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Journal Club for February 2021: Deformation, instability, and assembly of materials by dynamic solid-liquid interactions

Baoxing Xu's picture

Deformation, instability, and assembly of materials by dynamic solid-liquid interactions 

Yue Zhang, Qingchang Liu, and Baoxing Xu

Department of Mechanical and Aerospace Engineering, University of Virginia


Interaction of solid with liquid is ubiquitous in nature, and its underpinned physical and chemical characteristics have been leveraged to benefit almost every aspect of our daily life and industry over decades. This concept of solid-liquid interactions is recently reemerging with the ever-fasting demands of calling for innovative design principles and approaches in both design and manufacturing of advanced materials, structures and devices, where seamless integrations and assemblies of different material/phase components are required across multiple scales. Compared with interaction of solids with solids, liquid, an intrinsically deformation-free phase due to fluidity, provides a unique and tactful platform that helps release residual stress or/and avoid deformation mismatch with surrounding solid constraints during growth, self-assembly and manufacturing of solid phase materials and structures meanwhile remaining close contacts at their interface.

By leveraging solid-liquid interactions where either liquid is embedded into solid materials or solid is suspended in liquid matrixes, numerous achievements have been made. Examples in the design of functional materials and structures include, but not limited to, high-stiffness composite solids with liquid inclusions [1], large deformation-tolerance wearable electronics encapsulated on a microfluidic substrate [2], multifunctional liquid foam comprised of nanoporous particles suspended in a non-wetting liquid [3], microfluidics based switchable adhesion devices [4], and ultra-stretchable wicked membrane with fibrous membrane infused by a wetting liquid [5]. Discussion in this topic along with future opportunities and challenges can be found in our perspective article [6]. In parallel, solid-liquid interactions have innovated concepts and strategies of manufacturing, for example, self-folding of silicon films by liquid evaporation [7], self-assembly of pillars/fibers by liquid evaporation [8], non-etching liquid assistant transfer of 2D materials [9], mass production of 2D nanosheets by liquid exfoliation [10], and transfer of large-scale films in water [11]. Please refer to more excellent examples in this topic published in 2016 EML themed issue “Mechanics in Extreme Manufacturing” [12] and our perspective article [13].

In both material design and manufacturing approaches enabled by solid-liquid interactions, the establishment of mechanics theory that takes into account material deformation, interfacial wetting, molecular diffusion and intrusion, and/or their couplings at solid-liquid interfaces will be of critical importance. For example, solid materials can deform due to liquid fluid-induced forces and subsequently change flow fields of liquid, referred to as dynamic solid-liquid interactions. Here, we would like to highlight our recent two progresses in the manufacturing of functional structures enabled by dynamic solid-liquid interactions (I. Transfer of soft films, and II. Crumpling and self-assembly of deformable materials), with a focus on the development of mechanics theory (without chemical reactions involved, please refer to our paper with chemical reactions coupled [14]).

I. Transfer of soft films (for details please refer to reference [15])

Mechanical transfer of films is a process key to the fabrication of film-based functional structures, electronics and devices, but currently is established in the framework of solid native substrates. We developed a technique - capillary transfer - that relies on the capillary force at the interfaces of liquid-substrate-film to transfer films. Figure 1A shows the schematics of working principles for this capillary transfer technology. One end of receiver substrate is submerged in the liquid bath to form the initial contact line among substrate, soft film and liquid, referred to as the transfer front (orange dot line ‘P’). Given the fluidity of liquid that provides a compliant liquid-solid interface, the film is allowed to move upwards or downside the liquid substrate, referred to as pull-up transfer and push-down transfer accordingly. As a result, this opposite transfer direction enables a selective contact of the two film surfaces with the receiver substrates, i.e. with the pull-up direction, the bottom surface of film (in blue) will be in contact with the substrate; with the push-down direction, the top surface will be in contact with the substrate, which will be useful for transfer of films with different surface features and properties such as geometric patterns, roughness, and surface wettability.

Selection of transfer direction. We first established a mechanical model to help determine the selection of transfer direction. At the transfer front (Figure 1A), with a negligible elongation and local deformation of the soft film (see full model description in reference [15]), the geometric profile of the bent soft film is essentially led by the energy balance between elastic bending deformation and solid-liquid interactive energy, where the solid-liquid interactive energy relies on the dynamic contact angle of substrate to the liquid, and changes with the transfer direction. The height of the transfer front relative to the liquid surface (h) is formulated from the total energy to quantitatively determine the transfer direction. For example, the pull-up transfer requires h > 0 with the bending upwards of the film (i.e. ∅ < 90o ), and the push-down transfer requires h < 0 with bending downwards of the film (i.e. ∅ > 90o). h = 0 suggests that the capillary force is too small to bend the film (∅ = 90o), and the film cannot be transferred to the substrate. Figure 1B shows the theoretical phase diagram for the selection of transfer directions. A more hydrophobic transfer substrate and a higher capillary number will be beneficial for the push-down transfer. By contrast, a more hydrophilic transfer substrate will be beneficial for the pull-up transfer.

Success of continuous transfers. Once the transfer direction is determined, a successful transfer requires a continuous pass of film across the whole transfer front, and the criterion can be established in theory by predicting and comparing the transfer force with and without the transfer of film. At a steady state, the bending profile of the film keeps unchanged at the transfer front, which leads to a balance among the work done by the transfer force, the interfacial energy due to dynamic solid-liquid interactions, and the dissipated energy due to the viscous effect of liquid. This energy balance serves the foundation for predicting the transfer force. In particular, at capillary number Ca  Ca< 0.01, the effect of viscosity can be neglected, and the formulation of the transfer force can be greatly simplified. Figure 1C shows theoretical predictions of transfer force in both pull-up and push-down transfers with remarkable agreement with experimental measurements for a wide variety of film materials with different stiffness, thickness and surface wettability, transfer substrates and liquids under a series of moving velocities of transfer substrate. Similarly, the force without the transfer of film can also be obtained.

The difference in the magnitude of transfer force with and without the transfer of film is employed to formulate the criterion for a success of capillary transfer along both directions. Experiments on a wide variety of system materials for soft films, film thickness, transfer substrates and liquid media confirm theoretical results for both pull-up and push-down transfers


Figure 1: (A) Schematic illustrations of the capillary transfer. (B) Theoretical phase diagram on the choice of the transfer direction with respect to materials (bending stiffness of soft film B, the static contact angle of substrate to the liquid θsls) and loading conditions (capillary number CaCa). (C) The steady state transfer force of film normalized by the liquid surface tension (γl) and substrate width (bs), versus the capillary number.


Application demonstrations. Figure 2A shows the transfer of a PDMS/CNT composite film onto glass substrate by the pull-up transfer and onto PDMS substrate by push-down transfer at the transfer speed of 1mm/s. The pull-up transfer yields the top surface (labeled in T) of PDMS/CNT composite film exposed to air for use; the push-down transfer yields the bottom surface (labeled in B) of PDMS/CNT composite film exposed to air for use. The FEA strain analysis (inset) shows that the mechanical deformation fully recovers with zero strain after successful transfer onto both substrates, suggesting that both transfers will not cause any potential damage during transfer process or leave any residual strain to the film after transfer. Figure 2B shows the comparison of the electrical resistance of the PDMS/CNT composite films as fabricated and after transfer. The consistence between them further confirms the negligible mechanical deformation of film after transfer. In particular, when the CNT concentration is as high as 25%, the unchanged electrical resistance after transfer indicates potential applications in transfer of flexible/stretchable electronic devices with very sensitive microstructures. At the same time, the measured contact angles of water droplet on as-fabricated and after-transferred films (Figure 2C and D) are the same, independent of CNT concentrations, suggesting the unchanged surface wettability after transfer. These unaffected electrical function and surface wettability of films also indicates that the mechanical deformation associated with the drying out of possibly trapped water between film and substrate after transfer can be neglected, which can also be theoretical estimated.

Figure 2: (A) Capillary transfer of PDMS/CNT composite film from water surface to glass slide by pull-up transfer and to PDMS slide by push-down transfer. Inset is the FEA plane strain distribution on the film during both transfer processes. (B) Comparison of measured electrical resistance of PDMS/CNT composite film before and after transfer. (C) Comparison of measured contact angle of water droplet on the top (smooth) surface (via pull-up transfer) and bottom (rough) surface (via push-down transfer) of composite film before and after transfer. (D) Experimental images of water droplet on the top (smooth) surface (red “T”) and bottom (rough) surface (blue “B”) of PDMS/CNT composite film before (orange line) and after transfer (blue line).


II. Crumpling and self-assembly of deformable materials (for details please refer to reference [16])

The dynamic solid-liquid interaction at the transfer front in above example results from active movements of the solid film and substrate. On the other hand, the active flow of liquid will also lead to a dynamic interaction at solid-liquid interface and has been utilized in materials synthesis and processing, for example, evaporation-induced self-assembly [17]. Generally, nanoparticles in self-assembly are rigid and only aggregation is expected by inter-particle attraction to achieve dense-packed assembly of nanoparticles. When the suspended materials in liquid are intrinsically soft or structurally flexible, mechanical deformation will be expected along with self-assembly. For example, the 2D or 1D nanomaterials are expected to experience large deformation and even severe instability when assembled during their solution evaporation. Understanding of the evaporation-induced mechanical deformation, instability, and assembly of deformable materials, and their mutual competitions and coordination is of critical importance to develop structurally controllable manufacturing techniques (e.g. printing, aerosol processing) of mechanically deformable materials.

We developed a mechanics model to addresses liquid evaporation-induced large deformation, severe instability and self-assembly of deformable materials by taking two-dimensional (2D) graphene suspended in a droplet as an example, as summarized below.

A 2D sheet. When there is a single 2D sheet in a sufficient large droplet (Figure 3A), assume it stays on the surface of droplet (in practical experiments, its suspended status can be well controlled by liquid pH value or size of graphene sheet [18]), the energy competition between mechanical deformation of 2D sheet and solid-liquid interaction will lead to deformation of 2D sheet such as local wrinkling that may eventually transform to folding with the evaporation of droplet. These deformation evolutions with evaporation and their transition point can be determined using continuum mechanics analysis. On the other side, the deformation of local wrinkles and folds reflect the out-of-plane mechanical deformation of 2D sheets centered about a symmetric line (marked in red in Figure 3B) and can be modeled by a rotational spring connected with two planes, where the planes cannot be further deformed and are considered to be rigid. When the folded rigid parts get close to each other, their van der Waals interactions associated with interactive binding energy will emerge, and can be described by introducing a mechanical slider between two rigid planar sheets, as illustrated in Figure 3B. Therefore, the deformation of a single 2D sheet can be characterized by a rotational spring-mechanical slider mechanics model, where the parameters associated with both spring and slider models can be calibrated through quantitative comparison of their stored energy with corresponding determinations from continuum mechanical analysis.

Multiple 2D sheets. When there are multiple 2D sheets in the droplet, in addition to the mechanical deformation of individual 2D sheets, 2D sheets will be packed and assembled closely with the liquid evaporation. This assembly will lead to an emergence of interactive energy between 2D sheets, similar to the binding energy of the folded parts in a single 2D sheet and can also be described by a mechanical slider with the same as that in folded parts, referred to as inter-layer slider. Figure 3C shows the illustration of a network distribution of rotational spring-slider mechanics models for the assembly of multiple 2D sheets in droplet by evaporation. The total deformation, binding energy and assembly energy therefore can be determined by the superposition principle. Note that the specific formula of these energy changes at different evaporation stages, and their transition points can be determined by comparing the surface area of all 2D sheets and droplet. In particular, at the early stage of evaporation, the surface of droplet is not fully covered by the 2D sheets, and there is no overlap between 2D sheets, which leads to a zero assembly energy. We should note that when there will be no deformation within each 2D sheet and only assembly of 2D sheets among each other, this theoretical model will reduce to that of the assembly process of colloidal particles.

In both a single and multiple 2D sheets in droplet, the evaporation-induced pressure, including the vapor pressure, recoil force at the liquid-vapor interface, shear force by capillary flow, and capillary force will deform 2D sheets and its induced mechanical deformation to 2D sheet can be described by a potential energy.

Figure 3: (A) Schematic illustrations of the crumpling and assembling process of 2D material sheet by liquid evaporation. (B) Rotational spring-mechanical slider model for mechanical deformation of a single 2D sheet by liquid evaporation, where the parameters in the model can be calibrated from corresponding continuum mechanical analysis. (C) Rotational spring-mechanical slider network model for mechanical deformation and assembly of multiple graphene sheets by liquid evaporation. (D) Comparison of the deformation energy, binding energy and assembling energy of square-shaped graphene sheets between the theoretical predictions and simulations. (E) Comparison of dimension size of assembled particles after complete evaporation of liquid under different evaporation pressures, shapes and total areas of graphene sheets among theoretical predictions, CGMD simulations and available experiments.


Assembled particles. Once these energy components are determined, the crumpling and assembly of 2D sheets with the liquid evaporation can be elucidated by monitoring the variation of the total energy with the local curvature of deformed 2D sheets. Figures 3D shows the comparison of these energy variations with liquid evaporation between theoretical predictions and simulations for their crumpling and assembling of multiple square-shaped graphene sheets. The increase of deformation and binding energies (magnitude) reflects the crumpling of graphene sheets due to the generation and propagation of folded ridges; the appearance and increase of assembly energy represents the mechanical assembling among graphene sheets. The final stable stage of these energies suggests the arrival of crumpling or/and assembling process. That is, when the variation of the total energy with local curvature reaches zero, the crumpling and assembly stop, and the multiple 2D sheets will be crumpled into a particle stabilized by the van der Waals energy of adhesive self-folding and assembly. After the complete evaporation of liquid, the total energy of assembled particle can also be determined through this spring-slider model, thereby predicting both the size of assembled particle and the density of folding. Figure 3E gives the comparison of the overall radius of particles among our simulations, theoretical analyses, and experimental results from literature [19-24]. Remarkable agreement among them is observed. More importantly, our theoretical model closes the scale gap between molecular simulations with an emphasis on understanding of mechanism and experimental measurements with an emphasis on macro-phenomena and properties. In addition, the theoretical analysis on high local curvatures and overlap areas between individualson assembled particles can identify mechanisms of mechanical deformation on individuals and assembly among them.

This rotational spring-mechanical slider model can also be used to describe the deformation, instability, folding and self-assembly of other deformable materials such as 1D nanofibers and nanotubes by evaporation of droplet that is not fully suspended and could be on substrate such as printed droplet on substrate. Besides, when there are more than one solid phase materials in the droplet, their deformation and self-assembly could lead to well-organized composite structures by controlling dynamic solid-liquid interactions such as 1D folded-fiber core/2D crumpled-graphene core/shell structures [25]


Concluding remarks

In summary, in both demonstrated examples (I) and (II) above, the deformation, instability and self-assembly of materials rely strongly on dynamic solid-liquid interactions, where the transfer front in example (I) remains stable with a steady contact between solids and liquid during transfer process, and the solid-liquid interfaces in example (II) dynamically change for the generation of particles with liquid evaporation. These two types of dynamic solid-liquid interactions could also be combined in manufacturing, for example, self-assembly on a heated substrate by a solution shearing approach [26], and the development of theoretical models with the incorporation of mechanical deformation mechanism for their enabled applications in assembly of mechanical deformable materials will be critical. 

Looking forward, the fast thriving of material design and advanced manufacturing techniques enabled by dynamic solid-liquid interactions requires tremendous efforts on developments of precision theoretical models that need to integrate the complexity of interface boundaries both geometrically and physically. Besides, when active solids or/and responsive liquids to external environments or manipulation means such as heat, voltage, magnetism, force, and pH value are employed, the interlocked coupling of fluid dynamics of liquid and mechanical deformation of solids is expected at solid-liquid interfaces. Especially, when the dimension of solid materials such as porous materials or nanoparticles is down to nanosize, charge transfer, heat transfer and phase change at solid-liquid interfaces may become crucial, and additions/improvement of associated these factors to theoretical models will be very necessary. On the other hand, with synthesis of new solids and liquids along with creations of various manipulation approaches to solid-liquid interfaces, continuous understanding of and fully leveraging dynamic solid-liquid interactions will not only improve existing approaches and concepts of material design and manufacturing, but will also help create new ones toward the development of future material design and manufacturing approaches with intelligent and autonomous capabilities. 



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Yue, Qingchang, and Baoxing,

Great work on this important topic. Your findings are of great importance for the understanding of the interaction between dissimilar materials/components at the solid-liquid interface and will benefit the development of advanced materials and structures. I have two follow up questions and seek your insight.  

1. Is there any size effect on the energy balance between the soft film and the substrate? The fabrication example is at mm scale. Are your findings applicable to film transfer at a much larger or smaller sacle?

2. Does gas-phase play a role in the crumpling process? At the solid-liquid interface, all three phases, i.e. the gas, liquid, and solid phases, exist. Is this assembling process dominated by solid-liquid interaction only?

I look forward to reading your future works and your valuable comments. Thank you for sharing the journal club with us. 





Baoxing Xu's picture


Glad you like this discussion topic. Thanks much for raising these two excellent questions, and my thoughts are below

1. Yes, there might be a size effect when the thickness of film is down to several nanometers due to surface strain/stress of nanofilm. This size effect could influence both the calculation of mechanical energy such as bending and elongation energies and the measurement of contact angle between solid-liquid. However, it is not expected to affect the application of the energy balance theory between film and substrate at the steady state of transfer because of the associated constant bended configuration of film and the stable contact status with liquid at the transfer front.  In our experiments, we demonstrated the transfer of film with thickness as small as 1 um, and the transfer forces for both push-down and pull-up transfer directions showed good agreement with both theoretical predictions and FEA. For a very large film, this proposed transfer technology and mechanics theory are also expected to be applicable, but you need to ensure first that film floats on the surface of liquid before the start of transfer. 

From the theoretical point of view, when the film is extremely soft, the solid-liquid interaction at the transfer front is expected to lead to local mechanical deformation in film along its thickness direction, similar to the discussion in the cited literatures in our paper such as Phys. Rev. Lett. 122, 248004 (2019); Phys. Rev. Lett. 106, 186103 (2011); Nat. Commun. 6, 7891 (2015). This local deformation will also affect the measurement of contact angle that is one of important parameter for our transfer technology. Taking account of these local deformation of extreme soft films in theory will be helpful for broader applications of the proposed transfer technology, but could be minimized by selecting the most suitable liquid phase as a partner. 

In addition, a strong size effect is expected for solid-liquid interactions in a nanoconfined environment such as nanofluidics where the double layer structures are largely overlapped. This size effect is closely associated with nanoconfinemental environments and is usually independent of either solid or liquid phase. How to leverage this intrinsic size effect of solid-liquid interactions for future design and manufacturing of materials and structures is crucial.

2. Yes, gas phase is critical, for example, as the evaporation continues, the 2D sheets will be mitigated to the surface of droplet and are exposed to vapor, which leads to the appearance of capillary force at the solid-vapor-liquid interface. The evaporation also results in a recoil force at the liquid-vapor interface squeezing the interface toward the liquid side. In our mechanics analysis, because the capillary force changes its direction dynamically with the free motion of 2D sheets in droplet, along with other forces/pressures led by such as capillary flow, vapor recoil and vapor pressure, we developed a unified equivalent evaporation pressure model to describe their contributions to crumpling and assembling of 2D sheets.

The assembling process reflects the competition between solid-solid and solid-liquid interactions. For example, for multiple 2D solid sheets, at the early stage of evaporation, solid sheets are far from each other, and the assembling process is dominated by solid-liquid interactions. As the evaporation goes on, the solid sheets become closer, and the solid-solid interaction between 2D sheets (crumpled 2D sheets) will appear, i.e. the intra-layer slider in our rotational spring-mechanical slider model needs be to included. With the further evaporation of liquid, solid-solid interactions will become dominative for the assembling until a dry assembled solid particle forms after the complete evaporation of liquid where only solid-solid interactions exist. The competition between solid-solid and solid-liquid interactions and their dominative role can be determined by comparing the energies between rotational springs and mechanical sliders, as we did in the paper. Also note this assembling process is similar to that of rigid particles in traditional colloidal science, but our mechanical model takes into account mechanical deformation of particles (here 2D sheets) and can easily reduce to classic assembly theory when deformation is allowed (i.e. the constant of rotational spring in the spring model is sufficiently large) which we have demonstrated in our paper.    

Hope these explanations are helpful. Feel free to share your work, especially your work in solid-liquid interactions in nanoconfinements, by posting your relevant publications here if I forgot to cite them in the above references.  



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