Category: physics

Mechanics of biological tissue

The stiffness of a tissue reflects its cell-level properties. For instance, it is well known that increased stiffness can be an indicator a rapidly growing cancer in a tissue. At UPenn, I am using simulations and theory, supported by experiments with the Penn Science of Oncology Center, to develop the biophysical understanding of tissue mechanics.

On short time scales, biological tissues have the mechanical properties of polymer networks embedded with cells and fluid. This is a complicated composite from a materials physics perspective. Experiments study simplified tissues, with cells alone with no extracellular matrix, or of polymer networks with embedded cell-like particles. A long-term goal is to increase the utility of stiffness in characterizing the microscopic structure of the tissue.

We developed a 3D cell-vertex model based on the Voronoi tesselation to study the mechanical properties of confluent cell tissues. (TAS)	Modelling polymer networks embedded with beads helps interpret the experimental measurements of elastic moduli. (TAS)
Polymer matrix serves many mechanical roles in a tissue. In addition to providing mechanical support itself, it can also prevent cells from rearranging on long time scales. Simulations separate the two effects by choosing the topology of the fiber network. (TAS)	Confocal microscopy of liver tissue shows the changes in contact topology of the cells after straining. (TAS with LiKang Chin)

Contact properties of rough solids: all about the Gap

Essentially all materials have rough surfaces at small scales. This can be a problem for engineers designing electrical contacts, thermal contacts, or seals to prevent leaks, since the roughness creates gaps between the two contacting surfaces. By increasing the pressure, the two solids can be forced closer together, reducing the gaps somewhat, but some space between the surfaces usually still remains.

Two blocks in compression may appear to be in full contact, but there is nearly always microscopic roughness that leads to gaps. For randomly-rough geometries, only $\nabla h_{rms}$ , the root-mean-squared slope of the surface, determines the microscopic contact area, for an applied force on the block, $F_N$ , made of material with Young’s modulus $E$ .

Two blocks in compression may appear to be in full contact, but there is nearly always microscopic roughness that leads to gaps. For randomly-rough geometries, only $\nabla h_{rms}$ , the root-mean-squared slope of the surface, determines the microscopic contact area, for an applied force on the block, $F_N$ , made of material with Young’s modulus $E$ .

Work with Mark O. Robbins quantified how surprisingly resilient the gaps are, and how important they are in determining the contact properties. To analyze the problem, we had to overcome technical challenges related to the large range of length scales of random-rough geometry on typical material surfaces, usually millimeters down to atomic scales. The rough geometry can be modeled using the mathematics of random (self-affine) fractals. Large-scale parallel processing on a computing cluster is required to simulate and analyze contact between many random realizations of rough 3D cubes of length 1000 atoms on a side. We worked with Lars Pastewka to write C++ code for GPUs to do fast simulations using lattice Greens functions and continuum-elastic Greens functions, which will be described in a later post.

Small example region of contact at the atomic scale for two similar surfaces, a bent lattice and a stepped lattice. If the material can yield, stepped surfaces produce more plasticity and lead to increased contact area (dark atoms). The amount of increase is given by $\nabla h_{rms}$ . However, $\nabla h_{rms}$ must be very large to enter the limiting plastic regime, where the contact area is the applied force over the material yield stress.

We found that the large-scale features of the material’s surface get flattened more easily, but the small-scale roughness cannot be squeezed away even at considerable pressures; we found that the fraction of the surface that is in contact is well-defined and at common pressures is less than 20%. Within this regime the fractional contact area obeys a simple law: $f_A \approx \kappa \hspace{1mm} p/ (E' \nabla h_{rms})$ , where $p$ is the confining pressure, $E'=E/(1-\nu^2)$ is the material’s contact modulus, $\nu$ is the effective Poisson ratio, $E$ is the material’s Young’s modulus, and $\nabla h_{rms}$ is the root-mean-squared slope of the original roughness profile[1]. $\kappa$ is a dimensionless constant with value about 2.2.

Atoms on the surface of a solid are colored red if they experience a repulsive force from the opposing surface. The statistical properties of the contact and pressure spatial distributions change with atomic scale features.

We considered this problem with both continuum-scale and atomic-scale simulations. We found that the continuum results can often be applied to small scales, meaning that continuum analyses are often appropriate for nanotechnology applications[2]. At the atomic-scale, the concept of contact area gets replaced by the number of atoms that exert repulsive forces on the other surface. However, for very rough or crystalline surfaces, atomic-scale plasticity causes the value of $\kappa$ in the above equation to increase, up to a value of 10.0. We went on to analyze the transition from elastic to plastic contact with roughness.

The gap (here quantified by the mean surface separation $\bar{u}$ ) approaches zero as the log of applied force $F$ . When $\bar{u}$ is normalized by surface rms height $h_{rms}$ and $F$ is normalized by the product of macroscopic contact area and contact modulus, $A_0 E'$ , the result collapse for different symbols, corresponding to continuum and atomistic simulations with varying roughness parameters. The lowest forces correspond to effects from our simulations having limited size, while the regime with slope $\gamma$ corresponds to an exponential relationship between applied pressure and the gap size.

The gap (here quantified by the mean surface separation $\bar{u}$ ) approaches zero as the log of applied force $F$ . When $\bar{u}$ is normalized by surface rms height $h_{rms}$ and $F$ is normalized by the product of macroscopic contact area and contact modulus, $A_0 E'$ , the result collapse for different symbols, corresponding to continuum and atomistic simulations with varying roughness parameters. The lowest forces correspond to effects from our simulations having limited size, while the regime with slope $\gamma$ corresponds to an exponential relationship between applied pressure and the gap size.

Even if the contacting materials are very stiff, they can seem much less so because of the gap from roughness at the contacting surface; the rough topography can be compressed more easily than the solid material. For both atomic and continuum analyses, the average gap width, $\bar{u}$ decreases slowly with force. We quantified this and showed that at large confining force, $F$ , it follows $\bar{u} = -\gamma h_{rms} \text{ln(} F/A_0 E' \text{)}+ c$ where $h_{rms}$ is the root-mean-squared height of the original surface, $A_0$ is the apparent surface area. As a result, the stiffness of the interface is $k_I \approx 0.5 * F / h_{rms}$ . The stiffness of the interface is extremely low when there is no confining pressure, and so the surface roughness dominates the mechanical response of the whole two-block system.

References:
Chapter 5 of my thesis (papers in preparation)

Sampling of the related work that helped build up this picture:

 Hyun, S., et al. "Finite-element analysis of contact between elastic self-affine surfaces." Physical Review E 70.2 (2004): 026117.
 Pei, L., et al. "Finite element modeling of elasto-plastic contact between rough surfaces." Journal of the Mechanics and Physics of Solids 53.11 (2005): 2385-2409.
 Cheng, S. & Robbins, M.O. "Defining contact at the atomic scale."  Tribol Lett (2010) 39: 329. https://doi.org/10.1007/s11249-010-9682-5
 C Yang and B N J Persson 2008 J. Phys.: Condens. Matter 20 215214 http://iopscience.iop.org/article/10.1088/0953-8984/20/21/215214/
 Persson, B. N. J. "Relation between interfacial separation and load: a general theory of contact mechanics." Physical review letters 99.12 (2007): 125502.
 Mo, Yifei, Kevin T. Turner, and Izabela Szlufarska. "Friction laws at the nanoscale." Nature 457.7233 (2009): 1116.

Visualizing mathematical analysis

In this post, I’d like to muse about the visualizations that are used when thinking about math, and how they can become quite beautiful as well as useful

When adding $7+8$ , many people imagine stacking blocks and then see $5$ spilling over the $10$ mark. This visual thinking made the answer seem “obvious” after doing it many times. When doing calculus, simple picture tricks are taught less often. If confronted with $\int_0^1 x dx$ we might quickly follow the memorized rule (from the fundamental theorem of calculus) that you add $+1$ to the exponent and divide by that number. $\int_0^1 x dx = \frac{x^2}{2} |_0^1 = \frac{1}{2}-\frac{0}{2}=\frac{1}{2}$ . But one can also remember that integration finds the area under a curve. With a picture in mind like the one below, we know that the red line cuts the unit square into two parts, each with area $\frac{1}{2}$ . The visual understanding will sometimes be a useful tool.

This post is supposed to illustrate that visual thinking remains useful in more advanced math, especially mathematical analysis (real analysis, complex analysis, functional analysis, differential equations) which deals with objects you can imagine embedded in space (that is, with a metric). It’s just that the pictures become more and more rarely taught. That means it takes a motivated attitude and some patience, because often in classes you generally have to figure it out for yourself.

I’ve found that it is worth the effort. This post will give a couple of examples using partial differential equations (PDEs). PDEs are very useful in engineering disciplines, yet it is not often taught how to picture the solution.

For example, let’s consider Laplace’s equation.

$\Delta \phi (x,y,z) = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0$

In classical physics, the electrical potential $\phi$ satisfies Laplace’s equation in three dimensions everywhere in space that there is no electric charge.

Say we want to know the potential in a long square pipe (a square cylinder), and we are given boundary conditions that specify the potential of each of the four sides. We can visualize what this equation is saying. The equation is local and it says that at every point of the domain, the x-curvature of the potential must cancel with the y-curvature of the potential.

I picked some boundary conditions (parabolas on the ends and constants at the side) and plotted the solution for $\phi$ below using Wolfram Alpha.

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We can visually check that the curvature in x and y are equal and opposite at every point. Note how the solution looks like a stretched membrane. That’s because a stretched membrane also follows also follows Laplace’s equation. The solution is essentially the simplest surface you can draw that will connect the boundaries. If you can picture gluing an elastic sheet to those boundaries, you can picture the solution to Laplace’s equation.

Quantum probability current

In the example of the Schrödinger equation, the rate of change of the Hilbert state vector is determined by the Hamiltonian operator.

$i \hbar \frac{\partial}{\partial t} \left| \Psi(t) \right> = H \left| \Psi(t) \right>$

Operators and state vectors are abstract and we are sometimes interested in the positions basis specifically, since let’s face it, space seems to be a dominant aspect of reality. For the 1-dimensional problem of a free particle in the position basis we get the partial differential equation:

$\frac{\partial \Psi(x,t)}{\partial t} = i \hbar /2m \frac{\partial^2 \Psi(x,t)}{\partial x^2}$

$\Psi(x,t)$ is a complex field, in the sense that it has both real and imaginary components, and we can… really imagine it. The equation can be decomposed into two equations, one for the real part and one for the complex part of $\Psi(x,t) = \psi(x,t) + i \phi(x,t)$ , and we can plot it using a polar plot at each value of x. Again, the equation is local and says that the curvature is what matters. Where the curvature of the real part is positive, the imaginary part will grow. Where the curvature of the imaginary part is positive, the real part will shrink. If initially the field looks like a plane wave $\Psi(x,0) = e^{ipx} = \cos(px) + i \sin(px)$ where $p$ is a constant, the equation is telling us the corkscrew will turn.

Animation created with Python and ImageMagick

Now for the last part of this example, let’s look at the probability current. Usually when this is taught it seems to be a terribly opaque abstract construct. The probability current $\bold{j}$ is essentially the quantity that, together with the probability $P(x,t) = \Psi^* \Psi$ , satisfies the continuity equation $\frac{\partial}{\partial t} P(x,t) + \nabla \cdot \bold{j} = 0$ when $\Psi$ follows the Schrödinger equation. In one dimension it is

$j(x,t) = \hbar / 2mi (\Psi^*(x,t) \frac{\partial \Psi(x,t)}{\partial x} - \Psi(x,t) \frac{\partial \Psi^*(x,t)}{\partial x})$

It is difficult to picture what this equation is telling us! And yet wiki pages and textbooks* everywhere just state it and move on, letting the student grapple with it in the problem set. Let us stubbornly pursue a better picture. Using the polar form of the complex quantity $\Psi = A(x) e^{i \theta(x)}$ and working through a little math we get an equation for the flow of that is much simpler to picture.

$j(x) /P(x) = \hbar/m \: \frac{d}{dx}\theta$

The fraction of probability leaving location x is simply the derivative of the phase. In three dimensions, $\frac{d}{dx}\theta \rightarrow \nabla \theta( \bold{x} )$ which is saying that the probability current is just “flowing uphill” of the phase. That’s what the phase of a wave function has always meant, in case, like me, you weren’t told. This simpler version gives insight that might even be useful to a new wave of physics students.

Using the example of the corkscrew above, simply stated, the angle $\theta$ of the corkscrew increases constantly to the right, so our equation tells us there is uniform probability current to the right.

Here are other relations that are useful for visualizing these single-particle dynamics. Here I use shorthand for the derivatives so $\frac{d}{dt} P \rightarrow \dot P$ and $\nabla \theta \rightarrow \theta'$ . In the following, ignore the constant factors of $2m/\hbar$ which could be swallowed into the definition of the phase. The phase at a point x will decrease if it is sloped and it will increase due to curvature of A $\dot \theta 2m/\hbar = (\frac{A''}{A} - \theta'^2)$ . Meanwhile the amplitude A at the point x decreases from phase curvature or from slopes $\frac{\dot A}{A} \frac{2m}{\hbar}= -2 \frac{A'}{A} \theta' - \theta''$ . The probability increases from amplitude curvature (“smoothing out humps”) and decreases wherever the phase is sloped $\dot P = \nabla \cdot \bold{j} = 2 A'' - 2 \theta'^2$ . Hope this helps you picture some quantum dynamics.

The point of this post is just some examples of looking at partial differential equations to get a feeling for what the solutions look like. Locality of a PDE is a powerful property, something that college students often don’t appreciate in the standard science/engineering curriculum. My intuition dramatically increased once I began writing numerical codes to solve PDEs.

So I encourage the new mathematical scientists and engineers out there, don’t be afraid to break down your math problem and visualize it! It is worth the effort.

*Nearly all textbooks I have found stop at the abstract form of the probablility current. I eventually found however that Sakurai Modern Quantum Mechanics 1994 contains the same simple equation for $\bold{j}$ as above.