---
abstract: 'A new experimental technique to control crystal growth in confinement and pressure of a solid wall on the growing crystal has been developed and applied to calcite. At low contact pressure a cavity forms, the growth rim undergoes a transition from smooth to stepwise dynamic causing fast, wobbling growth at the confined surface. When contact pressure is increased to 10 kPa the wobbling growth stops, the growth rim becomes smooth again and the growth of the confined surface relaxes to a rate below the detection limit of the measurements. A new, complete theoretical description of the process is presented.'
author:
- 'Lei Li, Felix Kohler, Anja R[ø]{}yne, Dag Kristian Dysthe'
bibliography:
- 'limits.bib'
title: Limits to crystallization pressure
---

Introduction
============

Crystallization pressure has been studied over 150 years since Jean Lavalle firstly observed a growing crystal exert a pressure along the growing direction in 1853. [@Jean1853] Field observations have also indicated that growing crystals are able to exert a pressure which could lead to break mineral rocks and building stones. [@Watts1978] The crystallization force is so important in weathering engineer [@Scherer2004a; @Rodriguez-Navarro1999], cement formation [@Flatt2007] and geophysics [@Maliva1988; @Fletcher2001]. But the process of crystal growth under confinement and cavity deformation on the confined surface is still not clear enough.

In 1905, encouraged by Lavalle’s research, Becker and Day succeed showing that a centimeter sized Alum crystal could raise one kilogram weight for hundreds of micrometers when the crystal grows in a saturated solutions subject to evaporation. [@Becker1905], unfortunately, a quantitative measurements of crystallization surface stress are not feasible due to the difficulty of measuring the area of growth rims in their setup. Later Correns first derived a expression of crystallization pressure as a function of solution supersaturations. [@Correns1949] In their results, crystals exert a 7 mega pascals pressure as growing in a double saturated solution. Since Correns published his study, people test different crystals to observe the crystallization pressure. Additional experiment were carried out by several authors with important conclusions.In order to grow a crystal upon its loaded surface continuously, a nano-scaled solution film must exist to separate the loaded face from its constraint, otherwise deposition of matter and growth in the region of contact is impossible. The nano-scaled solution film is the important diffusion path for exchange of ions between the loaded crystal surface and solution. [@Taber1916; @Weyl1959; @Scherer1999] R[ø]{}yne and Dysthe [@Royne2012b] finished the latest experiment with crystal sodium chloride, which shows the average stress on the crystal surface is 10 to 1 millions times weaker than the Correns’ theoreticall equilibrium stress. [@Royne2012b]

![Left: The pattern used in the experiment. It has five inlets I-V to vary the CaCO$_3$ concentration which help to nuclei and grow the calcite. Inlet VI is the upper layer, which used to control the pressure on the calcite. Right: the cross section of the channel.[]{data-label="fig:calcitesetup"}](CrystalPress.png){width="8cm"}

Materials and Methods
=====================

We report experiments designed to control calcite growth in situ with a stable supersaturation and a high degree of control of the pressure at the confined surface. The topography of the confined calcite surface is recorded during the whole growth process by high resolution reflection interference contrast microscopy. The CaCO$_3$ concentration in the experiments was 0.801$\pm$0.002 mM and the saturation index $\sigma=0.44$ was calculated as previously reported [@Li2017b]. The microfluidic network, flow control system and the calcite growth solutions have been described in detail [@Li2017b]. The details of reflection interference contrast microscopy and imaging have been explained in detail  [@Li2018]. The microfluidic device is exactly as described previously except for the addition of a pressure control channel.

Microfluidic device with pressure control channel
-------------------------------------------------

![A,B: The interference images of calcite loaded with different control fluid pressures $P_2$. C: the height of calcite surface along the white line in the image A and B. Add legend, change “depth” to “h”[]{data-label="fig:Crystaltilt"}](Crystaltilt1.png){width="8cm"}

The microfluidic device consists of two layers as shown in Figure \[fig:calcitesetup\], one layer for fluid flow and one layer for pressure control. The lower layer has the same layout as earlier, but is only 29 $\pm$ 0.3 $\mu$m deep. It includes 5 inlets I-V and one outlet to control nucleation and growth of a calcite crystal in the fluid flow. CaCl$_2$, H$_2$O and Na$_2$CO$_3$ solutions are pumped into the channel from inlets I,II,III and mixed by diffusion before reaching the growing calcite crystal. Inlets IV and V are used for higher concentration of CaCl$_2$ and Na$_2$CO$_3$ solutions during the calcite nucleation. Over the lower calcite growth channel, the upper channel is used to press the calcite by bending a 6$\pm$0.3$~\mu$m thick PDMS membrane. The upper channel is filled with saturated calcium carbonate solution.

When the calcite is located and growing on the PDMS membrane in the lower channel, we start to increase the control pressure $P_2$ slowly until a corner of the calcite reaches the cover glass at $P_2=19$ kPa as shown in Figure \[fig:Crystaltilt\] A. The calcite is at this pressure tilted along the white line with a maximum distance of 560nm. In order to bring the calcite surface parallel to the cover glass surface, P$_2$ is increased to 20 kPa as shown in Figure \[fig:Crystaltilt\] B. The average distance along the white line is then 57 nm and the average distance $h$ of the whole surface is 30 nm. As documented previously [@Li2017b] the disjoining pressure $p$ between calcite and a glass surface at distance $h=30$ nm is $p=20\pm 10$ Pa. This indicates that almost all of the applied fluid pressure $P_2$ is used to deform the PDMS membrane enough to achieve full contact between the calcite and the glass. The pressure change $\Delta P_2=1$ kPa caused an average displacement of the crystal of 280 nm, thus the differential pressure loss in deforming the membrane is $dP_2/dh\approx 3$ Pa/nm. Further fluid pressure increase can displace the crystal at most 30 nm over which the pressure loss in deforming the membrane will be no more than about 90 Pa.

![Confined calcite surface evolution at low pressure and high pressure. Top 6 images: evolution from $t=0$ when saturation index of the solution is increased from 0 to 0.44, applied pressure is 20 kPa and contact pressure is 20-200 Pa. Bottom 6 images: evolution from $t=t_1=90$ min when applied pressure is increased to 30 kPa and contact pressure is $p>100$ kPa.[]{data-label="fig:calcitepress"}](Progress1.png "fig:"){width="9cm"} ![Confined calcite surface evolution at low pressure and high pressure. Top 6 images: evolution from $t=0$ when saturation index of the solution is increased from 0 to 0.44, applied pressure is 20 kPa and contact pressure is 20-200 Pa. Bottom 6 images: evolution from $t=t_1=90$ min when applied pressure is increased to 30 kPa and contact pressure is $p>100$ kPa.[]{data-label="fig:calcitepress"}](ProgressA.png "fig:"){width="9cm"}

Fluid film thickness, disjoining pressure and diffusion
-------------------------------------------------------

The images of the confined crystal interface have local intensity $I$ that depends on the fluid film thickness or local distance $h$ between the glass surface and the crystal: $I=I_0+I_1cos(4\pi hn/\lambda + \pi)$, where $n=1.33$ is the refractive index of water, $\lambda=550$ nm is the wavelength of our light source, $I_0$ is the background intensity and $I_1/I_0$ is the contrast. Because $I_0$ varies across the image due to refractions at other surfaces of the crystal, there is an uncertainty of about $\pm 10$ nm in the determination of contact ($h=0$). The accuracy of the determination of upwards growth $\Delta z=z(t+\Delta t)-z(t)$ equals the precision of $\pm 1$ nm.

In appendix A we have used available disjoining pressure data and diffusion data to obtain empirical fits that we may use to interpret our data. The disjoining pressure fit is $$p(h)=9\cdot 10^6 (h-h_0)^{-8} + 150e^{(h_0-h)/\delta},$$ where the effective minimum distance between the rough glass surface and an atomically flat calcite surface is $h_0=11.3$ nm and $\delta=7.85$ nm is the Debye length and $p$ is given in in Pa. The fit of the calcite diffusion coefficient $D_c$ is $$D_c/D_{c,\infty} =1-\exp(-(h-0.8)/3.2)$$ where $D_{c,\infty}=7\cdot 10^{-10}$ m$^2$/s is the bulk value. The measurement of fluid film thickness thus allows us to calculate the disjoining pressure $p(h)$ and diffusion $D_c(h)$ in the crystal-glass contact.

Experimental results
====================

![The calcite size increase along the growth process on 4 edges and Z direction.[]{data-label="fig:calcitesize"}](growthrates.png){width="9cm"}

We have succeed to nucleate and grow calcite crystals attached to the deformable membrane in several experiments. We have proceeded by increasing the control pressure $P_2$ to form a confined calcite-glass interface under pressure. All experiments have given qualitatively the same results but we focus here on the experiment where the contact stresses could be determined quantitatively and thereby be analysed properly.

While bringing the crystal into contact with the glass, the flowing fluid composition is kept at $c_{CaCO_3}$=0.05 mM, $\sigma$=0. Figure \[fig:Crystaltilt\] shows how the crystal is brought into contact with the glass surface with a fluid control pressure $P_2$=200 mbar=20 kPa and pressure transmitted to the upper calcite surface is $P_t=20\pm 10$ Pa. The flowing fluid composition is changed to $c_{CaCO_3}$=0.08 mM, $\sigma$=0.44 at time $t=0$. Figure \[fig:calcitepress\] shows the evolution of the confined crystal surface when growing outwards and downwards (perpendicular to the glass surface and image plane). The area $A$ of the crystal parallel to the glass surface grows continuously.

At time $t=0$ the calcite surface was flat (Figure \[fig:calcitepress\] A, top) and glass-calcite contact pressure $p=P_t=20\pm 10$ Pa is the same as the pressure transmitted to the upper calcite surface. The calcite surface was confined and the Ca$^{2+}$ and CO$_3$$^{2-}$ diffusion is limited in the confined solution film and the first contact growth rim was growing Figure \[fig:calcitepress\] B. As the calcite grows at the confined surface, a cavity appears on the calcite surface, a confined growth transition that we have already explained in detail [@Li2017b; @Kohler2018]. Along the rim of the crystal the crystal grows and the dark part of the growth rim signifies a small distance $h$ and that this area, $A_c$, of the crystal is load bearing. One observes in images C-F that less than 10% of the crystal area is load bearing, that is $A_c/A<0.1$. This means that the contact pressure increases at least tenfold from $p=20$ Pa to $p=P_tA/A_c\geq 200$ Pa. The growth rim is divided into different domains separated by large steps. Which domains of the rim that are load bearing is changing with time. This causes the crystal to “wobble upwards” in a manner already reported [@Li2018].

![The probability of confined solution film thickness during the experiment. Blue: the probability of confined film thickness after the calcite stop to grow upwards during 140 minutes to 180 minutes. Green: the probability of confined solution film thickness when the calcite is growing upwards during 1-140 minutes.[]{data-label="fig:crystalGhist"}](crystalG_filmhist.png){width="9cm"}

After 1.5 hours growth with saturation index $\sigma =$0.44, the pressure in the upper channel is increased from $P_2=20$ kPa to 30 kPa and the saturation index is kept constant. Since the PDMS membrane did not move the crystal upon this pressure change the transmitted pressure increased by $\Delta P_t=\Delta P_2=10$ kPa and the contact pressure is increased by $\Delta p=\Delta P_tA/A_c\geq$100 kPa to $p\geq$100 kPa.

During the three hours with high saturation index, $\sigma$=0.44, the outer rims of the calcite crystal grew at a constant rate as can be seen in the upper plot of Figure \[fig:calcitesize\]. The upwards growth shown the lower plot of Figure \[fig:calcitesize\], however goes through three distinct phases: The first 13 minutes the confined surface grows to accommodate the contact and tilting the crystal slightly, then there is a steady upwards growth of 2.6 nm/min. The pressure increase pushes the crystal 3 nm downwards and then at contact pressure $p\geq$100 kPa the vertical growth slows down exponentially and comes to a complete halt. We continued to let the crystal grow under this load for 12 hours more, but there was no further growth upwards within the accuracy of our technique of $\pm 0.5$ nm. The lower 6 images of the crystal in Figure \[fig:calcitepress\] shows that the growth rim that was split up in domains with steps between grows to form a smooth calcite rim in contact with the glass surface all around the crystal rim. During the first 30 minutes of this reformation of the rim the crystal growth is still pushing the crystal up against the applied load, after that the rim in contact widens as the outer edges of the crystal continue to grow. We have measured the distance $h$ between the growth rim and the glass along the outer perimeter during the low pressure and high pressure periods. Figure \[fig:crystalGhist\] shows the dramatically different distribution of fluid film thicknesses in the two periods. The confined film thickness curve at low pressure has two peaks, one is around 10-20 nm and another is around 40-50 nm. That is the ’wobble’ growth style we mentioned in our previous paper [@Li2017b]. And the confined film is supported by the disjoining pressure. When the pressure was increased the confined film thickness is less than 10 nm as shown in the blue curve.

Roughness of glass support
--------------------------

![AFM topographic image of 10x10 $\mu$m$^2$ and the corresponding histogram of heights with a Gaussian fit with width $0.2$ nm.[]{data-label="fig:glassrough"}](glass_rough_comp.png){width="9cm"}

The calcite surface grows by spreading molecularly flat unit planes one step at the time. The surface can thus be atomically flat over areas of several $\mu$m$^2$, but we do also observe growth instabilities resulting in large steps on the calcite surface. The glass, however is an amorphous silica that is inherently rough on the atomic scale. Figure \[fig:glassrough\] shows the results of AFM measurements on a glass coverslip as used for the microfluidic chambers. The 10x10 $\mu$m$^2$ image shows a rough surface with less than one nanometer peaks and throughs. The corresponding histogram of heights from the measurement is fit with a Gaussian distribution with width $0.2$ nm.

Theoretical model of experiments
================================

![Force balance and mass balance for force of crystallization. Constant force $F$ (solid) and constant growth rate $dz/dt$ (dashed) lines are drawn in the space of fluid film thickness $h$ and growth rim width $w$. Constant force contours are marked with $\log_{10}(F)$ and constant growth rate contours are marked with growth rate in nm/min.[]{data-label="fig:modelFoX"}](modelFoX.png){width="9cm"}

Thermodynamics
--------------

The saturation index $\sigma$ is related to the chemical potential of the solution: $$\sigma=\frac{\Delta\mu}{kT}=\ln(\frac{a_{Ca^{2+}}a_{CO_3^{2-}} }{K_{sp}}),$$ where $a_{Ca^{2+}}$ and $a_{CO_3^{2-}}$ are the ion activities and the solubility product $K_{sp}=10^{-8.54}$ was found by @Teng2000 [@Teng2000] to correspond to when spirals on the 10$\bar{1}$4 surface stopped growing. We have used PHREEQC [@Charlton2011] to calculate $\sigma$.

A normal stress, or pressure, $p$ on a solid surface contributes with a factor $pv$ to the chemical potential of the solid, where $v$ is the molecular volume of the solid [@Dysthe2014]. Thus the chemical potential difference between the solid and the solution that drives either growth ($\Delta\mu>0$) or dissolution ($\Delta\mu<0$) is: $$\Delta\mu=kT\sigma - pv.$$ One may then immediately calculate that a solution with saturation index of 0.44 is in equilibrium with a calcite surface subject to a pressure of $p=35$ MPa. This is 350 times larger than the pressure $p=100$ kPa at the calcite-glass contact when the pressure was increased. This also signifies that the pressure increase only changed the chemical potential difference $\Delta\mu$ driving the growth at the confined surface by 0.3%. We may safely conclude that the growth rate did not change due to the change in thermodynamic driving force!

Kinetics
--------

![A: In situ imaging of 26 h of growth of Crystal A. B: The confined calcite surface. The cavity grows layer by layer on the confined growth rims. We could measure rim width $\bar{w}$. C: upper, the purple dots are the measured growth rim width from the confined surface. The dark line is the maximum width $\bar{w}$ of different weight. The red line is the minim $\bar{w}$ from the mass and force balance. Down: the purple dots are the measured height from the calcite surface to cover glass. the dark line is the calculated height by the mass and force balance. []{data-label="fig:crystalA"}](crystal12_merge-small3.png){width="8cm"}

In order to model the growth of the confined surface we will simplify the geometry. We assume that the crystal surface is square with sides of length $L$ and width $w$ of the growth rim. The contact area between the crystal and the glass is then $$A_c=4w(L-w).$$ The total force $F=F_w+F_t$ of the crystal on the glass has a contribution from the weight of the crystal $F_w=HL^2(\rho_c-\rho_f)g$, where $H$ is the height of the crystal, $\rho_c$ is the crystal density, $\rho_f$ is the fluid density and $g$ the gravitational acceleration and from the force transmitted by the membrane, $F_t=L^P_t$. The balancing force is supplied by the interaction between the calcite and the rough glass surface. Because the solution is bound to the charged calcite and glass surfaces the surfaces repel each other even when there is a fluid film of some thickness $h$ between the surfaces. $$4w(L-w)P(h)=F.$$ The ions that diffuse into the confined region of width are all consumed by the growth of the confined surface. Therefore there we may write the mass balance between the diffusion flow $I_D$, and the growth flow, $I_g$ $$I_D=-Dh\frac{\Delta c}{w}=w\rho \frac{dz}{dt}=I_g$$ where $D$ is the diffusion coefficient, $c$ is the concentration, $\rho$ is the solid density and $dz/dt$ is the growth rate. This relation gives another relation between $w$ and $h$ that depends on the growth rate and diffusion rate: $$w=\sqrt{\frac{\Delta cDh}{\rho dz/dt}}$$ The force balance and mass balance for our experiments are presented in Figure \[fig:modelFoX\]. The first 90 minutes the logarithm of the force on the crystal is $\log_{10}(F)=-8.24$. The initial state of the crystal at time $t=0$ is that the growth rim is $w=L/2=8.5~{\mu}m$. Since the growth rate is 2.6 nm/min the growth rim evolves towards $w=1.5$ ${\mu}$m and the $h\approx 20$ nm. This is in good accord with the observations in the images (see Figure \[fig:calcitepress\]). One observes in the model that for the low initial force there is a large range of fluid film thicknesses $h$ that can support the load. When the pressure is increased to $P_t=10$ kPa ($\log_{10}(F)=-4.49$ in Figure \[fig:modelFoX\]), however the force curve is steep and it does not matter much what the width of the growth rim is. This model can also be used to explain the observations that $h$ increases and decreases when the rim width changes in steps [@Li2017b]. Figure \[fig:crystalA\] shows the model prediction together with the width and height data for one of the crystals described in [@Li2017b].

We still need to specify how the growth rate $dz/dt$ is determined. We observe from Figure \[fig:calcitesize\] that the growth rate $dr/dt$ on the outer surfaces is 9-23 nm/min, that is $dr/dz=4-9$. This is in the same range as observed previously [@Li2017b]: $dr/dz=15\pm 9$. Observations of formation of new layers in confined growth for NaClO$_3$ indicates that although the rim width $w$ is diffusion limited the growth rate is controlled by the nucleation rate. The nucleation probability per unit area $p_n=\Omega^m/\tau$ depends on the supersaturation $\Omega=(c-c_0)/c_0\propto$ which is proportional to the saturation index at this low supersaturation and where $\tau$ is a time constant and $c_0$ is the equilibrium concentration. In order to find the nucleation rate and growth rate on the free and confined surfaces we need to integrate over the whole surface available for nucleation: $$\begin{aligned}
\frac{dr}{dt}&\propto&p_n(\Omega)\Omega Z= \frac{Z\Omega^{m+1}}{\tau}\\
\frac{dz}{dt}&\propto&\int_0^wp_n(\Omega)\Omega(r)dr=\frac{w\Omega^{m+1}}{(m+1)\tau},\end{aligned}$$ where $Z$ is the height of the crystal which is close to $L/2$ since the crystal grows much slower on the confined surface. Consequently we find that $$\frac{dr}{dz}=\frac{(m+1)L}{2w}.$$ In our experiment $L\approx 17$ ${\mu}$m, $w\approx 1.5~{\mu}m$ and for $m=2$ we obtain $dr/dz\approx 15$ which is reasonably close to the observed value.

It remains to explain why the upwards growth stops. After the pressure is increased at $t=t_1=90$ min the growth rim quickly widens to widths between 2 and 4 $\mu m$. From the plot of the force balance and mass balance we find that this would correspond to growth rates of 0.3 to 1 nm/min. The nucleation argument we just presented should not be affected by changing the mean distance of the two surfaces from $h=20$ to 13 nm and as already shown, the thermodynamic driving force $\Delta\mu$ is not affected by the increased pressure. The rough glass surface should allow enough fast diffusion pathways to stay close to the bulk diffusion value. The only thing that is really changed between the two situations of low and higher pressure is that at higher pressure many points of the calcite surface are in full contact with the asperities of the glass surface and deforms it elastically. For a non-reactive, flat surface this only slightly deforms the shape of the disjoining pressure curve. For a growing surface, however the surface can accommodate to the shape of the rough glass surface and the adhesive regions around the contact points will grow. These adhesive regions have $h<1$ nm and $D/D_\infty<1/10$. The disjoining pressure curve of two rough but conform surfaces will have an adhesive well that becomes a force opposing the separating the two surfaces. This is directly analogue to the fact that setting CaCO$_3$ cements develop much stronger adhesion and yield strength when it is setting under pressure and the reactive grains are pressed together.

Conclusions
===========

A new experimental technique to control crystal growth in confinement and pressure of a solid wall on the growing crystal has been developed and applied to calcite. At low contact pressure a cavity forms, the growth rim undergoes a transition from smooth to stepwise dynamic causing fast, wobbling growth at the confined surface. When contact pressure is increased to 10 kPa the wobbling growth stops, the growth rim becomes smooth again and the growth of the confined surface relaxes to a rate below the detection limit of the measurements. A new, complete theoretical description of the process is presented.
