Condensation in One-Dimensional Dead-End Nanochannels

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Condensation in One-Dimensional Dead-End Nanochannels

Junjie Zhong, Seyed Hadi Zandavi, Huawei Li, Bo Bao, Aaron H.

Persad, Farshid Mostowfi, David Sinton

Version post-print

Citation (published version)

Zhong, J., Zandavi, S. H., Li, H., Bao, B., Persad, A. H., Mostowfi, F., & Sinton, D. (2017). Condensation in one-dimensional dead-end nanochannels. ACS Nano, 11(1), 304–313. https://doi.org/10.1021/acsnano.6b05666

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Publisher’s Statement This document is the Accepted Manuscript version of a Published Work that appeared in final form in ACS Nano, copyright © American Chemical Society after peer review and technical editing by the publisher. To access the final edited and published work see doi.org/10.1021/acsnano.6b05666

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1

Condensation in One-Dimensional Dead-End Nanochannels

Junjie Zhong, †, ¶ Seyed Hadi Zandavi, †, ¶ Huawei Li, † Bo Bao, † Aaron Harrinarine Persad, † Farshid Mostowfi, ‡

David Sinton†,*

† Department of Mechanical and Industrial Engineering, University of Toronto, M5S 3G8 Canada

‡ Schlumberger-Doll Research, Cambridge, Massachusetts, 02139 USA

* corresponding author: [email protected]

¶ These authors contribute equally

Abstract

Phase change at the nanoscale is at the heart of many biological and geological phenomena. The

recent emergence and global implications of unconventional oil and gas production from

nanoporous shale further necessitate a higher understanding of phase behavior at these scales.

Here, we directly observe condensation and condensate growth of a light hydrocarbon (propane)

in discrete sub-100 nm (~70 nm) channels. Two different condensation mechanisms at this

nanoscale are distinguished - continuous growth, and discontinuous growth due to liquid

bridging ahead of the meniscus - both leading to similar net growth rates. The growth rates

agree well with those predicted by a suitably defined thermo-fluid resistance model. In contrast

to phase change at larger scales (~220 and ~1000 nm cases), the rate of liquid condensate

growth in channels of sub-100 nm size is found to be limited mainly by vapor flow resistance

(~70% of the total resistance here), with interface resistance making up the difference. The

condensation-induced vapor flow is in the transitional flow regime (Knudsen flow accounting

for up to 13% of total resistance here). Collectively, these results demonstrate that with

confinement at sub-100 nm scales, such as is commonly found in porous shale and other

applications, condensation conditions deviate from the microscale and larger bulk conditions

chiefly due to vapor flow and interface resistances.

TOC Graphic

KEYWORDS: condensation, condensate growth, nanochannel, nanofluidics, hydrocarbon, flow

resistance, Knudsen flow

Phase change and fluid flow at the nanoscale play important roles in the environment,1-3 water

desalination,4 microelectronics,5, 6 targeted drug delivery7 and unconventional oil and gas

recovery.8-10 The latter has resulted in a large shift in global energy as hydrocarbons in shale and

tight oil formations are being produced from nano-sized pores.9, 11 To understand these systems,

it is crucial to understand phase change of nanoconfined reservoir fluids, particularly as reservoir

pressurization and depressurization alter the phase state.12 Despite the emergence and global

importance of unconventional oil and gas recovery, the fundamental physics governing phase

2

change at the nanoscale remains poorly understood, and the available classical theories and

simulation techniques have proven to be inadequate.11

As system size decreases, the classical approach for studying phase change breaks down.13 For

the condensation process specifically, the condensate growth will drive vapor flow at the

nanoscale. When the characteristic length of the system (h) becomes close to the mean free path

of the vapor molecule (l), the flow behavior can deviate from the bulk behavior predicted by the

Navier-Stokes equations,14 and affect the condensate growth process. Furthermore, at very small

scales the saturation pressure can change as a result of liquid-gas curvature leading to capillary

condensation15.

Many investigations of nanoscale condensation were carried out by gravimetric measurements of

adsorption,16 in fixed bed geometries,17 by interferometry,15, 18 through surface force

apparatuses,19, 20 or using environmental scanning electron microscopy (ESEM).21-23 Recent

advances in nanotechnology allow the investigation of phase change in deterministic structures

(e.g., nanochannels24) and direct observation of the whole process. Previous phase change studies

in nanochannels have examined liquid-to-vapor phase transition through evaporation-induced

cavitation,25 initiated by localized heating26 or capillary evaporation.27 The condensation process,

on the other hand, is physically distinct as a vapor-to-liquid phase transition, and there is a

profound need for targeted experiments and theory to better understand this phenomenon and

inform analytical methods.11, 13 Optically accessible nanofabricated channels can provide a much

needed window into the relevant phase phenomena at these scales and has been used, for

example, to observe the capillary condensation of water.28

Here, we study condensation and condensate growth of a pure substance in sub-100 nm

confinement using propane as a model fluid. Two different condensation mechanisms at the

nanoscale are distinguished and compared with previous simulation results:27, 29 continuous

growth, and discontinuous growth due to liquid bridging ahead of the meniscus. The directly

observed effects of pressure and temperature on the condensate growth and growth rate are

compared with those of larger channels (220 nm and 1000 nm) and those predicted by a

developed thermo-fluid resistance model. In contrast to phase change at larger scales, where heat

and mass transfer processes at the liquid-vapor interface dominate the condensation and

condensate growth process,30, 31 the rate of liquid condensate growth is found to be limited

mainly by vapor flow resistance at this scale. The findings in this paper demonstrate that

condensation and condensate growth dynamics in sub-100 nm dead-end nanochannels deviate

from bulk conditions. The findings will be beneficial in understanding many natural processes

and in optimizing related applications.

Results and Discussion

To study condensation in dead-end nanochannels, we fabricated chips by etching them (high-

reactive-ionic-etching, DRIE) on a silicon wafer and anodically bonding with glass (see Methods

section for details on the fabrication process). A schematic of the experimental setup is shown in

Figure 1A. As shown, the test region consisted of 10 identical dead-end nanochannels in one

group (each channel is 35 mm × 5.5 μm). Channels of this width are large enough to visualize

using optical microscopy, and small enough to avoid channel collapse during bonding. The

3

length in-plane (35 mm) is short enough for wafer fabrication, and long enough to ensure locally

uniform temperature within the observation region. SEM images of the cross-section of a 70 nm

high channel are shown in Figure 1B and AFM images of a channel profile are shown in Figure

1C.

Figure 1. Diagram of the experimental setup and SEM/AFM characterizations of 70 nm high channels. (A) Schematic of

the experimental setup. (B) SEM images of the cross-section of one nanochannel. (C) Typical AFM results of a series of

nanochannels. Each channel has a width of 5.5 μm and a nominal height of 70 nm (actually 68.0 ± 1.1 nm). The left image

shows the height profile of a single channel; the right image is a topological scan of 3 adjacent channels.

During condensation experiments, the pressure and temperature were closely controlled. The

fluid pressure at the inlet of the nanochannel was assumed to be the same as the pressure of the

fluid reservoir (piston) since the pressure drop in the mainchannel was negligible compared to

the drop in the nanochannels (detailed calculation is provided in the Supporting Information,

section 5). The reservoir fluid pressure was maintained within ±0.5% of the desired value. To

promote condensation near the dead-end (observation region), a temperature difference across

the chip was induced by pairing a chiller and heater clamped on top of the chip (Figure 1A). The

chiller was positioned over the dead-end region and was separated by a 1-mm thick insulation

layer from the heater at the inlet region. The chiller's temperature was adjustable, within ± 0.3 K,

and the heater's temperature was kept fixed at 473 K (±1 K).

In each condensation experiment, the pressure at the inlet of the nanochannel was initially set far

below the saturation pressure at the temperature in the chiller region. After the temperature of the

system reached steady state, the pressure was increased to the target pressure (response within 10

seconds) and was kept in steady-state operation for 30 minutes to detect condensation (see

Methods section for details on experimental procedures). Vapor condensation and condensate

growth were visualized using an inverted microscope in the bright-field mode. The vapor and

liquid phases were clearly distinguished by light intensity: the vapor phase appeared bright while

the liquid phase appeared darker, as shown in Figure 2A and Figure 2B. It is noteworthy that in

microscale or larger channels the opposite is generally observed, that is, the vapor phase is

darker compared to the liquid phase due to refraction.32, 33 However, in the subwavelength silicon

nanochannel, the vapor-Si interface provides a larger refractive index contrast and thus reflects

more incident light, appearing brighter than the liquid-Si case, as observed elsewhere.25, 34 The

4

liquid-vapor interface position and the condensate length were extracted directly from image

sequences using an image processing algorithm. Both temperature and pressure were varied in

the experiments: temperature ranged from 314.0 K to 347.9 K in steps of ~5 K (the details of

temperature measurements are shown in the Supporting Information, section 3), and pressure

ranged from 1.4 MPa to 3.4 MPa in steps of 0.2 MPa (reading from the pressure transducer at the

inlet of the mainchannel). All the experimental conditions are plotted in a P-T diagram in the

Supporting Information, section 2.

Condensation in Nanochannels

Figure 2. Direct observation of two types of condensation (Type A and Type B) in a 70 nm high channel at (A) 2.8

MPa/343.7 K and (B) 2.8 MPa/314.0 K. The time interval between each frame in (A) and (B) is 0.2 s (0.02 s for the zoom-

in image in (B)). (C) The number of channels with Type A and Type B condensation in a group of 10 nanochannels and

three repeats as a function of supercooling at 2.8 MPa.

Nanochannels within the chiller region of the chip were monitored at each pressure step for

30 minutes to detect condensation. Image sequences of the vapor condensation and condensate

growth in a nanochannel versus time (with a sampling speed of 50 frames per second) are shown

in Figure 2. The condensed liquid phase generally advanced from left to right (i.e., from the

dead- end) and the corresponding vapor flowed from right to left (i.e., from the inlet). Two

different condensation and condensate growth mechanisms were observed, depending on

conditions. Figure 2A shows a typical Type A condensation where the condensation started at (or

close to) the dead-end, and the single liquid column grew continuously towards the inlet of the

nanochannel as a result of condensation, i.e., in Type A condensation only one liquid segment

was observed. In Type B condensation, the initiation of condensation was also close to the dead-

end point. However, the progression of the liquid-vapor interface was not continuous. Rather, it

consisted of a repeating pattern of growth followed by liquid bridging ahead of the liquid column

front. An example of this pattern is shown in Figure 2B. As seen in the inset of Figure 2B, a

darkening in the channel was observed prior to the formation of a liquid bridge. Once the liquid

bridge formed, the original interface stopped growing, but the liquid bridge continued to grow in

5

both directions and, eventually, the vapor bubble between the two liquid-vapor interfaces

disappeared. The supplemental video (Supporting Information, section 8) shows examples of

Type A and Type B condensation in the nanochannels.

The number of nanochannels with each type of condensate growth depended on the degree of

supercooling, ∆𝑇 = 𝑇𝑆 − 𝑇𝐶, where 𝑇𝑆 denotes the saturation temperature under bulk conditions

at the inlet pressure, and 𝑇𝐶 denotes the temperature of the channels in the chiller region. Figure

2C shows the distribution of nanochannels in a group of 10 nanochannels with Type A and Type

B condensation as a function of supercooling. The number of nanochannels with Type B

condensation increased as the supercooling increased.

Additional tests were performed with larger channels heights (220 nm and 1000 nm, see AFM

results of channel heights in Supporting Information, section 17), with full details provided in the

Supporting Information, section 15. In these cases multiple liquid bridges ahead of the liquid

column formed, distinct from the 70 nm case where a maximum of one liquid bridge per channel

was observed.

Figure 3. Schematic showing two different types of condensation and condensate growth in the axial direction of a dead-

end nanochannel, in the height (nanoscale) dimension. (A) Initiation of the liquid column and film formation on the glass

and silicon surfaces. (B) The growth of the liquid column and growth of the adsorbed/condensed layers in front of the

liquid-vapor interface. (C) Type A condensation, where the liquid column grows continuously. (D) Type B condensation,

where the liquid bridge happens ahead of the liquid column. It is noteworthy that the cross sectional area is assumed

rectangular to illustrate the axial movement of the meniscus. Note that the actual cross sectional area of the channel is

shown in Figure 1B.

Figure 3 shows a schematic of the expected behavior in the height (nanoscale) dimension in

Type A (Figure 2A) and Type B (Figure 2B) condensation. Initially, the vapor will adsorb and

grow on the smooth walls of the channel (Figure 3A), forming liquid column when the two

adsorbed layers interact and touch. In our experiments, it was commonly observed that the liquid

column initiated at or near the dead-end, as shown in Figure 2A and 2B. These observations are

in agreement with the simulation results reported in the literature in which condensation also

started from the dead-end point in nanoslit or cylindrical pores.27, 36, 37 Subsequent to the liquid

column formation (Figure 3A), the liquid-vapor interface moves with a relatively constant

velocity of 𝑉𝑥1 toward the channel entrance due to the accumulation of propane molecules at the

liquid-vapor interface (as seen in Figure 2A and Figure 3B). Simultaneously, the growth of liquid

6

films on the top and bottom channel walls is expected (𝑉𝑦1, Figure 3B). When the supercooling is

relatively low (i.e., temperature in the chiller region is relatively high or pressure at the inlet of

the nanochannel is relatively low), the growth rate of the adsorbed/condensed layers (𝑉𝑦1) in front

of the liquid column is slow. In this circumstance (i.e. Type A condensation), the liquid column

grows continuously towards the inlet of the nanochannel, as indicated in Figure 3C. As the

supercooling is increased (temperature is decreased or pressure is increased), there is a higher

rate of adsorption on all nanochannel surfaces. Type B condensation results if the adsorption rate

causes the films on the silica and silicon surfaces to coalesce prior to the arrival of the liquid

column (see Figure 3D). The growth of the adsorbing layer leading up to the formation of the

liquid bridge was captured in the images shown in Figure 2B. In the post-processed images,

visible darkening appears prior to the formation of the liquid bridge for ~0.1 s, attributed here to

the growth of liquid films on the order of tens of nanometers thick. The formation of the liquid

bridge entrap a bubble during the condensate growth process (Figure 3D). This bubble collapses

asymmetrically as the vapor phase is replaced by the liquid phase (Figure 2B). The asymmetry is

a product of the density difference between liquid and vapor phase which necessitates bulk

motion from right to left (𝑉𝑚 in Figure 3D).

Liquid bridging mechanism

The formation of the liquid bridge during capillary condensation has also been predicted through

simulations.27, 29 Liquid bridging has been predicted in large and long nanopores only at high

temperatures (i.e., less supercooling).27 This behavior was attributed to the periodic undulations

of the interface of the adsorbed/condensed film. In contrast, we directly observe liquid bridging

more frequently at more supercooled conditions (Figure 2C), as would be expected for higher

condensation rates. In another recent simulation study of adsorption and condensation in

nanopores,29 it was found that after the detachment of the meniscus at the capped end, the

formation of a liquid bridge occurred close to the pore inlet. The authors concluded that the

probability of bridge formation was higher near the pore opening, particularly for longer

channels. It was found that after the detachment of the meniscus at the capped end, the formation

of a liquid bridge occurred close to the pore inlet.

In contrast to previous simulations, we observe that a liquid bridge intermittently formed close to

the liquid-vapor interface of the liquid column (~100 μm from the interface) in the dead-end

region within our observation window (~1.5 mm x 1.5 mm). There are several potential factors

that could influence the location and frequency of liquid bridge formation in our experiments: (i)

a gradient in nanochannel height or roughness, (ii) gradients of temperature and pressure along

the nanochannel, (iii) the local thermal undulation of adsorbed/condensed liquid films and (iv)

corner flow. Each of these factors is discussed in detail below.

Firstly, with respect to the gradient of channel height, we measured different locations along the

nanochannel (~10 mm as the interval) by AFM. Results showed there was no unidirectional

height gradient along the nanochannel. Moreover, the surface roughness was small (~1 nm) and

equally distributed (see Figure 1C). Therefore, the channel height variation is not expected to

play a significant role in the liquid bridging location observed here.

7

Secondly, regarding the variation in temperature and pressure along the nanochannel, several

aspects are noteworthy. The latent heat generated by condensation can locally disturb the

temperature field by up to 0.004 K (see Supporting Information, section 9), but this effect is

negligible compared to other sources of uncertainty. Another potential source of temperature

variation comes from the experimental setup: the maximum gradient in the chiller region was 0.6

K over the 1 mm field of view (at 314.0 K) (see Supporting Information, section 3), and thus

negligible (only 0.06 K) over the typical distance between the interface and the observed liquid

bridges. Regarding the pressure, the maximum pressure gradient calculated in the chiller region

was 0.025 MPa over the 1-mm field of view (at 3.4 MPa, see Supporting Information, section

10), and thus ~0.0025 MPa over the typical distance between the interface and the observed

liquid bridges. Both the temperature and pressure gradients described above are too small to

cause significant effects on the thickness of the adsorbed/condensed layer. In addition, the effects

of temperature and pressure gradients on the thickness of the adsorbed/condensed layer act in the

opposite direction. Specifically, the effect of the pressure gradient dominates that of the

temperature (~0.07% compared to ~0.02%), and the former would cause liquid bridge formation

in the opposite direction (i.e., close to the inlet) to that observed in our experiments. Therefore,

gradients of temperature and pressure are not responsible for the nature and location of the liquid

bridging observed here.

Thirdly, regarding the thermal undulations of liquid films, previous experimental results in

microconfinement have shown liquid bridge formation as a result of unstable surface waves.38-40

Similarly, in our experiments the formation of the liquid bridge close to the liquid-vapor

interface could also be attributed, in part, to the local undulation of the adsorbed/condensed film.

Lastly, corner flow may also influence the liquid bridging. Liquid tends to flow along the corners

to minimize the interfacial energy,41 reaching out from the liquid column in thin films along the

edges.42 The effect is stronger for thinner channels with effectively sharper corners (70 nm

geometry in Figure 1).43 In our experiments, we found the cross-channel liquid bridging occurs

less frequently in these smaller channels (70 nm) and more frequently in larger channels (220

and 1000 nm), for the same experimental conditions (experiments shown in Supporting

Information, section 15). We speculate that corner flow would tend to promote the continuous

condensate growth over liquid bridging, since vapor molecules would condense more readily

onto the liquid films (condensation coefficient of unity), rather than stick onto the silicon

substrate (sticking coefficient of 2×10-5)44 to form a new liquid bridge.

In summary, considering the four potential mechanisms outlined above, we expect a combination

of, or competition between, surface film undulation and corner flow contribute to the distinct

liquid bridge phenomena observed here in sub-100 nm channels. The exact mechanisms

governing liquid bridging in nanochannels, however, remains an open question.

Effects of pressure and temperature on condensate growth

The total condensate length versus time in the 70 nm high channel at 326.7 K and five different

pressures is shown in Figure 4A. For Type B growth cases, the total condensate length was

calculated as the sum of the liquid lengths in the nanochannel. Linear relations between the

condensate length and time were found for all conditions within our observation window. This

8

result illustrates that, regardless of the type of condensation progression (Type A or B), the net

rate of condensate accumulation was roughly constant in time (again, only within our

observation window). Specifically, the maximum deviation of condensate growth rate between

Type A and Type B condensation at the same thermodynamic condition was under 13% for all

cases, and growth rates are plotted vs. pressure in Figure 4B. Figure 4C shows the total

condensate length versus time in the 70 nm high channel at 2.8 MPa for five different

temperatures. Similarly, linear relationships between the condensate lengths versus time were

found for all temperatures within the observation window. Figure 4D shows the isobaric

condensate growth rate at different temperatures. The condensate growth rates in Figure 4B and

4D are calculated by fitting a linear regression line to the condensate length of Figure 4A and 4C

(the total measured condensate length versus time at eight different isobaric and isothermal

conditions in the 70 nm high channels are provided in the Supporting Information, section 4).

Figure 4. Condensate length and condensate growth rate as a function of pressure and temperature in 70 nm high

channel. (A) Condensate length versus time at 326.7 K (𝑻𝑪) and different pressures. Each measurement point is a result of

the average of 3 repeated experiments in the 10 nanochannels. Representative error bars are shown for 8 of the

measurement points at most (error bars for remaining data points are omitted for clarity). The time interval for each of

the two data points plotted is 0.5 s. (B) Condensate growth rate as a function of pressure calculated from the condensate

length in (A). (C) Condensate length versus time at 2.8 MPa (𝑷𝑹) and different temperatures (time interval of 0.5 s

between data points). (D) Condensate growth rate as a function of temperature calculated from the condensate length in

(C).

Accumulation of vapor molecules at the liquid-vapor interface during the condensate growth

process necessitates vapor flow through the nanochannel. To analyze the condensate growth in

the nanochannel, we employed a resistance model (see Figure 5). During the condensation

process, we consider three major resistances in series between the nanochannel entrance and the

condensed liquid phase: (i) flow resistance in the heater region of the nanochannel (RH), (ii) flow

resistance in the chiller region of the nanochannel (RC), and (iii) the resistance to crossing the

liquid-vapor interface (RI).

9

For flow resistance in the nanochannels and nanopores of this size, the mean free path of the

vapor molecules (l) becomes comparable to the dominant scale of the system (h in our cases). In

our experiments, the Knudsen number (Kn = l/h) for our 70 nm high channels was between 0.03

and 0.053 (see Supporting Information, section 11), which is in the transitional flow regime

(0.01 < 𝐾𝑛 < 1).48 In this regime, the Knudsen flow (i.e., net flow due to diffusion interactions)

must be considered in addition to the viscous flow.

Figure 5. Flow resistance model for condensate growth in the dead-end nanochannel.

The vapor flow resistance in the heater and chiller regions of the nanochannel are the

combination of two parallel resistances48: the resistance induced by the viscous flow (RV) and the

resistance induced by the Knudsen flow (RK):

𝑅𝛽 =𝑅𝑉𝛽𝑅𝐾𝛽

𝑅𝑉𝛽+𝑅𝐾𝛽=

1

ℎ2

12𝜇𝛽𝑔

𝐿𝛽+

2√2ℎ

3𝜌𝛽𝑔

𝐿𝛽√

𝑀

𝜋𝑅𝑇𝛽

(1)

where 𝜇 𝑔 and 𝜌𝑔 are the viscosity and density of the vapor phase, respectively, ℎ is the height of

the nanochannel, L is the channel length in the region, T is the temperature inside the channel, M

is the molar mass of the fluid, and the additional subscript 𝛽 specifies to the heater (H) or the

chiller (C) regions.

For the liquid-vapor interface resistance (𝑅𝐼), the Hertz-Knudsen equation is used to express the

condensation mass flux (𝐽𝐼 ) in terms of the difference between the vapor pressure and the

equilibrium pressure:49

𝐽𝐼 = 𝑁𝛼√𝑀

2𝜋𝑅(

𝑃𝑉

√𝑇𝑉−

𝑃𝐸𝑞

√𝑇𝑙𝑣) (2)

where Nα is an empirical constant discussed in more detail below, 𝑅 is the universal gas constant,

𝑇𝑉 and 𝑇𝑙𝑣 are the temperatures in the vapor phase and at the liquid-vapor interface, respectively

(𝑇𝑉 = 𝑇𝑙𝑣 = 𝑇𝐶 in our experiments), 𝑃𝑉 is the vapor pressure at the liquid-vapor interface, and

𝑃𝐸𝑞 is the equilibrium pressure which can be calculated from the Kelvin equation.50

Following the procedure outlined in the Supporting Information (section 5), the total resistance

(𝑅𝑇𝑂𝑇) is:

10

𝑅𝑇𝑂𝑇 =𝜌𝐶

𝑙

𝜌𝐻𝑔

ℎ2

12𝜇𝐻𝑔

𝐿𝐻+

2√2ℎ

3𝐿𝐻√

𝑀

𝜋𝑅𝑇𝐻

+𝜌𝐶

𝑙

𝜌𝐶𝑔

ℎ2

12𝜇𝐶𝑔

𝐿𝐶+

2√2ℎ

3𝐿𝐶√

𝑀

𝜋𝑅𝑇𝐶

+𝜌𝐶

𝑙

𝑁𝛼√

2𝜋𝑅𝑇𝑐

𝑀 (3)

where 𝜌𝐶𝑙 is the liquid-phase density at 𝑇𝐶.

The first, second and third terms in eq 3 represent the resistances in the heater region (𝑅𝐻),

chiller region (𝑅𝐶) and at the interface (𝑅𝐼), respectively. The condensate growth rate (𝑉𝐿) can be

written in terms of the difference between the reservoir pressure (𝑃𝑅 ) and the equilibrium

pressure (𝑃𝐸𝑞):

𝑉𝐿 =𝑃𝑅−𝑃𝐸𝑞

𝑅𝑇𝑂𝑇 (4)

Equations 3 and 4 can be solved simultaneously to predict the condensate growth rate as a

function of temperature and pressure (see Supporting Information, section 5). For isothermal

conditions (constant 𝑇𝐶 and 𝑇𝐻 ), eq 3 and eq 4 show a relationship between the condensate

growth rate (𝑉𝐿) and the reservoir pressure (𝑃𝑅). For a constant reservoir pressure (𝑃𝑅), these

equations predict a constant growth rate, which is similar to the trend found in the experiments

within the observation window (Figure 4A). When pressure is increased, based on our

experimental conditions, the gas density (𝜌𝐻𝑔

and 𝜌𝐶𝑔

) will increase more significantly compared

to the increase in viscosity (𝜇𝐻𝑔

and 𝜇𝐶𝑔

), and the liquid density (𝜌𝐶𝑙 ) remains almost constant,

which result in decreases in both 𝑅𝐻 and 𝑅𝐶. Meanwhile, 𝑅𝐼 is not affected by the increase in the

reservoir pressure (𝑃𝑅), and the total resistance (𝑅𝑇𝑂𝑇) will decrease (eq 3). The equilibrium

pressure (𝑃𝐸𝑞) remains unchanged since it is a property that depends on the channel height (ℎ)

and the temperature (𝑇𝐶), thus 𝑃𝑅 − 𝑃𝐸𝑞 will increase. Therefore, eq 3 and eq 4 predict that the

condensate growth rate (𝑉𝐿 ) increases with an increase in 𝑃𝑅 . Figure 4B shows the model

prediction (solid line) along with the experimental results in an isothermal condition (constant 𝑇𝐶

and 𝑇𝐻). Close agreement between the experimental results and our predicted condensate growth

rate is achieved by fitting the value of Nα to be 7.4× 10-5 (see Supporting Information, section 5).

We note, however, that for the 70 nm case the fit is relatively insensitive to the value of Nα, as

the interfacial resistance component as a whole is not dominant (see Supporting Information,

section 16).

For isobaric conditions (constant 𝑃𝑅), eq 3 and eq 4 show a relationship between the condensate

growth rate (𝑉𝐿) and the temperature in the chiller region (𝑇𝐶). For a constant 𝑇𝐶, these equations

predict a constant growth rate, which is similar to the trend found in the experiments (Figure 4C).

When 𝑇𝐶 is increased, the gas density (𝜌𝐻𝑔

and 𝜌𝐶𝑔

) will increase more significantly compared to

the increase in viscosity (𝜇𝐻𝑔

and 𝜇𝐶𝑔

), and the liquid density (𝜌𝐶𝑙 ) decreases, which results in

decreases in 𝑅𝐻 , 𝑅𝐶 and 𝑅𝐼, and further results in a decrease in 𝑅𝑇𝑂𝑇. However, the equilibrium

pressure (𝑃𝐸𝑞) will increase due to the saturation pressure (𝑃𝑆) increase (i.e., 𝑃𝑅 − 𝑃𝐸𝑞 decrease).

As a result, eq 3 and eq 4 predict that the condensate growth rate (𝑉𝐿) will decrease with the

increase in 𝑇𝐶 in an isobaric system. The condensate growth rate (𝑉𝐿) predicted from the flow

resistance model (solid line in Figure 4D) is in agreement with the experimental results (with 𝑁𝛼

= 7.4 × 10−5 as before). The nonlinear dependence of condensate growth rates on temperature

predicted by our model agree closely with the experiments (Figure 4D). The nonlinear behaviors

11

are the result of both flow and interface resistances changing nonlinearly with temperature (via

density and viscosity of vapor phase).

We emphasize that the value of Nα is empirically determined from considering all experimental

conditions (15 experiments with temperatures and pressures that ranged between 314.0 and 343.7

K, and 2.0 and 3.4 MPa, respectively, see Supporting Information, section 5). The value of Nα

that gives the best agreement between our model-predicted condensation rates and those

measured in the 70 nm high channels was (7.4 ± 0.9) × 10-5 (details of the uncertainty calculation

of Nα are given in the Supporting Information, section 6). As an empirical parameter in our

experiments, Nα groups the effects of different physical phenomena that occur near the liquid-

vapor interface. For example, in nanoconfinement the sticking of the propane to silicon at the

three-phase line might contribute significantly to the condensation at the interface (details

discussed in the Supporting Information, section 14). Thus, Nα in our model does not represent

the classical condensation coefficient since it amalgamates the effects of the potential

condensation coefficient at the liquid-vapor interface and the sticking coefficient at the solid-

vapor interface44 in addition to potential effects of corner flow that result from the channel

geometry.

Figure 6. Contributions of the interface and flow resistances are shown as percentages of the total resistance (eq 3) for

both the highest and lowest condensation rates observed under isothermal and isobaric conditions. The interface, viscous

flow, and Knudsen flow resistance contributions are shown as green bars, blue bars, and hatches, respectively (see

Supporting Information, section 7).

The modeling results indicate that flow resistance (i.e., first two terms of eq 3) accounts for

~70% of the total resistance (𝑅𝑇𝑂𝑇) in our experiments. The effect is more pronounced when the

pressure ( 𝑃𝑅) or temperature (𝑇𝐶) is decreased. For example, the contribution of flow resistance

to the total resistance is increased from 71% at 3.4 MPa to 74% at 2.0 MPa in isothermal

conditions, and from 67% at 343.7 K to 76% at 314.0 K in isobaric conditions (Figure 6).

Therefore, it is essential to take both flow resistance and interface resistance into consideration

when studying condensate growth in sub-100 nanoconfinement. In contrast, when the scale of

the system is increased from 70 nm to 220 nm and further to 1000 nm, the maximum ratio of the

flow resistance to the total resistance decreases from 76% to 31% further to 7% (based on our

experimental conditions, see Supporting Information, section 11). Thus at the micro- or

macroscale, flow resistance can usually be neglected, and only heat and mass transfer processes

at the interface dominate the condensation process.30, 31 In addition, in the 70 nm channels, the

Knudsen flow contribution is ~10% of the total resistance (8% to 13% depending on the

12

experimental conditions, as shown in Figure 6). This contribution decreases dramatically as the

channel height increases: we found it to be only ~1% at 220 nm and less than 0.1% at 1000 nm

(see Supporting Information, section 11, for details).

The present results demonstrate a deviation from bulk or microscale condensation growth that is

not attributable to capillary condensation. Capillary condensation is the early onset of liquid

formation at the nanoscale due to multilayer adsorption and occurs at pressures below the

saturation pressure or at temperatures above the saturation temperature. However, the kinetics of

capillary condensation is limited by the growth rate of the adsorbed layers, which is very slow at

conditions close to saturation.28, 52, 53 Moreover, previous experimental results showed that the

capillary condensation of water in graphitic pores with openings less than 10 nm is slow and

continues to occur for almost 3 days at room temperature.54 For the size of nanochannels

considered in this study (70 nm), the equilibrium pressure calculated from the Kelvin equation is

~0.5% less than the saturation pressure (𝑃𝐸𝑞 ≅ 0.995 𝑃𝑠, see Supporting Information, section 12),

but we did not observe any capillary condensation within 30 minutes due to the limitation of our

pumping accuracy (±0.5% of the set value). In fact, supercooling was always required to observe

the initiation of condensation in all cases tested here (see the Supporting Information, Figure S2).

The 70 nm channel size studied here represents a middle-ground in the context of nanoporous

shale reservoirs which have, in general, pore sizes that vary from single nanometers to hundreds

of nanometers. For instance the Bakken shale from Williston basin55 and the Pierre shale from

Colorado basin56 (both are significant US shale reservoirs) have dominant pore size within the

range of 60-110 nm and 40-90 nm, respectively. Collectively this work demonstrates that, in

pores on the scale of tens of nanometers in shale and related applications, phase-change

conditions deviate from bulk conditions due primarily to flow and interface resistances, rather

than capillary effects.

Conclusion

In summary, the condensation and condensate growth of propane in dead-end nanochannels are

experimentally investigated. Two different types of condensation mechanisms are distinguished

based on the number of condensed liquid segments, which is shown to be a function of

supercooling. The directly observed effects of both pressure and temperature on the condensate

growth and growth rate show very good agreement with those predicted by a suitably defined

thermo-fluid resistance model. The rate of liquid condensate growth in sub-100 nm nanochannels

is limited by vapor flow resistance (~70% of the total resistance), in contrast to larger scales (220

nm and 1000 nm) where the vapor flow resistance effect is greatly reduced. The vapor flow in

our experiments with 70 nm high channels is in the transitional flow regime where Knudsen flow

contributes up to 13% of the total resistance. The findings in this paper illustrate the

condensation and condensate growth of pure substance in the dead-end nanochannels, which will

be beneficial in understanding many natural processes and optimizing related applications. The

specific experimental conditions target tens of nanometer porous shale gas/oil formations, where

we found condensation conditions that deviate significantly from bulk conditions due primarily

to vapor flow and interface resistances.

Methods

13

Fabrication Procedure of Nanofluidic Chip

In the nanofluidic chip, there were 10 groups of parallel dead-end nanochannels (10 separate

nanochannels in one group and each channel was 5.5 μm in width, 35 mm in length) placed

perpendicularly along the source microchannel. This identical layout was used for chips with 70

nm, 220 nm and 1000 nm channel heights. The inlet and outlet holes were at the two ends of the

source microchannel symmetrically, which was used to fill the chip with liquid propane and to

control the pressure inside the nanochannels.

To fabricate the nanofluidic chip, the pattern was first generated using AutoCAD and then

transferred onto a 5-inch chrome mask through a mask writer (Heidelberg μPG 501) with a

resolution of 1 μm. Second, the pattern on the mask was transferred to the S1818 photoresist on

top of the 1 mm thick silicon wafer by standard photolithography methods, as shown in Figure

S1A. Third, a high reactive ion etching (DRIE) instrument (Oxford Instruments PlasmaPro

Estrelas100 DRIE System) was used to etch the nanochannels with a certain height in the wafer,

as shown in Figure S1B. Next, the microchannel with a typical height of 50 μm was patterned

and etched into the same silicon wafer by repeating the standard photolithography and DRIE, as

shown in Figure S1C and S1D. Afterward, inlet and outlet holes were drilled through the silicon

substrate at both ends of the microchannel, as shown in Figure S1E. After cleaning the silicon

wafer and 2.2 mm thick borosilicate glass in the piranha solution (H2SO4:H2O2 = 3:1) for 20

minutes (Figure S1F), the silicon wafer and the borosilicate glass were bonded together through

anodic bonding (AML AWB-04 Aligner Wafer Bonder) to seal the channels, as shown in Figure

S1G. The bonding process was carried out for approximately 20 minutes at 400 ̊C with a voltage

of 600 V and a current of 4 mA. Finally, the chip was diced into the desired shape using a dicing

machine (Disco DAD3220 Automatic Dicing Saw).

Experimental Procedures

The assembled nanofluidic chip was installed on a customized manifold enabling appropriate

sealing for high pressures. The chip and manifold were connected to a sample source cylinder

(research grade propane, Praxair 99.99%), pump (TELEDYNE ISCO MODEL 260D), and

piston cylinder (HIP 70C3-10-P) via tubing and valves. The tubing, pump, piston chambers and

valves were cleaned thoroughly before connecting to the nanofluidic chip. The entire system was

vacuumed at 2 x 10-7 MPa (PFPE RV8) for 3 hours (~10 m3/h as the displacement rate) before

each set of experiments to minimize the residual air in the system (the mass fraction of the air

was less than 10−4% of the mass of propane in the system during the experiment, see Supporting

Information, section 13). Research-grade liquid propane was filled with high pressure (4 MPa)

into the nanochannels. The temperature and pressure were then set below the liquid-vapor

saturation line to ensure vaporization of liquid propane (a P-T diagram of propane is provided in

the Supporting Information, section 2). The vapor pressure was provided and controlled by the

pump and monitored by a pressure transducer (PX409-3.5KGUSBH) at the inlet of the

mainchannel. An isolation piston cylinder was used between the pump and chip to avoid

contamination of the sample (Figure 1A). The temperature in the chip was controlled by a copper

block heater (33 mm along the channel direction and 50 mm perpendicular to the channel

direction) and chiller (10 mm along the channel direction and 50 mm perpendicular to the

channel direction) clamped on the top of the chip.

14

The test region, the field of view, (~1.5 x 1.5 mm) focused on the dead-end portion of the group

of 10 nanochannels. To ensure condensation initiated within test region, we introduced a

temperature difference across the chip by pairing the heater and chiller. The entrance region of

the nanochannels was set at a high temperature (𝑇𝐻) using electric heaters (OMEGA CIR-1030,

temperature was constantly set to 473 ± 1 K) inserted in the block heater controlled by PID,

which covered the first 29 mm length of the nanochannels. The copper dead-end portion of the

nanochannel was set at a lower temperature (𝑇𝐶). The temperature in this region was controlled

by a refrigerated circulator (Cole-Parmer Polystat® Standard 6.5L Heated Bath, temperature is

controlled between 278.2 ± 0.3 K to 368.2 ± 0.3 K) circulating water in the copper block chiller,

which covered the last 5 mm length of the nanochannels (the total length of the chiller was 10

mm). Both the heater and the chiller were in close contact with the silicon side of the chip using

thermally conductive paste (OMEGATHERM@201) for minimizing contact thermal resistance.

An insulation layer (Alumina Oxide Fiber, ~1 mm thick) was placed between them. A high-

speed camera (PCO 1200S, 50 frames per second) connected to the inverted optical microscope

(LEICA DMI 6000B) with a 10 x objective was used to observe the condensation directly and to

record the condensation growth process in the test region of the nanochannels.

For the isothermal condensation experiments (constant 𝑇𝐶), the temperature of the water bath

was set constant at 318.2 K and we waited 1 hour to reach steady-state thermal conditions. The

pressure was increased from 1MPa directly to the target pressure (1.4 MPa to 3.4 MPa, in steps

of 0.2 MPa), and for each pressure, we waited 30 minutes to detect any condensation. For the

isobaric condensation experiments, the pressure was kept at 1 MPa first, and temperature of the

water bath was increased from 303.2 K to 343.2 K (5 K as an incremental step). For each

temperature, we waited 1 hour to reach steady-state thermal conditions. Then we increased the

pressure from 1 MPa to 2.8 MPa (within 10 seconds). We also waited 30 minutes to detect the

condensation. Figure S2 shows the experimental conditions in the phase diagram of propane. As

seen there, hollow symbols indicate that condensation was not detected within 30 minutes of

observation while the solid symbols indicate that condensation occurred within that period. As

can be seen, supercooling was required to detect condensation within the 30 minutes of

observation time. Note that the abscissa in Figure S2 is the temperature in the chiller region of

the nanochannels (𝑇𝐶 ). This temperature is not necessarily the circulated water temperature

because of heat losses and heat transfer in the chip.

Acknowledgement

The authors gratefully acknowledge support from Schlumberger Canada Ltd., Alberta Innovates-

Energy and Environment Solutions, and from the Natural Sciences and Engineering Council of

Canada through a Collaborative Research and Development Grant, as well as on-going research

funding through the Discovery Grants program and the Canada Research Chairs program. In

addition, infrastructure funding provided by the Canada Foundation for Innovation and Ontario

Research Fund is gratefully acknowledged.

Supporting Information Available: Nanofluidic chip fabrication procedure,

experimental conditions, temperature calibration in the nanochannels, effects of pressure and

temperature on condensate growth, derivation of flow resistances model, error analysis of the

condensation coefficient, flow resistance contributions to the total resistance, real time

15

observation of Type A and Type B condensation, temperature rise due to latent heat generated by

condensation, pressure gradient in the chiller region, contributions of Knusden flow and flow

resistance at isobaric and isothermal conditions for different scales, PEq/PS at different

experimental conditions, air remaining in the system, variation of empirical parameter, Nα, with

channel heights, discontinuous condensate growth, unity empirical parameter for model

prediction of the condensate growth rate in 70, 220, 1000 nm channel vs. experimental results, as

well as AFM result for 220 nm and 1000 nm channels. This material is available free of charge

via the Internet at http://pubs.ACS.org.

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