Mono-to-Multilayer Transition (MTMT) Dynamics

• MTMT is a threshold phenomenon where a monolayer transitions to multilayer structures due to mechanical compression in bacterial colonies and interlayer coupling in TMDCs.
• In bacterial systems, MTMT is marked by localized cell extrusion driven by compressive stress and modulated by stochastic cell division, characterized by parameters like critical area and time.
• In TMDCs, increasing layer number alters Raman modes, band gaps, and excitonic dynamics, highlighting symmetry breaking and distinct nonlinear optical properties.

Mono-to-Multilayer Transition (MTMT) denotes a transition from a planar monolayer state to a configuration with vertical or interlayer structure. In the literature surveyed here, the term is used in two technically distinct settings. In confluent bacterial colonies, MTMT is the sudden extrusion of cells out of the plane of the monolayer, forming a second layer; it is the first irreversible step toward three-dimensional biofilm formation and a mechanical release of in-plane compressive growth stresses (You et al., 2018). In layered transition-metal dichalcogenides (TMDCs), MTMT refers to the evolution from monolayer to bilayer, few-layer, and bulk regimes, where interlayer coupling, symmetry changes, and environmental pinning reorganize phonons, band topology, nonlinear absorption, phase-transition kinetics, and excitonic pseudospin dynamics (Zhang et al., 2015).

1. Terminological scope and principal observables

The term MTMT does not identify a single universal mechanism. Rather, across the cited arXiv literature it denotes a class of transitions in which adding a vertical degree of freedom changes the governing physics. In bacterial microcolonies, the relevant observables are the critical area $A_c$, critical time $t_c$, first-extrusion position, compressive stress, and the appearance of a visible second layer. In layered TMDCs, the relevant observables include Raman-active interlayer shear and breathing modes, Davydov splittings, direct-to-indirect band-gap crossover, layer-dependent nonlinear absorption, thickness-dependent phase-transition barriers, and in-plane exciton $g$ factors.

Domain	MTMT definition	Principal observables
Confluent bacterial colonies	First extrusion from a planar monolayer into a second layer	$A_c$, $t_c$, stress threshold, first-event statistics, hydrodynamic fields
Layered TMDCs	Evolution from monolayer to multilayer/bulk, or monolayer-versus-bilayer thickness dependence	Raman modes, band gaps, nonlinear absorption, switching barriers, pseudospin dynamics

A plausible implication is that MTMT is best understood as a geometry-changing threshold phenomenon: in one case, growth-induced compression drives out-of-plane buckling; in the other, interlayer coupling and symmetry breaking create new collective degrees of freedom. This shared structural motif is explicit in both the biofilm and van-der-Waals-materials literatures, although the microscopic variables are entirely different (You et al., 2018, Zhang et al., 2015, Manchanda et al., 2020).

2. Deterministic and stochastic onset in bacterial colonies

In growing bacterial colonies, the canonical MTMT problem is formulated mechanically. A bacterium is modeled as a spherocylinder of fixed diameter $d_0$ and time-dependent cylindrical length $l_i(t)$, growing until division. Growth and division generate an in-plane compressive active stress. In a laterally confined one-dimensional chain of $N$ cells of total length $L$, the stress profile takes the virial form

$$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$

with $t_c$0 and $t_c$1. Vertical restoring forces arise from cell-substrate adhesion or overlying compression and are modeled as a Hookean force of magnitude $t_c$2 per unit vertical displacement (You et al., 2018).

The local tipping instability of a horizontal cell is obtained from torque balance. For a cell under axial force $t_c$3, the threshold condition yields the critical compressive force

$t_c$4

This establishes that extrusion is localized and mechanically deterministic once the local compressive stress exceeds the adhesion-limited threshold. The transition is therefore not a global Euler buckling mode of the entire colony, but a local instability of individual cells (You et al., 2018).

Asynchronous cell division adds stochasticity. Because cell lengths fluctuate between $t_c$5 and $t_c$6, extrusion can occur when a cell divides inside the region where the local stress exceeds the minimal critical stress $t_c$7. This region, the “P-zone,” is defined by

$t_c$8

with

$t_c$9

If the average number of cells in the P-zone is

$g$0

then the first division in that region is approximated by a nonhomogeneous Poisson process with instantaneous rate

$g$1

The first-event time distribution is therefore

$g$2

The rate $g$3 is identified as the order parameter of the transition, highlighting its mixed deterministic-stochastic character: deterministically, the stress threshold defines when extrusion becomes mechanically possible; stochastically, random division events determine the precise time and place of the first extrusion (You et al., 2018).

3. Phenotypic noise, hydrodynamics, and transport at MTMT

A complementary formulation treats MTMT in surface-attached, non-motile E. coli colonies through a two-dimensional continuum model with packing fraction $g$4, growth rate $g$5, drag $g$6, and pressure $g$7. The governing equations are

$g$8

Extrusion at the colony center occurs when $g$9, which yields the closed-form expressions

$A_c$0

where $A_c$1 is the cell length at division, $A_c$2 is a material length scale, $A_c$3 is the inoculum area, and $A_c$4 (Dhar et al., 2021).

This formulation separates the statistics of size and time. The critical area $A_c$5 is nearly temperature independent, whereas $A_c$6. Single-cell traits such as $A_c$7 and aspect ratio $A_c$8, and colony-scale traits such as $A_c$9 and $t_c$0, are all reported as log-normally distributed. Noise is quantified by the normalized variance

$t_c$1

Experimentally, $t_c$2 spans two to three orders of magnitude: $t_c$3–$t_c$4 for $t_c$5 and $t_c$6, $t_c$7 for colony doubling times $t_c$8, $t_c$9 for $d_0$0, and $d_0$1 for $d_0$2. From the closed-form expression for $d_0$3, one obtains

$d_0$4

As temperature and thus $d_0$5 increase, the variance of $d_0$6 is suppressed as $d_0$7 even though $d_0$8 and $d_0$9 remain large. The paper describes this as a trade-off between growth-rate noise and geometry noise that pins down $l_i(t)$0 to within $l_i(t)$1 in all conditions (Dhar et al., 2021).

The same study connects MTMT to synchronized hydrodynamics. The continuum model predicts that at $l_i(t)$2 not only $l_i(t)$3 but also gradients of $l_i(t)$4, including vorticity and divergence, peak synchronously, in agreement with PIV measurements. Particle-tracking simulations in PIV-derived velocity fields produce an effective diffusion $l_i(t)$5 that peaks at or just after $l_i(t)$6. The enhancement ratio $l_i(t)$7, where $l_i(t)$8 is the Stokes-Einstein Brownian diffusivity, can exceed $l_i(t)$9 for micron-sized cargo in visco-elastic media with $N$0–$N$1. Over a finite window after MTMT, the Péclet number

$N$2

satisfies $N$3. Experiments with $N$4 polystyrene beads confirm temperature-dependent de-clustering of micrometer-scale aggregates and enlargement of the bead cloud area. The colony is consequently described as a “multifield” topological system in which structural topology, namely nematic micro-domains and $N$5 defects, and hydrodynamic topology, namely vorticity patches, co-emerge and buffer population-scale transport against single-cell variability (Dhar et al., 2021).

4. Substrate-stiffness control of MTMT in nascent biofilms

Substrate mechanics alters both the timing and morphology of MTMT. On soft agarose pads with Young’s modulus $N$6 and $N$7, the MTMT time $N$8 increases with stiffness: $N$9 at $L$0 and $L$1 at $L$2. The critical colony size at transition also increases with stiffness: the mean radius rises from $L$3 at $L$4 to $L$5 at $L$6, corresponding to colony areas

$L$7

This is summarized as a $L$8 increase in area. Softer substrates promote distinct, multilayered colony structures; harder substrates first support growth up to large monolayers before MTMT (Rani et al., 1 Aug 2025).

Boundary morphology changes concurrently. Box-counting on segmented colony boundaries gives a fractal dimension

$L$9

Larger $$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$0 on softer substrates corresponds to higher boundary roughness, whereas stiffer substrates approach nearly smooth circular colonies (Rani et al., 1 Aug 2025).

The associated biomechanical model introduces drag explicitly. At the single-cell scale,

$$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$1

and at the colony scale,

$$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$2

with $$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$3 and indentation depth

$$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$4

so that $$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$5. In the one-dimensional continuum reduction,

$$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$6

with $$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$7 and $$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$8, one obtains

$$\sigma_{xx}(x)=\sigma_m\left[1-\left(\frac{2x}{L}\right)^2\right], \qquad |x|\le L/2,$$9

so the maximum stress is $t_c$00. MTMT occurs when $t_c$01, giving

$t_c$02

The dimensionless control parameter is

$t_c$03

with the transition at $t_c$04. Mapping $t_c$05 from indentation theory onto $t_c$06 reproduces the reported trends $t_c$07 and $t_c$08 (Rani et al., 1 Aug 2025).

5. Phonons, symmetry, and dimensional crossover in TMDCs

In semiconducting TMDCs, the monolayer-to-multilayer transition is tracked most systematically through Raman spectroscopy. The basic effect of increasing layer number is a change in symmetry and the emergence of interlayer vibrational modes. In bulk $t_c$09-$t_c$10 with point group $t_c$11, the prominent first-order Raman modes are $t_c$12 and $t_c$13. In monolayer $t_c$14-$t_c$15 with point group $t_c$16, these become $t_c$17 and $t_c$18. For MoS$t_c$19, $t_c$20 softens nearly monotonically by $t_c$21–$t_c$22 from bulk to monolayer, while $t_c$23 hardens by $t_c$24–$t_c$25. An empirical fit for $t_c$26 is

$t_c$27

capturing the layer-number dependence of the mode separation (Zhang et al., 2015).

Low-frequency rigid-layer modes are the distinctive markers of multilayers. In an $t_c$28-layer crystal there are $t_c$29 doubly degenerate shear modes and $t_c$30 breathing modes. In the monatomic-chain model they follow

$t_c$31

and

$t_c$32

with $t_c$33 and $t_c$34 the bilayer frequencies. These modes provide a substrate-free fingerprint of layer number and of interlayer coupling (Zhang et al., 2015).

MoTe$t_c$35 offers a particularly complete example of the crossover from quasi-two-dimensional to bulk behavior. High-resolution Raman measurements on $t_c$36-layer $t_c$37-MoTe$t_c$38 resolve low-frequency interlayer shear modes (LSM) and layer-breathing modes (LBM), as well as layer-dependent Davydov splittings of mid-frequency $t_c$39 and $t_c$40 modes. No rigid-layer mode appears in the monolayer. In the bilayer, the LSM is observed at $t_c$41 and the LBM at $t_c$42. As $t_c$43 increases, the LSM branches stiffen toward $t_c$44 and the LBM branches evolve toward $t_c$45. Mid-frequency Davydov splittings converge to bulk partners $t_c$46 and $t_c$47, while high-frequency $t_c$48 and $t_c$49 modes show much smaller splittings. A force-constant model with intralayer nearest-neighbor constant $t_c$50, interlayer nearest-neighbor constant $t_c$51, second-neighbor constants $t_c$52 and $t_c$53, and surface modifications $t_c$54 and $t_c$55 reproduces the full set of branches from $t_c$56 to the bulk limit (Froehlicher et al., 2015).

6. Electronic and nonlinear-optical crossovers in few-layer TMDCs

The electronic counterpart of MTMT in TMDCs is the thickness-driven reordering of direct and indirect gaps. Few-layer MoTe$t_c$57 is anomalous within this family. Low-temperature micro-reflectance and photoluminescence measurements give the direct A-exciton energies

$t_c$58

The integrated photoluminescence yield, normalized to the monolayer, is

$t_c$59

These measurements are reported as fully consistent with monolayer and bilayer MoTe$t_c$60 being direct-gap semiconductors, trilayers having nearly identical direct and indirect gaps, and tetralayers being indirect-gap semiconductors. This differs from MoS$t_c$61, WS$t_c$62, WSe$t_c$63, and MoSe$t_c$64, where only monolayers are found to be direct-gap semiconductors (Gutiérrez-Lezama et al., 2015).

Layer number also controls nonlinear absorption. In a two-level ground-state-absorption/excited-state-absorption framework,

$t_c$65

with steady-state populations

$t_c$66

and saturation intensity

$t_c$67

At low intensity, the sign of $t_c$68 determines the regime: saturable absorption (SA) occurs when $t_c$69, while reverse saturable absorption (RSA) occurs when $t_c$70. The layer-dependent band gap $t_c$71 sets whether a fixed excitation energy favors one-photon absorption or two-photon absorption. The cited work attributes SA-RSA transitions to the number of layers, temperature, and defects, because these modify the band gap and therefore the relative roles of GSA, ESA, and two-photon channels (Neupane et al., 2018).

Taken together, these results establish that MTMT in TMDCs is not solely a structural classification by thickness. It is a crossover in symmetry class, density of states, excitonic hierarchy, and allowed optical pathways. This suggests that “few-layer” should be treated as a distinct physical regime rather than as a perturbation of the monolayer limit (Gutiérrez-Lezama et al., 2015, Neupane et al., 2018).

7. Excitonic pseudospin, phase-transition kinetics, and mechanics in multilayers

Multilayers can acquire dynamical degrees of freedom absent in monolayers. Time-resolved Faraday ellipticity measurements on mono- and multilayer WSe$t_c$72 and MoSe$t_c$73 in in-plane magnetic fields $t_c$74 up to $t_c$75 provide a clear example. In monolayers, resonant excitation of the $t_c$76 exciton yields traces well fitted by

$t_c$77

with $t_c$78, $t_c$79 in WSe$t_c$80, and $t_c$81 in MoSe$t_c$82. Turning on $t_c$83 produces no detectable oscillations or change in decay rates, implying $t_c$84 within experimental uncertainty. In multilayers, by contrast, the ellipticity signal for $t_c$85 is fitted by

$t_c$86

with

$t_c$87

The extracted values are $t_c$88 for multilayer WSe$t_c$89 and $t_c$90 for multilayer MoSe$t_c$91, very close to reported out-of-plane exciton $t_c$92 factors. The proposed interpretation is pseudospin quantum beats caused by spin- and pseudospin-layer locking in H-type stacked multilayers, yielding ultrafast pseudospin rotations in the GHz-to-THz range (Raiber et al., 2022).

Thickness also changes phase-transition thermodynamics and kinetics. In hydrogenated MoTe$t_c$93, hydrogen adsorbs more favorably on the metallic distorted octahedral $t_c$94 phase than on the semiconducting $t_c$95 phase, thereby stabilizing $t_c$96. The free-energy difference $t_c$97 crosses zero at $t_c$98 for the monolayer but only at $t_c$99 for the bilayer when adsorption occurs on the top sheet. This shift is attributed to substrate friction or interfacial pinning in the bilayer. The activation barriers for the $g$00 transition are $g$01 in pristine monolayer, $g$02 in monolayer at $g$03, $g$04 in pristine bilayer, and $g$05 in bilayer at $g$06. Using

$g$07

the calculated transition times at $g$08 are $g$09 for the hydrogenated monolayer and $g$10 for the hydrogenated bilayer, so that $g$11 (Manchanda et al., 2020).

Mechanical properties also undergo systematic layer-number evolution. Fully atomistic molecular-dynamics simulations on WSe$g$12 and MoSe$g$13 show that single layers deposited on silicon substrates have larger friction coefficients than $g$14, $g$15, and $g$16 layered structures. In WSe$g$17, the sliding force decreases from about $g$18 at $g$19 to $g$20 at $g$21, then rises slightly to $g$22 and $g$23 at $g$24 and $g$25; MoSe$g$26 shows the same trend at slightly lower force. Peel-off energies likewise drop sharply from monolayer to bilayer, consistent with the inequality

$g$27

that is, substrate-layer binding exceeds layer-layer binding. Fracture is chirality dependent, with crack propagation preferentially perpendicular to W(Mo)-Se bonds and faster for zigzag-like defects (Jaques et al., 2018).

Across these studies, multilayers are not simply thicker monolayers. They introduce interlayer vibrational branches, altered symmetry selection rules, new electronic orderings, modified nonlinear-optical pathways, strong thickness dependence of phase-switching kinetics, and coherent pseudospin dynamics that are absent or strongly suppressed in monolayers. In the bacterial literature, MTMT marks the first irreversible entry into three-dimensional colony architecture; in the TMDC literature, it marks the onset of genuinely interlayer physics.