiffness and strength. This ensures the structural stability
of the thin-walled extruded structure and sufficient load-bearing capacity to
withstand the dead weight of the successively added layers. However, while
rapid strength development of the concrete mix is necessary for 3D printing,
it may also negatively affect the bond between two neighboring filaments [10,
11, 12].

After the extruded filament is deposited, its exposed surface dries, forming a
solid crust over time [13]. Unless the top filament surface is promptly
covered with new concrete, the crust thickens and results in a porous region
that weakens the interface between the already dry filament and the
sequentially added and fully water-saturated filament [14]. The presence of
this inhomogeneity in the stack of extruded filaments leads to an internal
structural defect, commonly denoted as a “cold joint” which compromises the
long-term durability of the extruded structural element [15, 16, 17, 18].

The origins of crust formation are related to material drying, which
inherently arises from two sources. Water is consumed internally by cement
hydration but also evaporates through the unprotected structural surface [19],
see Figure 1(b). As the microstructure develops, the smallest pores will be
emptied first, leaving voids in the material, which results in self-
desiccation. In addition, cement contact with water triggers a series of
spontaneous and highly exothermic reactions, liberating over 500 J as heat per
gram of cement over the whole reaction [20], as schematically shown in Figure
1(c). The released thermal energy leads to a progressive temperature increase
within the concrete structure [21]. Since the cement paste matrix exhibits
rather low thermal conductivity, a high-temperature gradient forms between the
heating structural core and the exposed outer surface of the concrete
filament, accelerating water transport through the pore network. While all
unprotected sides of the extruded structure facilitate water evaporation, the
drying of the top surface of the uppermost filament is critical, creating a
potential risk for cold joint formation.

Cold joint formation challenges the durability of extruded concrete
structures. Hence, understanding their formation and subsequent mitigation is
of great interest. As shown by Mechtcherine et al. [19], multiple input
factors influence cold joint formation. While their experimental
quantification is challenging, computational models provide a feasible
alternative to studying their magnitude. Several multiphysics models have been
developed, focusing on predicting the temperature increase in massive concrete
structures [22, 23, 24] and simulating early-age cracking [25, 26]. However,
predictive models focusing on drying kinetics of hydrating concrete for
extruded structures, which lie at the core of the cold joint formation, are
yet missing.

Here, we formulate a computational model to investigate the factors that lead
to cold joint formation in extruded concrete structures. The model is based on
solving a coupled thermo-hydro conduction phenomenon over a planar domain,
representing a full cross-section of an extruded structure, see Figure 1(b).
The proposed approach captures the kinetics of cold joint formation,
accounting for concrete composition, structural dimensions, and interaction
with the ambient environment. We analyze the effect of different factors
quantitatively, allowing us to determine the most important ones for cold
joint formation, which aids with the material design in practice.

##  2 Model formulation

Origins of cold joint formation point to premature surface drying of hydrating
concrete. This physical phenomenon can be mathematically described by a set of
differential equations, formulated here as a coupled thermo-hydro transport
problem (the theoretical derivation can be found in Appendix A). The governing
equations take the form of

$\displaystyle\nabla\cdot(\lambda_{mat}(\alpha)\,\nabla
T_{mat})+\frac{\partial Q_{h}}{\partial t}$
$\displaystyle=\rho_{mat}\,c_{mat}(\alpha)\,\frac{\partial T_{mat}}{\partial
t},$ (1a) $\displaystyle\nabla\cdot(D_{H}\,\nabla H_{mat})-\frac{\partial
H_{s}}{\partial t}+\kappa\,\frac{\partial T_{mat}}{\partial t}$
$\displaystyle=\frac{\partial H_{mat}}{\partial t},$ (1b)

where $\alpha$ denotes the degree of cement hydration, described in Section
2.1, and $T_{mat}$, and $H_{mat}$ the concrete temperature and relative
humidity, respectively. All three unknown quantities then act as field
variables. In Eq. (1a), $\lambda_{mat}$, $\rho_{mat}$, and $c_{mat}$ denote
the heat conductivity, mass density, and heat capacity of concrete,
respectively, and $\frac{\partial Q_{h}}{\partial t}$ the rate of hydration
heat released, see Section 2.1. In Eq. (1b), $D_{H}$ represents the
diffusivity coefficient and $\kappa$ the hygrothermic coefficient ensuring a
weak coupling between temperature and humidity, see Appendix A.2. The sink
term $\frac{\partial H_{s}}{\partial t}$ accounts for the self-desiccation of
the material due to hydration.

While Eq. (1) describe a general transient heat and humidity conduction over a
domain and thus form a backbone of the computational model, we will now
introduce three aspects to focus its broad applicability to early-age
hydrating concrete. In particular, we focus on i) the rate of hydration heat,
which acts as an internal heat source in the concrete structure, followed by
ii) the description of an analytical model to characterize microstructural
development, and finally arriving at iii) the description of thermophysical
properties of concrete and their link to microstructural evolution.

###  2.1 Internal heat source from cement hydration

Upon contact with water, cement particles undergo hydration reactions
resulting in a measurable heat release. Isothermal calorimetry permits
continuous measurements of the cumulative heat of hydration
$Q_{h}(t,T_{\infty},H_{\infty})$ in [J/kg] released for a given time $t$ [s].
Heat production depends on ambient temperature $T_{\infty}$ [∘C] and relative
humidity $H_{\infty}$ [-] history, and can be expressed as

$\displaystyle
Q_{h}(t,T_{\infty},H_{\infty})=\alpha\,Q_{\text{tot}}(t\to\infty),\quad\alpha\in\langle
0,1\rangle,$ (2)

where $\alpha$ [-] denotes the achieved state of microstructural development,
described here as the degree of cement hydration, and
$Q_{\text{tot}}(t\to\infty)$ in [J/kg] denotes the latent hydration heat of
cement. The latter is unique for each cement as it depends on its
mineralogical composition. It can, however, be estimated as a weighted average
of enthalpies of the main clinker constituents at complete hydration [27, p.
232].

Focusing on the rate of hydration heat originating from cement dissolution
$\frac{\partial Q_{h}}{\partial t}$ in [J/(s m3)], we find

$\displaystyle\frac{\partial Q_{h}}{\partial t}=\frac{\partial\alpha}{\partial
t}\,Q_{\text{tot}}(t\to\infty),$ (3)

where the rate of microstructural development

$\displaystyle\frac{\partial\alpha}{\partial
t}=\tilde{A}(t,T_{\infty},H_{\infty})\,,$ (4)

is described by $\tilde{A}(t,T_{\infty},H_{\infty})$ in [s-1], the chemical
affinity, and is defined as the change of microstructural development in time.
This term was introduced by Ulm and Coussy [28] and conjugates the gradient of
free enthalpy between the reactants and products of the chemical reactions
occurring during hydration [29, 30]. Consequently,
$\tilde{A}(t,T_{\infty},H_{\infty})$ is a time-, temperature-, and humidity-
dependent quantity. The time specification will be replaced by $\alpha$ in
$\tilde{A}(\alpha,T_{\infty},H_{\infty})$, recalling that $\alpha$ defines the
resultant state of the microstructural evolution, regardless of when and under
which conditions it was achieved.

The rate of hydration heat defined by Eq. (3) physically represents an
internal heat source within the concrete structure. This term, therefore,
appears in the governing equation for transient heat conduction, see Eq. (1a).
Eq. (4) describing microstructural development serves as the final governing
equation, together with Eq. (1).

###  2.2 Affinity-based hydration model

Experimental determination of the hydration heat rate for a given cement type,
mixture composition, and variable temperature requires repeated measurements,
which are time-intensive and, hence, impractical for modeling purposes. In
addition, fluctuations in curing temperature and/or ambient humidity affect
the reaction kinetics of cement hydration. To account for both phenomena
separately, we use dimensionless correction factors $\beta_{\text{T}}$ and
$\beta_{\text{H}}$ [30]. These will be used to scale the cement hydration
kinetics from a reference state defined by $\tilde{A}_{\text{ref}}$ as follows

$\displaystyle\tilde{A}(\alpha,T_{\infty},H_{\infty})=\tilde{A}_{\text{ref}}(\alpha)\,\beta_{\text{T}}(T_{\infty})\,\beta_{\text{H}}(H_{\infty})\,.$
(5)

Both scaling factors are discussed separately below.

For a computational implementation, we consider a phenomenological affinity-
based hydration model that provides an analytical relationship between
microstructural development in a hydrating paste and concurrent hydration
heat. This model was initially formulated by Cervera et al. [29] and later
modified by Jendele et al. [31]. We adopt the latter formulation, which
defines $\tilde{A}_{\text{ref}}$ as

$\displaystyle\tilde{A}_{\text{ref}}=B_{1}\left(\frac{B_{2}}{\alpha_{\infty}}+\alpha\right)(\alpha_{\infty}-\alpha)\exp\left(-\eta\,\frac{\alpha}{\alpha_{\infty}}\right),$
(6)

where $\tilde{A}_{\text{ref}}$ needs to be determined at a reference
temperature, $B_{1}$ in [s-1], $B_{2}$ [-] and $\eta$ [-] are best-fit
parameters characterizing the kinetics of cement hydration and
$\alpha_{\infty}$ represents the maximum-reachable degree of hydration of the
cement paste. The initial three parameters need to be first calibrated for the
specific cement. In this paper, we use isothermal calorimetry measurements for
calibration, but if such results are not available, suitable hydration models
can be used instead (e.g., CEMHYD3D [32, 20]). The term $\alpha_{\infty}$ then
depends on the fresh mix design and curing conditions [33], see Appendix B for
details.

As cement hydration is a thermally activated process, an Arrhenius-type
equation can be used to correct $\tilde{A}_{\text{ref}}$ for temperature
fluctuation in Eq. (5). Consequently, we consider the coefficient
$\beta_{\text{T}}$ [-] as

$\displaystyle\beta_{\text{T}}(T)=\exp\left[\frac{E_{a}}{R}\left(\frac{1}{T_{\infty}}-\frac{1}{T_{mat}}\right)\right],$
(7)

where $T_{\infty}=293.15$ K represents the reference ambient temperature for
which $\tilde{A}_{\text{ref}}$ is determined, $R=8.31441$ J/(mol K) is the
universal gas constant, and $E_{a}=38.3$ kJ/mol is the apparent activation
energy based on measurements from Kada-Benameur et al. [34]. The latter
quantity is a mixture- and temperature-dependent parameter that characterizes
the binder’s integrated and simultaneous chemical reactions.

Cement hydration encompasses a series of chemical processes, all requiring a
constant supply of liquid water from the capillary pore network. This
facilitates the transport of ionic species and the subsequent formation of
reaction products. However, the rate of hydration progressively decreases with
the dropping of pore relative humidity and eventually stops upon reaching
$\text{H}\approx 0.80$ [35]. As a remedy, we recall the coefficient
$\beta_{\text{H}}$ [-] coined in [36] into Eq. (5), which accounts for the
reduction of hydration rate for decreasing the capillary humidity as

$\displaystyle\beta_{\text{H}}(H)=\frac{1}{1+{\left(a-a\,H_{mat}\right)}^{4}},$
(8)

where $a=7.5$ [-] is a material parameter introduced in [36].

###  2.3 Thermophysical properties of concrete constituents

The parameters $\rho_{mat}$, $c_{mat}(\alpha)$, and $\lambda_{mat}(\alpha)$
postulated in Eq. (1) characterize the physical and thermal properties of
concrete. For the development of the model, we consider $\rho_{mat}$ as
constant, unaffected by ongoing hydration or water evaporation from the
exposed structural surface. However, $c_{mat}(\alpha)$ and
$\lambda_{mat}(\alpha)$ evolve in time, reflecting the temperature- and
humidity-dependent microstructural development.

The specific heat capacity of a material, $c_{mat}$, is defined as the amount
of heat necessary to change the temperature of the material by one degree,
normalized per mass of the material. An expression for the heat capacity of
hydrating cement paste, $c_{paste}(\alpha)$, should account for the initial
paste composition given by the water-to-cement mass ratio $w/c$, ongoing
microstructural development, and curing conditions [37]. We here consider

$\displaystyle c_{paste}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{c_{water}\,w/c+c_{cement}}{1+w/c}\,(1-0.26\,(1-\exp(-2.9\,\alpha))),$
(9)

where $c_{water}=4.18$ J/(g K) and $c_{cement}=0.75$ J/(g K) stand for the
heat capacities of water [38] and anhydrous cement [39] measured at room
temperature, respectively.

The heat capacity of concrete, $c_{concrete}$, analogously depends on the
properties of both cement paste and aggregates and can be estimated [37] using
a simple rule of mixtures as

$\displaystyle
c_{concrete}=m_{f,aggr}\,c_{aggr}+m_{f,paste}\,c_{paste}(\alpha),$ (10)

where $m_{f,phase}$ stands for a mass fraction of the given phase, and
$c_{aggr}=0.8$ J/(g K) [40].

The heat conductivity $\lambda_{mat}$ is defined as the ratio of heat flux to
the temperature gradient [41]. For cement paste, $\lambda_{paste}\approx
0.9-1.1$ W/(m K) remains virtually unchanged throughout hydration regardless
of the mix composition and curing conditions. This has been experimentally
verified in [37, 42] and computationally corroborated in [43, 44]. A virtually
constant $\lambda_{paste}=1.0$ W/(m K) can be anticipated based on a
comparison of heat conductivities of constituting solid phases in the paste
and their envisioned relative proportions throughout hydration
($\lambda_{clinker}=1.55$ [37], $\lambda_{C-S-H}=0.98$, $\lambda_{CH}=1.32$
(both from [43]), $\lambda_{water}=0.604$ W/(m K) [37]). While a minor
decrease of $\lambda_{paste}$ with $w/c$ ratio was observed in [45], the
differences may be deemed practically negligible for the considered range of
$w/c\in 0.25-0.4$ [-].

The initial concrete conductivity $\lambda_{concrete}$ is largely governed by
the relative proportions of aggregates and paste in the mixture. Bentz [37]
proposed an estimate to evaluate the initial $\lambda_{concrete}^{init}$ of
fresh concrete using a Hashin-Shtrikman homogenization scheme, with
$\lambda_{concrete}$ as an arithmetic mean from the upper and lower bounds
given as

$\displaystyle\lambda_{concrete}^{upper}$ $\displaystyle=$
$\displaystyle\lambda_{aggr}+\frac{\phi_{paste}}{\frac{1}{\lambda_{paste}-\lambda_{aggr}}+\frac{\phi_{aggr}}{3\,\lambda_{aggr}}},$
(11) $\displaystyle\lambda_{concrete}^{lower}$ $\displaystyle=$
$\displaystyle\lambda_{paste}+\frac{\phi_{aggr}}{\frac{1}{\lambda_{aggr}-\lambda_{paste}}+\frac{\phi_{paste}}{3\,\lambda_{paste}}},$
(12) $\displaystyle\lambda_{concrete}^{init}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{concrete}^{upper}+\lambda_{concrete}^{lower}}{2},$
(13)

where $\lambda$ and $\phi$ denote the heat conductivity and volume fraction of
a given component (paste or aggregates); here, we consider
$\lambda_{aggr}=3.0$ W/(m K) [46]. Note that Eq. (13) assumes that
$\lambda_{aggr}>\lambda_{paste}$, which is valid for most siliceous and
calcitic aggregates but excludes lightweight aggregates.

Experimental measurements by Chen et al. [47] indicate that concrete heat
conductivity further evolves with ongoing cement hydration and is strongly
affected by curing conditions. Based on their experimental results, we
consider the evolution of $\lambda_{concrete}$ under sealed conditions as

$\displaystyle\lambda_{concrete}^{sealed}(\alpha)$ $\displaystyle=$
$\displaystyle\lambda_{concrete}^{init}\,(1+0.22\,\alpha)\,.$ (14)

Note that while the experimentally measured concrete conductivity increases
under sealed conditions in time, the opposite trend was observed for saturated
conditions. Chen et al. [47] attribute this observation to the differences in
the microstructural development during hydration, particularly in the presence
of water within the overall pore network, from capillary pores to the
interfacial transition zone around aggregates. During sealed curing, the
material experiences self-desiccation as capillary pores are progressively
emptied. As the solid phase fraction increases, the moisture content
decreases, so the overall conductivity of the material becomes proportional to
the conductivity of the solid skeleton matrix. In comparison, water remains
omnipresent within the pore network during the saturated curing, thus
effectively lowering the overall conductivity. However, the only former curing
regime applies in realistic conditions for extruded concrete structures.

##  3 Numerical implementation

Surface drying of hydrating concrete can be described by a set of governing
Equations given by Eq. (1) and (4). To find an approximate solution, we employ
the Finite Element Method implemented in the open-source FEniCS package [48].
First, we spatially discretize a rectangular domain representing a 2D slice of
the extruded concrete structure. To this end, we consider a full structural
cross-section, grouping individual filaments into a materially homogeneous
domain as shown in Figure 1(b). Next, we resolve the coupled transient
conduction phenomena over the entire domain by discretizing the time
derivatives using a finite difference approximation with a backward difference
quotient, yielding an implicit Euler method. We solve for the sought $\alpha$,
$T$, and $H$ fields, which act as field variables.

Solving the set of governing equations given in Eq. (1) and (4) requires
suitable initial and boundary conditions, reflecting the interaction of the
extruded structure with the surrounding environment. The initial condition
(IC) denotes the starting value of the degree of hydration, temperature, and
humidity fields, assumed herein as $\alpha(x,0)=0$ [-], $T_{mat}(x,0)=20$∘C ,
and $H_{mat}(x,0)=1.0$ [-] over the entire domain. We model the heat exchange
between the structural surface and its environment with Robin’s boundary
condition (BC) as ${\overline{q}_{T}}(t)=h_{T}(T_{mat}(x,t)-T_{\infty})$,
where $h_{T}$ the heat transfer coefficient in [W/(m2 K)], $T_{mat}(x,t)$ the
surface temperature of the material, and $T_{\infty}=20$∘C is the remote
(ambient) temperature. An analogous form of Robin BC is also used for humidity
transport with ${\overline{q}_{H}}(t)=h_{H}(H_{mat}(x,t)-H_{\infty})$ with
$h_{H}$ as the hygric exchange coefficient in [m/s], $H_{mat}(x,t)$ the
surface humidity of the material, and $H_{\infty}=0.60$ [–] as the remote
(ambient) humidity. The listed values represent the default settings used for
model evaluation and are applied unless noted otherwise.

##  4 Model verification

###  4.1 Calibration of the affinity-based hydration model

Here, we demonstrate the affinity-based model’s descriptive capability for
assessing the hydration kinetics of various cement pastes. To this end, we
consider experimental results [20] from isothermal calorimetry tests of
pastes, which differ in cement type and $w/c$ ratio used. Notably, all
considered pastes were free from chemical admixtures that could alter the
hydration kinetics. We plot the experimental results, interpreted by
$\tilde{A}_{\text{ref}}$ as a function of the achieved microstructural
development $\alpha$, and fit the analytical form from Eq. (6) using a damped
least-squares algorithm. This method provides the sought set of parameters
$B_{\text{1}}$, $B_{\text{2}}$, and $\eta$, summarized in Appendix B, which
are independent for each paste tested and characterize their hydration
kinetics.

The affinity-based hydration model captures the main hydration peak and
subsequent decrease of reaction kinetics with acceptable accuracy, as shown in
Figure 2(a). We then use Eq. (2) to calculate the time-evolution of cumulative
heat for each paste and compare the calculated results to experiments in
Figure 2(b). The analytical model shows a satisfactory agreement with the
measured data regardless of the cement paste composition. Thus, it provides an
analytical link between microstructural formation and corresponding heat
release. This achievement enables us to evaluate the temperature increase in
concrete structures caused by cement hydration, as discussed next.

Figure 2: Application of the analytical affinity-based hydration model to
quantify cement hydration. (a) Calibration of the model to experimental data
from isothermal calorimetry. (b) Quantitative evaluation of heat release after
model calibration.

###  4.2 Temperature increase in hardening concrete

We analyze the effect of hydration heat on the temperature increase within a
massive concrete block, as recorded by Šmilauer and Krejčí [23]. In this
experiment, a cubic block with a side length of 1 m was cast from self-
compacting concrete and kept in a protective shelter, shielding it from direct
sun radiation and rain. Monitoring gauges were embedded into the block,
recording the temperature evolution at the core and 3 cm from its outer
surface, as well as ambient temperature, see Figure 3(a). The block was
separated from a solid concrete foundation by a 2 mm thick impregnated paper.
A 15-mm-thick plywood formwork protected its sides, whereas the top surface
was exposed to the ambient environment.

We simulate the experiment as a coupled temperature and humidity problem
within a 2D cross-section, representing a central slice of the specimen as
shown in Figure 3(a). We considered the kinetics of a generic cement with a
common mineralogical composition, see Appendix C, since the reported data in
[23] lacked the necessary inputs for precise calibration of the hydration
model. We prescribe a variable heat and humidity flow to each block boundary,
as shown in Figure 3(a), to account for the fluctuating $T_{\infty}$ but
suppose $H_{\infty}=0.6$ [-] since it was not reported.

Computed results show a satisfying prediction of temperature evolution,
correctly capturing the main peak, as well as an acceptable cooling regime,
see Figure 3(b). The discrepancy in surface temperature observed after 2 days
from casting can probably be attributed to the removal of formwork, which
would affect the chosen value of the prescribed transfer coefficients.
However, such an event was not explicitly documented in [23], and therefore,
we kept $h_{T}$ and $h_{H}$ constant throughout the simulation. Nonetheless,
we consider the simulation results satisfactory and emphasize the correctly
calculated peak temperature and initial cooling stage.

Figure 3: Simulation of temperature increase in a hydrating concrete block.
(a) Modeled 2D cross-section and considered heat and humidity transfer
coefficients. (b) Computed results shown as solid lines and circles indicate
measurements from [23]. The dashed line shows recorded ambient temperature
fluctuation, which was imposed as a boundary condition.

We note that the computed results were based solely on empirical knowledge of
the specimen (geometry, material composition) and its interaction with the
environment (surface treatment and ambient temperature). We chose the values
for the transfer coefficients ($h_{T}$ and $h_{H}$) and ambient humidity
$H_{\infty}$ in line with common considerations for modeling transport
phenomena in concrete in a protected environment [21, 26, 23, 49]. While a
mathematical optimization of the used values could provide an even closer
agreement with the measured data, such analysis is beyond the scope of this
paper.

###  4.3 Drying of concrete elements

We verify the suitability of the presented model for predicting concrete
drying by analyzing experimental measurements carried out by Vinker and Vítek
[50, 51]. Their data contains measured short-term self-desiccation of a
standardized concrete cube over 2 months and long-term relative humidity
measurements in a massive concrete element over 3 years. We will use our model
to simulate the humidity evolution in each specimen separately to account for
the differences in specimen size, curing regime, and at different depths from
the specimen surface and compare with the experimental data.

The first specimen assessed was a standardized concrete cube with a side of
150 mm that was wrapped in foil and sealed in a protected environment upon
demolding. The internal humidity decrease from self-desiccation was recorded
at the cube’s core at irregular intervals over two months; see points in
Figure 4(a). To model this experiment, we considered a coupled temperature and
humidity simulation over a 2D domain representing the central slice of the
cube. We prescribed a constant ambient humidity as
$H_{\infty}=0.955\leavevmode\nobreak\ [-]$ to account for the sealed
conditions and transport coefficients as shown in Figure 4(a), as well as the
cement kinetics parameters summarized in Appendix C.

Figure 4: Simulation of self-desiccation within a standardized concrete cube.
(a) Modeled 2D cross-section and considered heat and humidity transfer
coefficients. (b) Computed results shown as a solid line and circles indicate
measurements from [50, 51].

Simulation results capture the drop in internal relative humidity at the cube
core with acceptable accuracy for engineering purposes, differing only by
single units of % from experiments. We attribute the measured rapid drop in
humidity down to $H_{mat}\approx 0.96$ [-] during the first week to specimen
demolding and subsequent short-term exposure to low ambient humidity before
wrapping; refer to early measured values of $H_{\infty}$ in Figure 5(b). We
will show in Section 6.1 that even a short exposure to dry ambient conditions
leads to a rapid acceleration of material drying. Deplorably, the demolding
event is not explicitly documented in [50, 51], yet it plays a crucial role in
$H_{mat}$ development. As a conclusion, exposure to a sub-saturated ambient
environment accelerates the humidity exchange with the environment, leading to
material drying.

The second experiment involved continuous humidity measurements within a
massive concrete cube with a side length of 800 mm [50, 51]. The material
composition was identical to the standardized cube described above, but the
element’s surface was kept uncovered and thus exposed to a variable
$H_{\infty}$; see Figure 5(C). Internal humidity measurements were
continuously recorded over 800 days at different depths within the specimen.
We simulated the experimental setup by considering a 2D planar cross-section
representing the central slice of the element. Given the measurement duration
of over 800 days, any temperature-induced hydration effects and
microstructural development in such material become negligible after the first
few weeks. Consequently, we carried out only a transient humidity simulation.
To simulate the drying of a mature concrete element, we set the initial
material humidity as $H_{mat}=0.955$ [-] in line with the self-desiccation
measurements of concrete reported above. We then applied a fluctuating
$H_{\infty}$ as the BC on the top and sides of the model; the rest of the
numerical inputs are summarized in Figure 5(a).

Figure 5: Drying simulation of a massive concrete element over 800 days. (a)
Modeled 2D cross-section and considered humidity transfer coefficients. (b)
Computed results are shown as solid lines, and points indicate measurements
from [50, 51]. (c) Recorded ambient humidity fluctuation, which was imposed as
a boundary condition.

Simulation results shown in Figure 5(b) indicate that the fluctuation of
ambient humidity $H_{\infty}$ affects the internal humidity $H_{mat}$ within
the concrete element. At shallow depths from the surface (here depths of 15
and 30 mm), $H_{mat}$ evolution mirrors the fluctuation of $H_{\infty}$. For
depths below 100 mm, the influence of $H_{\infty}$ on $H_{mat}$ vanishes, as
experimentally measured in [50, 51] and obtained computationally herein. As a
general observation, the continuous drying of the entire specimen follows from
the long-term exposure to a sub-saturated ambient environment, as accurately
simulated by the computational model. This successful model verification
serves as a stepping stone to quantify a surface humidity threshold
potentially leading to cold joint quantification, as described below.

##  5 Surface humidity threshold for cold joint formation

While concrete surfaces naturally dry over time, we focus on identifying a
humidity threshold that would indicate a potential threat of cold joint
formation. To this end, we consider the experimental data measured by Keita et
al. [11], who quantified the drying rate of fresh concrete under controlled
conditions. In their work, they exposed freshly cast specimens to a forced
airflow in a climatic chamber and recorded the exposure time. Measured results
indicate that surface drying after 10 minutes of exposure had detrimental
consequences on the material’s mechanical performance.

We simulate the aforementioned drying experiment from Keita et al. [11] to
determine a surface humidity value leading to a potential cold joint
formation. For model evaluation, we consider a central 2D slice of the tested
specimen, see Figure 6(a), and perform a coupled temperature and humidity
simulation with cement kinetics parameters summarized in Appendix C. The
specimen geometry and considered BC are shown in Figure 6(a). The computed
relative humidity at the top specimen surface after 10 minutes of forced
drying amounted to $H_{mat}=0.978$ [-], see result in Figure 6(b). From now
on, we will consider this value as a humidity threshold in Section 6 to
consistently compare the factors affecting surface drying and assess their
severity under given exposure conditions.

Figure 6: Determination of surface humidity threshold leading to a cold joint
formation. (a) Modeled 2D cross-section and considered heat and humidity
transfer coefficients. (b) The computed surface humidity as it evolves over
time.

Given the lack of experimental data on surface humidity measurements leading
to cold joint formation, we simulate the well-defined drying experiment from
[11] and obtain a calculated humidity threshold value. We acknowledge that our
assessment will rely on a single data point and thus cannot be proclaimed a
universally applicable criterion for cold joint formation. Nonetheless, it
allows for a qualitative assessment of the various input factors. More
experimental data on surface humidity evolution should be collected before any
concluding recommendation can be made.

##  6 Factors affecting cold joint formation

Various factors potentially influence the surface drying of extruded concrete.
To evaluate the risk of cold joint formation, we first envision a fictitious
concrete structure with a realistic geometry, extruded within a common ambient
environment, as schematically shown in Figure 7. Next, we perform a coupled
heat and humidity transport simulation and evaluate the time $t_{cj}$
necessary for the humidity on the top surface of the structure to reach the
threshold, potentially leading to a cold joint formation. For the listed
combination of inputs, the computed time amounts to $t_{cj}=501$ min. While
such duration may seem excessive and far from threatening in a typical use
case, a combination of the factors listed below can drastically reduce the
time to units of minutes, raising concerns in all but the fastest extrusions
of structures.

Figure 7: A fictitious concrete structure used to assess the drying severity
for a given structure’s geometry, its interaction with the environment, and
ambient conditions.

To assess the severity of drying for each factor separately, from the state of
material development to structural geometry over to the influence of the
ambient environment, we sequentially vary the value for each factor within
experimentally achievable bounds and compare the resulting times. To this end,
we introduce

$\displaystyle\chi=\frac{t_{i}}{t_{cj}},$ (15)

with $t_{i}$ denoting the simulated time to reach the threshold humidity for a
given variation in the input factor. Eq. (15) provides a relative scaling
factor that indicates either accelerated ($\chi<1$) or hindered ($\chi>1$)
time for cold joint formation and allows the assessment of the severity of
surface drying. In the following subsections, we discuss each factor’s
influence and provide recommendations for possible cold joint mitigation.

###  6.1 Ambient conditions

The influence of the surrounding environment on the concrete structure
comprises the contributions of ambient temperature and humidity, both of which
fluctuate jointly throughout the day and season. While their effects on the
structure are coupled in real-life scenarios, the computational model enables
us to dissect their impact and consequences separately, see Figure 8 and 9.

Figure 8: Influence of the ambient temperature on the concrete surface drying
rate. (a) Schematic illustration of the affecting temperature. (b) Computed
relative time to cold joint formation $\chi$ for a variation in ambient
temperature. Figure 9: Influence of the ambient humidity on the concrete
surface drying rate. (a) Schematic illustration of the affecting humidity. (b)
Computed relative time to cold joint formation $\chi$ for a variation in
ambient humidity.

We consider an ambient temperature fluctuation in the range
$T_{\infty}\in[6,50]^{\circ}$C, corresponding to globally foreseeable extreme
temperatures encountered during 3D concrete printing. Calculated results for
$\chi$ indicate two drying regimes, see Figure 8(b). For typical temperatures
in the range between 6 to 30∘C, we observe no influence on the surface drying
rate within $\pm 5\%$ of $t_{cj}$. Such temperatures are commonly present if
the structure is shielded from direct sun exposure, as is usual in production
halls and laboratories. For ambient temperatures above $30^{\circ}$C, we
observe $\chi<1$, denoting an accelerated cold joint formation rate. In
particular, $\chi$ decreases by $\approx 3\%$ per $5^{\circ}$C. In summary,
calculated results demonstrate that isolated temperature fluctuation within
common temperature limits does not impact the kinetics of cold joint
formation.

Ambient humidity has a strong effect on surface drying kinetics. The
considered humidity range $H_{\infty}\in[0.3,0.7]\leavevmode\nobreak\ [-]$
covers the common limits encountered throughout the year. Calculated results
shown in Figure 9(C) indicate a linear dependence between $H_{\infty}$ and
$\chi$, namely $\chi\approx 10\%$ per 5% $H_{\infty}$. As a consequence, 3D
printing in a dry ambient poses a significant challenge regarding accelerated
surface drying, effectively shortening the time to cold joint formation.

Summarizing the effect of the ambient environment, low ambient humidity and
high ambient temperature accelerate surface drying and increase the risk of
cold joint formation. While shielding the structure from the sun helps to
lower the environmental temperature, active control of ambient humidity on the
building site or laboratory is probably just reserved for niche applications.

###  6.2 Humidity and heat exchange with environment

The uncovered surface of extruded concrete structures facilitates the transfer
of heat and humidity from the structure to the environment, see illustration
in Figure 10. Similar to Section 6.1, we here separately investigate the
contribution of each phenomenon to the surface drying rate.

Figure 10: Influence of the heat and humidity exchange between the structure
and its environment on the concrete surface drying rate. (a) Schematic
separation of both phenomena. (B & C) Computed relative time to cold joint
formation for a variation of heat and humidity transfer coefficients,
respectively.

We characterize the temperature exchange between the structure and its
environment using the heat transfer coefficient $h_{T}$, see Figure 10(a). For
exposed sides, $h_{T}\in[1,20]$ W/(m2 K), which represent typical values used
for unshielded concrete freely exposed to the environment [22, 23, 31, 24].
Calculated results of $\chi$ shown in Figure 10(b) indicate no noticeable
influence on the surface drying rate for $h_{T}>5$ W/(m2 K), implying that
heat exchange from an uninsulated concrete surface does not accelerate cold
joint formation. A time prolongation in their formation can be effectively
achieved by a reduction of $h_{T}$ below 5 W/(m2 K), indicating that the
addition of a heat-insulating layer atop the extruded filament surface would
help slow down surface drying. Assuming $h_{T}=10$ W/(m2 K) to represent the
heat exchange between the naked concrete structure and the ambient
environment, the addition of at least 4 mm of polystyrene ($\lambda=0.033$
W/(m K)) would be sufficient to delay their formation.

For humidity exchange, we consider a transfer coefficient
$h_{H}\in[0.25,8.00]$ mm/day. The lower end of the $h_{H}$ range corresponds
to gradual drying under stagnant conditions, e.g. structures positioned in a
protected environment. The upper range of the $h_{H}$ interval describes a
surface exposed to a moving airflow. Results of drying simulations shown in
Figure 10(C) indicate a bilinear drying regime with a transition around
$h_{H}=2-3$ mm/day. Under stagnant environmental conditions ($h_{H}<2$
mm/day), the rate of surface drying is strongly affected by the chosen $h_{H}$
with $\chi\approx 10\%$ per 0.1 mm/day. For extrusion with structures exposed
to forced airflow, rapid drying rate becomes a major threat. In such
conditions, a stable $\chi\approx 0.05-0.15$ denotes a very fast cold joint
formation. We therefore emphasize that precise quantification of $h_{H}$ is
crucial for accurate modeling surface drying rate.

As a practical recommendation, our modeling results indicate that any
protective measure applied to the exposed concrete surface strongly affects
its drying rate. Sealing the open surface with an impenetrable foil or
spraying an evaporation retardant will efficiently increase $\chi$, prolonging
the time before a cold joint forms. 3D printing under ambient conditions with
active air circulation should be avoided or, if imperative, the extrusion
should be carried out rapidly, minimizing the time between deposition of
individual layers.

###  6.3 Material state

Hydrating concrete generates measurable heat from the reacting binder, which
is concurrently accompanied by the development of a pore network on the
microstructural scale. Here we investigate how the material temperature and
the humidity diffusivity through the porous microstructure influence the
surface drying rate, see Figure 11(a).

Figure 11: Influence of a material state described by the material temperature
and diffusivity on the rate of cold joint formation. (a) Schematic separation
of both phenomena. (B & C) Computed relative time to cold joint formation for
a variation in material temperature and diffusivity, respectively.

In our fictitious concrete structure, the initial material temperature
$T_{mat}$ corresponds to a temperature resulting from raw material mixing,
pumping, extrusion, and partial cement hydration. This processing history
results in an increase of $T_{mat}$, offsetting it from $T_{\infty}$ at the
model evaluation time. We vary $T_{mat}\in[10,50]^{\circ}$C in our drying
simulations while maintaining $T_{\infty}=20^{\circ}$C. Calculated results
indicate a linear relationship between $T_{mat}$ and $\chi$, see Figure 11(b),
with $\chi\approx 10\%$ for every 5∘C in $T_{mat}$. Notably, it is interesting
to point out that if the concrete temperature is cooler than the ambient
environment, this effectively prolongs the time necessary for cold joint
formation. This can be conveniently achieved by cooling down water with ice
before mixing, rendering this recommendation easily applicable to every
building site or laboratory.

The exposed concrete surface can be re-saturated by water diffusion through
the internal pore network. To account for this phenomenon, we use the
coefficient $D_{1}$, which describes the rate of humidity diffusion through
the complex pore structure. We note that $D_{1}$ depends on the cement
fineness distribution and w/c ratio used [20, 52] and is affected by the
ongoing microstructural development, but consider a constant $D_{1}$ value for
simplicity in the simulations.

For the model evaluation, we assume a value $D_{1}\in[20,200]$ mm/day. Low
values of $D_{1}$ indicate a dense microstructure with unconnected fine pores,
hindering humidity transport from the structural core to the exposed outer
surface. This characterizes a microstructure of mature concrete. The opposite
spectrum of $D_{1}$ values corresponds to a highly porous medium with
interconnected pores, i.e. a fresh concrete state, whose microstructure
facilitates humidity transport within the material. Simulated results indicate
an almost linear relationship between $D_{1}$ and $\chi$. Low values of
$D_{1}$ result in a rapid formation of a dry, thin surface layer as the
material cannot transport the internal humidity, shortening the time to cold
joint formation. High values of $D_{1}$ denote a highly permeable material
that allows for a steady redistribution of internal water to the drying
surface, effectively prolonging the time to cold joint formation. However,
providing recommendations for selecting a unique value of $D_{1}$ for a
particular generic extruded structure is beyond the scope of this manuscript.

In summary, the chosen value for material diffusivity $D_{1}$ has a pronounced
effect on the rate of surface drying. However, since many factors influence
the diffusivity through the pore network, it becomes impossible to set a
recommendable initial value for model predictions. The chosen value of $D_{1}$
serves only for qualitative assessment of the cold joint formation but can be
adapted if a specific value becomes obtainable by other means.

###  6.4 Structural geometry

Extruded concrete structures typically exhibit a wall-like geometry with a
very large height-to-thickness ratio. While the extrusion nozzle typically
determines the cross-sectional width (i.e., wall thickness), the structural
height gradually increases following the sequential deposition of individual
filaments. We present the results of simulated drying kinetics for variable
structural geometry below, see Figure 12.

Figure 12: Influence of the structural geometry on the rate of cold joint
formation.

In our numerical study on the fictitious concrete structure, we varied both
structural dimensions over expectable ranges achievable in the construction
practice, i.e. $W\in[0.1,0.4]$ m and $H\in[0.7,2.0]$ m. Calculated results
show an approximately linear dependency of both structural dimensions with
$\chi$, in particular, $\chi\approx 3\%$ for $\Delta H=0.1$ m and $\chi\approx
13\%$ for $\Delta W=0.1$ m. Based on purely geometric considerations, an
increase in any structural dimension proportionally increases the exposed
surface area and cement content within the bulk concrete; see discussion in
Sections 6.2, 6.3. Both these factors shorten the time to form a cold joint.

In summary, the effect of structural geometry on the drying rate is non-
negligible. However, we acknowledge that the geometry of extruded structures
is typically constrained by civil engineering and architectural requirements
or technological limitations. The presented model, however, allows a rapid
assessment of any arbitrary structure with a wall-like geometry, making it a
suitable tool for assessing the threat of cold joint formation for a given
design.

##  7 Conclusions

Cold joints in extruded concrete structures form once the exposed surface of a
deposited filament dries prematurely and gets sequentially covered by a layer
of fresh concrete. This creates a material heterogeneity which lowers the
structural durability and shortens the designed service life.

This paper introduces a computational model for assessing the risk of cold
joint formation that accounts for the extruded structural geometry, structural
interaction with the environment, and ambient conditions. The model provides,
as an output, the time for cold joint formation, which allows us to assess the
severity of surface drying for a given combination of input parameters.

The main conclusions from our study can be summarized as follows:

  * •

Ambient conditions: Low ambient humidity $H_{\infty}$ and high ambient
temperature $T_{\infty}$ shorten the time to cold joint formation at a rate of
$10\%$ for $\Delta H_{\infty}=5\%$ and $3\%$ for $\Delta
T_{\infty}=5^{\circ}$C if $T_{\infty}>30^{\circ}$C, respectively. While
shielding the structure from direct sun exposure helps to control the ambient
temperature, a regulated increase of ambient humidity on the building site or
laboratory is probably reserved for niche applications.

  * •

Heat exchange with environment: Isolated heat exchange between a naked
concrete structure and its environment does not accelerate the surface drying
rate. Adding a thermal insulation layer (e.g., at least 4 mm of polystyrene)
atop the exposed surface effectively delays the time for cold joint formation.

  * •

Humidity exchange with environment: Surface drying is strongly affected by
humidity exchange between the structure and the environment. The humidity
coefficient $h_{H}$ increases the drying rate by 10% for $\Delta h_{H}=0.01$
mm/day in a stagnant environment. Sealing the open concrete surface with water
vapor barriers, temporarily adding impenetrable foils, or regular spraying
with water provides an effective means for keeping the concrete surface from
prematurely drying. 3D printing outdoors (under “open air” conditions) should
be avoided or, if imperative, carried out at least uninterrupted, minimizing
the time between the extrusion of individual layers.

  * •

Material temperature: Concrete temperature $T_{mat}$ strongly affects the
surface drying and shortens the time to cold joint formation at a rate of
$10\%$ for $\Delta T_{mat}=5^{\circ}$C. Cooling the mixing water before
extrusion provides a very efficient and effective means of prolonging the time
for cold joint formation.

  * •

Material pore structure: The material diffusivity remains an outcome of the
microstructural development rather than an input parameter for a concrete mix
design. Concretes with a denser microstructure exhibit a lower diffusivity,
translating into a faster cold joint formation due to the constrained water
supply from the structural interior onto the drying surface.

  * •

Structural geometry: Structural geometry affects the drying rate as $3\%$ for
$\Delta H=0.1$ m and $13\%$ for $\Delta W=0.1$ m. The presented model allows
the user to verify and compare the time for cold joint formation for any wall-
like structural geometry and provides a relative time to cold joint formation.

##  8 CRediT authorship contribution statement

Michal Hlobil: Conceptualization, Methodology, Formal analysis, Investigation,
Data curation, Writing - Original Draft, Visualization; Luca Michel: Data
curation, Formal analysis, Writing - Review & Editing; Mohit Pundir: Software,
Writing - Review & Editing; David S. Kammer: Writing - Review & Editing,
Resources, Supervision, Project administration, Funding acquisition

##  9 Declaration of Competing Interests

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  10 Data availability

The code used for the numerical simulations is available on ETHGitlab and the
generated simulation data has been deposited in the ETH Research Collection
database under accession code 10.3929/ethz-b-000675875.

##  11 Acknowledgement

D. S. K., M. H., and L. M. acknowledge support by the Swiss National Science
Foundation under grant 200021_200343. We are thankful to Yaqi Zhao and Dr.
Antoine Sanner for the helpful discussion and to Mikey Ronen for his feedback
on the manuscript.

##  Appendix A Derivation of governing equations

###  A.1 Transient heat conduction in concrete

The temperature field within a hydrating concrete structure is highly non-
uniform. While cement hydration internally generates heat throughout the
structure, the exposed surface is cooled down by heat exchange with the
ambient environment. This phenomenon may be conveniently modeled by
formulating a transient heat conduction problem with an internal heat source
over a given domain. The governing equation for the thermodynamic equilibrium
defined for a differential element of the structure requires

$\displaystyle\nabla\cdot\overline{\bm{q}}_{T}+\frac{\partial Q}{\partial
t}=0,$ (16)

where $\overline{\bm{q}}_{T}$ is the heat flux emerging from heat conduction
in [J/(s m${}^{2})]$ and $Q$ is the heat accumulated within a unit volume of
concrete in [J/m3].

For statistically homogeneous and isotropic materials, such as plain concrete
investigated in the present study, the heat flux vector
$\overline{\bm{q}}_{T}$ can be expressed using Fourier’s law of heat
conduction as a negative vector gradient of temperature, namely as

$\displaystyle\overline{\bm{q}}_{T}=-\lambda_{mat}(\alpha)\,\nabla T,$ (17)

where $\lambda_{mat}(\alpha)$ represents the heat conductivity of concrete in
[J/(s m K)], reflecting the microstructural development described by $\alpha$.

The total heat $Q$ accumulated within a unit volume comprises two heat rate
contributions: accumulated from conduction $Q_{c}$ and released from hydration
$Q_{h}$, both given in [J/m3], given as

$\displaystyle\frac{\partial Q}{\partial t}=\frac{\partial Q_{c}}{\partial
t}-\frac{\partial Q_{h}}{\partial t}=\frac{\partial Q_{c}}{\partial
T}\,\frac{\partial T}{\partial t}-\frac{\partial Q_{h}}{\partial
t}=c_{v}\,\frac{\partial T}{\partial t}-\frac{\partial Q_{h}}{\partial t},$
(18)

where $c_{v}$ stands for the volumetric heat capacity of concrete in [J/(m3
K)]. This follows from multiplying the specific heat capacity of the material
$c_{mat}$ in [J/(kg K)] by its volumetric mass density $\rho_{mat}$ in
[kg/m3].

###  A.2 Transient humidity conduction in concrete

Free (chemically unbound) water in the capillary pore network plays a central
role in microstructural formation. Here, we consider the relative humidity
$H_{mat}$ as an equilibrated humidity of the gaseous phase and the
interstitial liquid phase present within the pore network of the cementitious
microstructure. After the initial stages of cement hydration, the ongoing
microstructural development becomes controlled by water availability from
mutually unconnected pores, with water diffusion serving as a driving force
for the supply of ionic species. Thus, we formulate the conduction of humidity
as a transient diffusion problem with $H_{mat}$ acting as the driving
potential for the moisture field and with negligible moisture convection
through pores, given as

$\displaystyle\nabla\cdot\overline{\bm{q}}_{H}+\frac{\partial
H_{mat}}{\partial t}=-\frac{\partial H_{s}}{\partial t}+\kappa\,\frac{\partial
T_{mat}}{\partial t},$ (19)

where $\overline{\bm{q}}_{H}$ represents the humidity flux in [m2/s], the sink
term $H_{s}$ accounts for self-desiccation of the material, and
$\kappa=\frac{\partial H_{mat}}{\partial T_{mat}}$ is the hygrothermic
coefficient. Bažant and Najjar [36] propose

$\displaystyle\kappa=0.0135\,H_{mat}\frac{1-H_{mat}}{1.25-H_{mat}}\,.$ (20)

The self-desiccation follows as a drop in moisture content $w_{h}$ due to
ongoing cement hydration, given by

$\displaystyle w_{h}=w_{c,pot}\,m_{c}\,\alpha,$ (21)

where $w_{c,pot}$ stands for the amount of chemically-bound water necessary
for complete cement hydration, expressed as $w_{c,pot}=0.23$ kg of water per
kg of cement used [41], and $m_{c}$ stands for the mass of cement used in 1 m3
of concrete. The corresponding decrease of relative humidity due to self-
desiccation $H_{s}$ amounts

$\displaystyle H_{s}=\frac{w_{h}}{k},$ (22)

where $k=\frac{\partial w}{\partial H_{mat}}$ stands for the moisture capacity
of the material in [kg/m3] determined as the slope of the desorption isotherm,
assumed here as $320$ kg/m3.

The constitutive law defines the humidity flux as a negative vector gradient
of relative humidity $H_{mat}$ given as

$\displaystyle\overline{\bm{q}}_{H}=-D_{H}\nabla H_{mat},$ (23)

where $D_{H}$ represents the diffusivity coefficient in [m2/s]. Several
formulations for $D_{H}$ have been proposed, as summarized by the fib Model
Code 2010 [53]. Here, we consider the formulation proposed by Bažant and
Najjar [36] in the form of

$\displaystyle
D_{H}=D_{1}\,\left(\gamma_{0}+\frac{1-\gamma_{0}}{1+{\left(\frac{1-H_{mat}}{1-H_{c}}\right)}^{n}}\right),$
(24)

with parameters taken from [54] as $D_{1}=200$ mm2/day as the value of
diffusivity at full pore saturation ($H_{mat}\to 1$),
$\gamma_{0}=\frac{D_{0}}{D_{1}}=0.05$ [–] with $D_{0}$ as the minimum $D_{H}$
for $H_{mat}\to 0$, $H_{c}=0.75$ as the relative pore humidity for
$D_{H}=0.5\,D_{1}$, and $n=12$ characterizing the drop in spread of $D_{H}$.
The value of $D_{1}$ can be estimated for mature concrete as a function of its
average strength $f_{cm}$ [53] as

$\displaystyle D_{1}=\frac{10^{-8}}{f_{cm}-8},$ (25)

which provides an effective range of $D_{1}$ between $[200,20]$ mm2/day for
typical concrete strengths $f_{cm}$ between $[12,50]$ MPa. The impact of
$D_{1}$ on simulation results is shown in detail in Section 6.

##  Appendix B Parameters for the affinity-based hydration model

The analytical expression for the affinity-based hydration model given in Eq.
(6) uses $\alpha_{\infty}$ as the maximum-reachable degree of hydration in
cement paste. This parameter depends on the initial water-to-cement ratio
$w/c$ and the curing regime considered [55]. As such, $\alpha_{\infty}$ takes
the form of

$\displaystyle\alpha_{\infty}=\begin{cases}\frac{w/c}{0.42}\leq
1&\quad\text{(sealed)},\\\ \frac{w/c}{0.36}\leq
1&\quad\text{(saturated)}.\end{cases}$ (26)

However, the practically reachable limit for $\alpha_{\infty}$ for ordinary
Portland cements under common curing conditions and in reasonable time-scales
(up to years of hydration under normal temperature) amounts to $\approx 0.85$
[-], which will be used for model evaluation.

Table 1 summarizes the parameters of the affinity-based hydration model that
best fit the experimental data from isothermal calorimetry, as shown in Figure
2.

Table 1: Calibrated parameters for affinity-based hydration model for isothermal calorimetry measurements of various cement pastes. | CEM I Ladce | CEM I Ladce | CEM II B/LL | CEM I Ladce
---|---|---|---|---
| (391 m2/kg) | (391 m2/kg) + 20% quartz | 32.5 R Hranice | (273 m2/kg)
$B_{1}\leavevmode\nobreak\ [s^{-1}]$ | $142\cdot 10^{-6}$ | $109\cdot 10^{-6}$ | $67\cdot 10^{-6}$ | $104\cdot 10^{-6}$
$B_{2}\leavevmode\nobreak\ [-]$ | $408\cdot 10^{-5}$ | $1210\cdot 10^{-5}$ | $1829\cdot 10^{-5}$ | $477\cdot 10^{-5}$
$\eta\leavevmode\nobreak\ [-]$ | $610\cdot 10^{-2}$ | $550\cdot 10^{-2}$ | $332\cdot 10^{-2}$ | $608\cdot 10^{-2}$
$\alpha_{\infty}\leavevmode\nobreak\ [-]$ | 0.85 | 0.85 | 0.85 | 0.85
$Q_{tot}\leavevmode\nobreak\ [J/g]$ | 532 | 426 | 330 | 512

##  Appendix C Parameters for computational model evaluation

Table 2 summarizes the parameters used for the analytical hydration model,
which are used for model verification in Sections 4.2, 4.3, and 5.

Table 2: Used parameters for affinity-based hydration model for simulations. | Section 4.2 | Section 4.3 | Section 5
---|---|---|---
$B_{1}\leavevmode\nobreak\ [s^{-1}]$ | $134\cdot 10^{-6}$ | $142\cdot 10^{-6}$ | $142\cdot 10^{-6}$
$B_{2}\leavevmode\nobreak\ [-]$ | $689\cdot 10^{-5}$ | $408\cdot 10^{-5}$ | $408\cdot 10^{-5}$
$\eta\leavevmode\nobreak\ [-]$ | $573\cdot 10^{-2}$ | $610\cdot 10^{-2}$ | $610\cdot 10^{-2}$
$\alpha_{\infty}\leavevmode\nobreak\ [-]$ | 0.85 | 0.85 | 0.83
$Q_{tot}\leavevmode\nobreak\ [J/g]$ | 520 | 532 | 532

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DMUS-MP-21/06

Large-$N$ limit as a second quantization

Masanori Hanada

Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH,
UK

###### Abstract

We propose a simple geometric interpretation for gauge/gravity duality, that
relates the large-$N$ limit of gauge theory to the second quantization of
string theory.

Essay written for the Gravity Research Foundation 2021 Awards for Essays on
Gravitation.

Gauge/gravity duality à la Maldacena [1] is a promising framework for the
study of non-perturbative aspects of quantum gravity. An important open
problem is how the bulk geometry of string theory is encoded in gauge theory.
Given that the duality can work even for the matrix model, there is no doubt
that the color degrees of freedom play crucial roles.

In a somewhat different context, a natural mechanism of emergent geometry is
known. When $(p+1)$-dimensional maximally-supersymmetric SU($N$) Yang-Mills
theory is regarded as the low-energy effective theory describing $N$
D$p$-branes and open strings between them, $9-p$ adjoint scalar fields
$X_{I}(I=1,\cdots,9)$ contain the information of transverse $\mathbb{R}^{9-p}$
(Fig. 1) [2]. The action contains the interaction term $\int d^{p+1}x{\rm
Tr}[X_{I},X_{J}]^{2}$ that favors the simultaneously-diagonalizable
configurations. When all scalars are close to diagonal, i.e., $X_{I,ij}\simeq
x_{I,i}\delta_{ij}$, the $(i,i)$-elements
$\vec{x}_{i}=(x_{1,i},\cdots,x_{9-p,i})\in\mathbb{R}^{9-p}$ are regarded as
the coordinates of $i$-th D-brane. The off-diagonal elements correspond to the
open-string excitations, i.e., if the $(i,j)$-elements are large then a lot of
strings are excited between the $i$-th and $j$-th D-branes.

The Matrix-Theory conjecture [3] pushes this interpretation further in the
case of $p=0$, and claims this ‘eigenvalue = location’ picture defines the
target space in M-theory. If $M\times M$ sub-matrices take nontrivial values
and do not commute with each other, they would describe extended objects such
as a black hole. In the large-$N$ limit, various many-body states are realized
as block-diagonal matrix configurations. It suggests a natural possibility of
relating the large-$N$ limit of the matrix model and the second quantization
of the gravity dual.

Figure 1:  Emergent space from gauge-theory degrees of freedom, when
$(p+1)$-dimensional maximally-supersymmetric SU($N$) Yang-Mills theory is seen
as the low-energy effective action of D$p$-branes and open strings [2].
D$p$-branes are extended along $(x_{0},x_{1},\cdots,x_{p})$ in
$\mathbb{R}^{1,9}$. The transverse directions $\mathbb{R}^{9-p}$, parametrized
by $(x_{p+1},\cdots,x_{9})$, are described by adjoint scalars
$X_{1},\cdots,X_{9-p}$.

There was widespread skepticism regarding the applicability of this
interpretation to gauge/gravity duality, based on (i) a well-known argument
against the ‘eigenvalue = location’ picture [4, 5, 6], and (ii) the lack of an
apparent mechanism for the emergence of block-diagonal matrices. Furthermore,
(iii) ‘nothing’ and ‘something’ cannot be distinguished, because the sizes of
the blocks have to sum up to $N$; in this sense, although the Matrix-Theory
interpretation admits various multi-body states, it is not a genuine second
quantization yet. In this essay, we resolve these issues and propose a
geometric picture for gauge/gravity duality that relates the large-$N$ limit
of gauge theory to a second quantization of gravity.

The logic of Refs. [4, 5, 6] was as follows. Consider the action of the form

$\displaystyle S=\int d^{p+1}x{\rm
Tr}\left(\frac{1}{2}F_{\mu\nu}^{2}+\frac{1}{2}(D_{\mu}X_{I})^{2}+\frac{g^{2}}{4}[X_{I},X_{J}]^{2}+{\rm
fermion\ part}\right),$

where $F_{\mu\nu}$ is the field strength and $D_{\mu}X_{I}$ is the covariant
derivative. We focus on the ’t Hooft large-$N$ limit ($\lambda=g^{2}N$: fixed,
the energy $E\sim N^{2}$). Via the ’t Hooft counting, we obtain $\langle{\rm
Tr}X_{I}^{2}\rangle\sim N^{2}$. Hence typical eigenvalues of $X_{I}^{2}$ and
$X_{I}$ should scale as $N$ and $\sqrt{N}$, respectively. This scaling is not
expected from a dual gravity point of view; for $p<3$, the entire bulk region
where the weakly-curved gravity picture is valid is covered, and for $p=3$,
the sub-AdS region is covered. Furthermore, a similar argument shows that the
commutator $[X_{I},X_{J}]$ is far from zero. Hence D-branes and open strings
form a big fuzz filling a large portion of the bulk geometry. This argument
uses only the ’t Hooft counting, and hence, it applies to the low-energy
states including the ground state.

Ref. [7] showed this argument is not correct, and resolved the issue (i). Let
us take the D0-brane theory as an example. (Generalization to other theories
is straightforward.) The Hamiltonian is given by

$\displaystyle\hat{H}={\rm
Tr}\left(\frac{1}{2}\hat{P}_{I}^{2}-\frac{g^{2}}{4}[\hat{X}_{I},\hat{X}_{J}]^{2}+{\rm
fermion\ part}\right).$

Each ‘matrix’ $\hat{X}_{I=1,\cdots,9}$ consists of $N^{2}$ operators, and
$\hat{X}$ and conjugate momentum $\hat{P}$ satisfy the canonical commutation
relation. The states mapped to each other via SU($N$)-transformations are
identified.*** This is equivalent to considering the gauge-invariant Hilbert
space. See the following discussion and Ref. [7].  Refs. [4, 5, 6] assumed the
existence of c-number matrices $X_{I=1,\cdots,9}$ that satisfy ${\rm
Tr}X_{I}^{2}=\langle{\rm Tr}\hat{X}_{I}^{2}\rangle\sim N^{2}$. Implicitly,
they assumed a coordinate-eigenstate $|X\rangle$ that satisfies
$\hat{X}_{I,ij}|X\rangle=X_{I,ij}|X\rangle$ for all $I,i,j$. This assumption
is not legitimate, because low-energy states are not coordinate-eigenstates.
Rather, we should consider wave packets in $\mathbb{R}^{9N^{2}}$. The
‘matrices’ that carry the information of D-branes and strings are the center
of a wave packet. The location of the wave packet can move in
$\mathbb{R}^{9N^{2}}$ via gauge transformation, but the size and shape do not
change. The distribution of D-branes determined by the center of the wave
packet is well localized. In particular, there is a ground state in which all
D-branes are at the origin of $\mathbb{R}^{9}$, that is gauge-invariant and
localized around the origin of $\mathbb{R}^{9N^{2}}$. Various well-localized
wave packets around different points in $\mathbb{R}^{9N^{2}}$ exist, and the
overlap between two wave packets remains small if the centers of wave packets
are separated more than an $O(N^{0})$ value.

Figure 2:  A visual sketch of wave packets (disks with light colors) in
$\mathbb{R}^{9N^{2}}$. Although the center of a wave packet (points with thick
colors) can move in $\mathbb{R}^{9N^{2}}$ via the gauge transformation, the
shape does not change. There are SU($N$)-invariant states, including a single-
bunch ground state in which all D-branes are sitting at the origin of
$\mathbb{R}^{9}$. Wave packets related via the SU($N$) transformation, which
are regarded as the same state, are shown with the same color.

To resolve the issues (ii) and (iii), a close connection [8] between color
confinement in the large-$N$ gauge theory and Bose-Einstein condensation [9]
plays a crucial role. The canonical partition function of gauge theory with
gauge group $G$ at temperature $T$ is

$\displaystyle Z(T)=\frac{1}{{\rm vol}(G)}\int_{G}dg{\rm
Tr}(\hat{g}e^{-\hat{H}/T}),$

where the trace is taken over the extended Hilbert space containing non-
singlet modes. (The micro-canonical partition function can be obtained by
constraining the energy.) The operator $\hat{g}$ is the gauge transformation
corresponding to $g\in G$, and $\frac{1}{{\rm vol}(G)}\int_{G}dg\hat{g}$ is
the projector to the singlet sector.††† The integral is taken over the Haar
measure. When $G$ is discrete, a sum is taken instead of the integral. Quantum
mechanics of $N$ indistinguishable bosons is a special case of gauge theory,
whose gauge group SN is permutations of particles [8, 10]. In the free limit,
the energy-einegstates are products of one-particle eigenstates, schematically
expressed as $|\phi_{1},\phi_{2},\cdots,\phi_{N}\rangle$. A permutation $g\in
G={\rm S}_{N}$ transforms such a state to
$|\phi_{g(1)},\phi_{g(2)},\cdots,\phi_{g(N)}\rangle$. Generic excited states
are not invariant under any nontrivial permutation, and hence, only
$g=\textbf{1}$ has a nonzero contribution to the partition function. On the
other hand, the ground state is invariant under any permutation, and there is
an enhancement factor $N!$ that corresponds to ${\rm vol}(G)$. This
enhancement triggers the Bose-Einstein condensation [9], and the Bose-Einstein
condensate is characterized by the invariance under SN [10]. In the same
manner, the confining ground state of ${\rm SU}(N)$ gauge theory is invariant
under $G={\rm SU}(N)$, and receives the enhancement factor ${\rm vol}(G)$ that
triggers the color confinement [8].‡‡‡ More precisely, $G$ is the set of all
${\rm SU}(N)$-valued functions $g(\vec{x})$, where $\vec{x}$ is spatial
coordinates in QFT.

This mechanism of color confinement, combined with the ‘eigenvalue = location’
interpretation, naturally resolves the issues (ii) and (iii). For
concreteness, let us consider a large-$N$ confining gauge theory with the
first-order confinement/deconfinement transition, e.g., 4d Yang-Mills on S3.
To deconfine all colors, the energy of order $N^{2}$ is needed. What if
$E\sim\epsilon N^{2}$, where $\epsilon$ is of order $N^{0}$ but much smaller
than $1$? Then the excitations coalesce to form an $M\times M$ block, where
$M\sim\sqrt{\epsilon}N$. This is because a large symmetry SU($N-M$) can be
preserved in this way, which leads to a large enhancement factor ${\rm
vol}({\rm SU}(N-M))$. This SU($M$)-partial-deconfinement [11, 12, 13] can
occur both at weak- [14] and strong-coupling regions [15]. In gauge/gravity
duality, the partially-deconfined phase is a natural counterpart of the small
black hole [11]. What if there are multiple excitations with smaller energy
(say $E\sim N^{1}$ or $N^{0}$), which are localized in separate locations in
the emergent space (and, possibly, also in the spatial dimensions of QFT)?
Then each of them should clump to a small block in the color space, such that
a large enhancement factor appears, leading to the block-diagonal structure.
It is natural to regard each block as an extended object such as fundamental
string or small black hole, that moves in the emergent space and interacts
with others. In the large-$N$ limit, various many-body states can be realized
as configurations with multiple excited blocks. The number of blocks and their
sizes can take arbitrary values. Therefore, the large-$N$ limit of gauge
theory can serve as a second quantization of the gravity side.

The above reasoning is based on generic kinematical features of gauge theory.
To establish a quantitative correspondence between gauge theory and gravity,
we have to solve theory-specific strongly-coupled dynamics. In addition to
traditional quantum Monte Carlo simulations (e.g., Refs. [16, 15]), new
technologies such as deep learning [17] and quantum simulation [18] might be
useful. If the second-quantization picture is valid indeed, we might be able
to decode the hologram and understand nontrivial processes such as the
formation and evaporation of black hole (Fig. 3).

Figure 3: Formation and evaporation of a small black hole, in the second-
quantized picture proposed in this article. Similarly to Matrix Theory [3],
the block-diagonal structure is manifest after gauge fixing. Small excited
blocks describe small objects such as gravitons. They merge together and form
a large resonance, which is interpreted as a black hole. Eventually black hole
decays by emitting small objects.

Acknowledgments

The author would like to thank Hidehiko Shimada and Bo Sundborg for extremely
useful comments. He was supported by the STFC Ernest Rutherford Grant
ST/R003599/1.

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